Integrand size = 25, antiderivative size = 25 \[ \int x^3 \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Int}\left (x^3 \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)),x\right ) \] Output:
Defer(Int)(x^3*(c^2*x^2+1)^(1/3)*(a+b*arcsinh(c*x)),x)
Leaf count is larger than twice the leaf count of optimal. \(119\) vs. \(2(28)=56\).
Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.76 \[ \int x^3 \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {3 \sqrt [3]{1+c^2 x^2} \left (3 b c x \left (11-16 c^2 x^2\right ) \sqrt {1+c^2 x^2}+56 a \left (-3+c^2 x^2+4 c^4 x^4\right )+56 b \left (-3+c^2 x^2+4 c^4 x^4\right ) \text {arcsinh}(c x)\right )+405 b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},-c^2 x^2\right )}{3136 c^4} \] Input:
Integrate[x^3*(1 + c^2*x^2)^(1/3)*(a + b*ArcSinh[c*x]),x]
Output:
(3*(1 + c^2*x^2)^(1/3)*(3*b*c*x*(11 - 16*c^2*x^2)*Sqrt[1 + c^2*x^2] + 56*a *(-3 + c^2*x^2 + 4*c^4*x^4) + 56*b*(-3 + c^2*x^2 + 4*c^4*x^4)*ArcSinh[c*x] ) + 405*b*c*x*Hypergeometric2F1[1/6, 1/2, 3/2, -(c^2*x^2)])/(3136*c^4)
Leaf count is larger than twice the leaf count of optimal. \(2540\) vs. \(2(28)=56\).
Time = 2.79 (sec) , antiderivative size = 2540, normalized size of antiderivative = 101.60, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.960, Rules used = {6223, 262, 262, 235, 214, 233, 833, 760, 2418, 6227, 262, 235, 214, 233, 833, 760, 2418, 6213, 235, 214, 233, 833, 760, 2418}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {1}{7} \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{14} b c \int \frac {x^4}{\sqrt [6]{c^2 x^2+1}}dx+\frac {3}{14} x^4 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{7} \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \int \frac {x^2}{\sqrt [6]{c^2 x^2+1}}dx}{14 c^2}\right )+\frac {3}{14} x^4 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{7} \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \int \frac {1}{\sqrt [6]{c^2 x^2+1}}dx}{8 c^2}\right )}{14 c^2}\right )+\frac {3}{14} x^4 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 235 |
| \(\displaystyle \frac {1}{7} \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}-\frac {1}{2} \int \frac {1}{\left (c^2 x^2+1\right )^{7/6}}dx\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {3}{14} x^4 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 214 |
| \(\displaystyle \frac {1}{7} \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}-\frac {1}{2} \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt [3]{c^2 x^2+1} \int \frac {1}{\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}d\frac {x}{\sqrt {c^2 x^2+1}}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {3}{14} x^4 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {1}{7} \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \int \frac {\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{4 c^2 x}+\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {3}{14} x^4 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {1}{7} \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\int \frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right )}{4 c^2 x}+\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {3}{14} x^4 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {1}{7} \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (-\int \frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}\right )}{4 c^2 x}+\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {3}{14} x^4 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \int \frac {x^2}{\sqrt [6]{c^2 x^2+1}}dx}{8 c}-\frac {3 \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx}{4 c^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \int \frac {1}{\sqrt [6]{c^2 x^2+1}}dx}{8 c^2}\right )}{8 c}-\frac {3 \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx}{4 c^2}\right )\) |
| \(\Big \downarrow \) 235 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}-\frac {1}{2} \int \frac {1}{\left (c^2 x^2+1\right )^{7/6}}dx\right )}{8 c^2}\right )}{8 c}-\frac {3 \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx}{4 c^2}\right )\) |
| \(\Big \downarrow \) 214 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}-\frac {1}{2} \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt [3]{c^2 x^2+1} \int \frac {1}{\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}d\frac {x}{\sqrt {c^2 x^2+1}}\right )}{8 c^2}\right )}{8 c}-\frac {3 \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx}{4 c^2}\right )\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \int \frac {\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{4 c^2 x}\right )}{8 c^2}\right )}{8 c}-\frac {3 \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx}{4 c^2}\right )\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\int \frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{8 c}-\frac {3 \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx}{4 c^2}\right )\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\int \frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{8 c}-\frac {3 \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx}{4 c^2}\right )\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{8 c}-\frac {3 \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{2/3}}dx}{4 c^2}\right )\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{8 c}-\frac {3 \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {3 b \int \frac {1}{\sqrt [6]{c^2 x^2+1}}dx}{2 c}\right )}{4 c^2}\right )\) |
| \(\Big \downarrow \) 235 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{8 c}-\frac {3 \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {3 b \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}-\frac {1}{2} \int \frac {1}{\left (c^2 x^2+1\right )^{7/6}}dx\right )}{2 c}\right )}{4 c^2}\right )\) |
| \(\Big \downarrow \) 214 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{8 c}-\frac {3 \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {3 b \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}-\frac {1}{2} \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt [3]{c^2 x^2+1} \int \frac {1}{\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}d\frac {x}{\sqrt {c^2 x^2+1}}\right )}{2 c}\right )}{4 c^2}\right )\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{8 c}-\frac {3 \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {3 b \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \int \frac {\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{4 c^2 x}\right )}{2 c}\right )}{4 c^2}\right )\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{8 c}-\frac {3 \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {3 b \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\int \frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right )}{4 c^2 x}\right )}{2 c}\right )}{4 c^2}\right )\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{8 c}-\frac {3 \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {3 b \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\int \frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right )}{4 c^2 x}\right )}{2 c}\right )}{4 c^2}\right )\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {3}{14} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^4-\frac {3}{14} b c \left (\frac {3 x^3 \left (c^2 x^2+1\right )^{5/6}}{14 c^2}-\frac {9 \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{14 c^2}\right )+\frac {1}{7} \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2}{8 c^2}-\frac {3 b \left (\frac {3 x \left (c^2 x^2+1\right )^{5/6}}{8 c^2}-\frac {3 \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{8 c^2}\right )}{8 c}-\frac {3 \left (\frac {3 \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {3 b \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )}{2 c}\right )}{4 c^2}\right )\) |
Input:
Int[x^3*(1 + c^2*x^2)^(1/3)*(a + b*ArcSinh[c*x]),x]
Output:
(3*x^4*(1 + c^2*x^2)^(1/3)*(a + b*ArcSinh[c*x]))/14 - (3*b*c*((3*x^3*(1 + c^2*x^2)^(5/6))/(14*c^2) - (9*((3*x*(1 + c^2*x^2)^(5/6))/(8*c^2) - (3*((3* x)/(2*(1 + c^2*x^2)^(1/6)) + (3*((1 + c^2*x^2)^(-1))^(1/3)*Sqrt[-((c^2*x^2 )/(1 + c^2*x^2))]*(1 + c^2*x^2)^(5/6)*((-2*Sqrt[-1 + x^3/(1 + c^2*x^2)^(3/ 2)])/(1 - Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))*Sqrt[(1 + x^2/(1 + c ^2*x^2) + (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (c^2*x^ 2)/(1 + c^2*x^2))^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - (1 - (c^2*x^2) /(1 + c^2*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3)) ], -7 + 4*Sqrt[3]])/(Sqrt[-1 + x^3/(1 + c^2*x^2)^(3/2)]*Sqrt[-((1 - (1 - ( c^2*x^2)/(1 + c^2*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2*x^2) )^(1/3))^2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*(1 - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))*Sqrt[(1 + x^2/(1 + c^2*x^2) + (1 - (c^2*x^2)/(1 + c^2*x ^2))^(1/3))/(1 - Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))^2]*Ellipti cF[ArcSin[(1 + Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[- 1 + x^3/(1 + c^2*x^2)^(3/2)]*Sqrt[-((1 - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/ 3))/(1 - Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))^2)])))/(4*c^2*x))) /(8*c^2)))/(14*c^2)))/14 + ((3*x^2*(1 + c^2*x^2)^(1/3)*(a + b*ArcSinh[c*x] ))/(8*c^2) - (3*b*((3*x*(1 + c^2*x^2)^(5/6))/(8*c^2) - (3*((3*x)/(2*(1 ...
Int[((a_) + (b_.)*(x_)^2)^(-7/6), x_Symbol] :> Simp[1/((a + b*x^2)^(2/3)*(a /(a + b*x^2))^(2/3)) Subst[Int[1/(1 - b*x^2)^(1/3), x], x, x/Sqrt[a + b*x ^2]], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b }, x]
Int[((a_) + (b_.)*(x_)^2)^(-1/6), x_Symbol] :> Simp[3*(x/(2*(a + b*x^2)^(1/ 6))), x] - Simp[a/2 Int[1/(a + b*x^2)^(7/6), x], x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) *s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x ] && NegQ[a]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && NegQ[a]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Sinh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f* x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int [(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Not integrable
Time = 1.82 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int x^{3} \left (c^{2} x^{2}+1\right )^{\frac {1}{3}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )d x\]
Input:
int(x^3*(c^2*x^2+1)^(1/3)*(a+b*arcsinh(x*c)),x)
Output:
int(x^3*(c^2*x^2+1)^(1/3)*(a+b*arcsinh(x*c)),x)
Not integrable
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int x^3 \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} x^{2} + 1\right )}^{\frac {1}{3}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{3} \,d x } \] Input:
integrate(x^3*(c^2*x^2+1)^(1/3)*(a+b*arcsinh(c*x)),x, algorithm="fricas")
Output:
integral((b*x^3*arcsinh(c*x) + a*x^3)*(c^2*x^2 + 1)^(1/3), x)
Not integrable
Time = 24.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int x^3 \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x^{3} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \sqrt [3]{c^{2} x^{2} + 1}\, dx \] Input:
integrate(x**3*(c**2*x**2+1)**(1/3)*(a+b*asinh(c*x)),x)
Output:
Integral(x**3*(a + b*asinh(c*x))*(c**2*x**2 + 1)**(1/3), x)
Not integrable
Time = 0.17 (sec) , antiderivative size = 174, normalized size of antiderivative = 6.96 \[ \int x^3 \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} x^{2} + 1\right )}^{\frac {1}{3}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{3} \,d x } \] Input:
integrate(x^3*(c^2*x^2+1)^(1/3)*(a+b*arcsinh(c*x)),x, algorithm="maxima")
Output:
3/56*a*(4*(c^2*x^2 + 1)^(7/3)/c^4 - 7*(c^2*x^2 + 1)^(4/3)/c^4) + 1/3136*b* (3*(56*(4*c^4*x^4 + c^2*x^2 - 3)*(c^2*x^2 + 1)^(1/3)*log(c*x + sqrt(c^2*x^ 2 + 1)) - 3*(16*c^4*x^4 - 17*c^2*x^2 - 33)*(c^2*x^2 + 1)^(1/3))/c^4 - 3136 *integrate(3/56*(4*c^2*x^2 - 3)*(c^2*x^2 + 1)^(1/3)/(c^4*x + sqrt(c^2*x^2 + 1)*c^3), x))
Exception generated. \[ \int x^3 \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(c^2*x^2+1)^(1/3)*(a+b*arcsinh(c*x)),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Not integrable
Time = 3.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x^3 \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (c^2\,x^2+1\right )}^{1/3} \,d x \] Input:
int(x^3*(a + b*asinh(c*x))*(c^2*x^2 + 1)^(1/3),x)
Output:
int(x^3*(a + b*asinh(c*x))*(c^2*x^2 + 1)^(1/3), x)
Not integrable
Time = 0.49 (sec) , antiderivative size = 158, normalized size of antiderivative = 6.32 \[ \int x^3 \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {56 \left (\sqrt {c^{2} x^{2}+1}+c x \right )^{\frac {2}{3}} \left (\int \left (c^{2} x^{2}+1\right )^{\frac {1}{3}} \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{4}+12 \left (\sqrt {c^{2} x^{2}+1}\, c x +c^{2} x^{2}+1\right )^{\frac {2}{3}} a \,c^{4} x^{4}+3 \left (\sqrt {c^{2} x^{2}+1}\, c x +c^{2} x^{2}+1\right )^{\frac {2}{3}} a \,c^{2} x^{2}-9 \left (\sqrt {c^{2} x^{2}+1}\, c x +c^{2} x^{2}+1\right )^{\frac {2}{3}} a}{56 \left (\sqrt {c^{2} x^{2}+1}+c x \right )^{\frac {2}{3}} c^{4}} \] Input:
int(x^3*(c^2*x^2+1)^(1/3)*(a+b*asinh(c*x)),x)
Output:
(56*(sqrt(c**2*x**2 + 1) + c*x)**(2/3)*int((c**2*x**2 + 1)**(1/3)*asinh(c* x)*x**3,x)*b*c**4 + 12*(sqrt(c**2*x**2 + 1)*c*x + c**2*x**2 + 1)**(2/3)*a* c**4*x**4 + 3*(sqrt(c**2*x**2 + 1)*c*x + c**2*x**2 + 1)**(2/3)*a*c**2*x**2 - 9*(sqrt(c**2*x**2 + 1)*c*x + c**2*x**2 + 1)**(2/3)*a)/(56*(sqrt(c**2*x* *2 + 1) + c*x)**(2/3)*c**4)