Integrand size = 25, antiderivative size = 25 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\text {Int}\left (\frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x^2},x\right ) \] Output:
Defer(Int)((c^2*x^2+1)^(1/3)*(a+b*arcsinh(c*x))/x^2,x)
Leaf count is larger than twice the leaf count of optimal. \(211\) vs. \(2(28)=56\).
Time = 5.52 (sec) , antiderivative size = 211, normalized size of antiderivative = 8.44 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {1}{3} \left (-\frac {3 a \sqrt [3]{1+c^2 x^2}}{x}+2 a c^2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {3}{2},-c^2 x^2\right )+b c \sqrt [3]{1+c^2 x^2} \left (-\frac {9 \sqrt [6]{1+\frac {1}{c^2 x^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{6},\frac {7}{6},-\frac {1}{c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}-\frac {3 \text {arcsinh}(c x) \left (1+c^2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {5}{6},1,\frac {4}{3},1+c^2 x^2\right )\right )}{c x}+\frac {\sqrt {\pi +c^2 \pi x^2} \operatorname {Gamma}\left (\frac {2}{3}\right ) \, _3F_2\left (\frac {5}{6},\frac {5}{6},1;\frac {4}{3},\frac {11}{6};1+c^2 x^2\right )}{2^{2/3} \operatorname {Gamma}\left (\frac {4}{3}\right ) \operatorname {Gamma}\left (\frac {11}{6}\right )}\right )\right ) \] Input:
Integrate[((1 + c^2*x^2)^(1/3)*(a + b*ArcSinh[c*x]))/x^2,x]
Output:
((-3*a*(1 + c^2*x^2)^(1/3))/x + 2*a*c^2*x*Hypergeometric2F1[1/2, 2/3, 3/2, -(c^2*x^2)] + b*c*(1 + c^2*x^2)^(1/3)*((-9*(1 + 1/(c^2*x^2))^(1/6)*Hyperg eometric2F1[1/6, 1/6, 7/6, -(1/(c^2*x^2))])/Sqrt[1 + c^2*x^2] - (3*ArcSinh [c*x]*(1 + c^2*x^2*Hypergeometric2F1[5/6, 1, 4/3, 1 + c^2*x^2]))/(c*x) + ( Sqrt[Pi + c^2*Pi*x^2]*Gamma[2/3]*HypergeometricPFQ[{5/6, 5/6, 1}, {4/3, 11 /6}, 1 + c^2*x^2])/(2^(2/3)*Gamma[4/3]*Gamma[11/6])))/3
Not integrable
Time = 0.90 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
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.
| \(\displaystyle \int \frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x^2} \, dx\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx+b c \int \frac {1}{x \sqrt [6]{c^2 x^2+1}}dx-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx+\frac {1}{2} b c \int \frac {1}{x^2 \sqrt [6]{c^2 x^2+1}}dx^2-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx+\frac {3 b \int -\frac {c^2 x^8}{1-x^{12}}d\sqrt [6]{c^2 x^2+1}}{c}-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx-\frac {3 b \int \frac {c^2 x^8}{1-x^{12}}d\sqrt [6]{c^2 x^2+1}}{c}-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx-3 b c \int \frac {x^8}{1-x^{12}}d\sqrt [6]{c^2 x^2+1}-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\) |
| \(\Big \downarrow \) 825 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx-3 b c \left (\frac {1}{3} \int \frac {1}{1-x^4}d\sqrt [6]{c^2 x^2+1}+\frac {1}{3} \int -\frac {\sqrt [6]{c^2 x^2+1}+1}{2 \left (x^4-\sqrt [6]{c^2 x^2+1}+1\right )}d\sqrt [6]{c^2 x^2+1}+\frac {1}{3} \int -\frac {1-\sqrt [6]{c^2 x^2+1}}{2 \left (x^4+\sqrt [6]{c^2 x^2+1}+1\right )}d\sqrt [6]{c^2 x^2+1}\right )-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx-3 b c \left (\frac {1}{3} \int \frac {1}{1-x^4}d\sqrt [6]{c^2 x^2+1}-\frac {1}{6} \int \frac {\sqrt [6]{c^2 x^2+1}+1}{x^4-\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}-\frac {1}{6} \int \frac {1-\sqrt [6]{c^2 x^2+1}}{x^4+\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}\right )-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx-3 b c \left (-\frac {1}{6} \int \frac {\sqrt [6]{c^2 x^2+1}+1}{x^4-\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}-\frac {1}{6} \int \frac {1-\sqrt [6]{c^2 x^2+1}}{x^4+\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{c^2 x^2+1}\right )\right )-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx-3 b c \left (\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{x^4-\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}-\frac {1}{2} \int -\frac {1-2 \sqrt [6]{c^2 x^2+1}}{x^4-\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [6]{c^2 x^2+1}+1}{x^4+\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}-\frac {3}{2} \int \frac {1}{x^4+\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{c^2 x^2+1}\right )\right )-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx-3 b c \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [6]{c^2 x^2+1}}{x^4-\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}-\frac {3}{2} \int \frac {1}{x^4-\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [6]{c^2 x^2+1}+1}{x^4+\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}-\frac {3}{2} \int \frac {1}{x^4+\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{c^2 x^2+1}\right )\right )-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx-3 b c \left (\frac {1}{6} \left (3 \int \frac {1}{-x^4-3}d\left (2 \sqrt [6]{c^2 x^2+1}-1\right )+\frac {1}{2} \int \frac {1-2 \sqrt [6]{c^2 x^2+1}}{x^4-\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}\right )+\frac {1}{6} \left (3 \int \frac {1}{-x^4-3}d\left (2 \sqrt [6]{c^2 x^2+1}+1\right )+\frac {1}{2} \int \frac {2 \sqrt [6]{c^2 x^2+1}+1}{x^4+\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{c^2 x^2+1}\right )\right )-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx-3 b c \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [6]{c^2 x^2+1}}{x^4-\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}-\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{c^2 x^2+1}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [6]{c^2 x^2+1}+1}{x^4+\sqrt [6]{c^2 x^2+1}+1}d\sqrt [6]{c^2 x^2+1}-\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{c^2 x^2+1}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{c^2 x^2+1}\right )\right )-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}-3 b c \left (\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{c^2 x^2+1}-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (-\sqrt [6]{c^2 x^2+1}+x^4+1\right )\right )+\frac {1}{6} \left (\frac {1}{2} \log \left (\sqrt [6]{c^2 x^2+1}+x^4+1\right )-\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{c^2 x^2+1}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{c^2 x^2+1}\right )\right )\) |
| \(\Big \downarrow \) 6209 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{2/3}}dx-\frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}-3 b c \left (\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{c^2 x^2+1}-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (-\sqrt [6]{c^2 x^2+1}+x^4+1\right )\right )+\frac {1}{6} \left (\frac {1}{2} \log \left (\sqrt [6]{c^2 x^2+1}+x^4+1\right )-\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{c^2 x^2+1}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{c^2 x^2+1}\right )\right )\) |
Input:
Int[((1 + c^2*x^2)^(1/3)*(a + b*ArcSinh[c*x]))/x^2,x]
Output:
$Aborted
Not integrable
Time = 0.74 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int \frac {\left (c^{2} x^{2}+1\right )^{\frac {1}{3}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{x^{2}}d x\]
Input:
int((c^2*x^2+1)^(1/3)*(a+b*arcsinh(x*c))/x^2,x)
Output:
int((c^2*x^2+1)^(1/3)*(a+b*arcsinh(x*c))/x^2,x)
Not integrable
Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {1}{3}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \] Input:
integrate((c^2*x^2+1)^(1/3)*(a+b*arcsinh(c*x))/x^2,x, algorithm="fricas")
Output:
integral((c^2*x^2 + 1)^(1/3)*(b*arcsinh(c*x) + a)/x^2, x)
Not integrable
Time = 6.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \sqrt [3]{c^{2} x^{2} + 1}}{x^{2}}\, dx \] Input:
integrate((c**2*x**2+1)**(1/3)*(a+b*asinh(c*x))/x**2,x)
Output:
Integral((a + b*asinh(c*x))*(c**2*x**2 + 1)**(1/3)/x**2, x)
Not integrable
Time = 0.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {1}{3}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \] Input:
integrate((c^2*x^2+1)^(1/3)*(a+b*arcsinh(c*x))/x^2,x, algorithm="maxima")
Output:
integrate((c^2*x^2 + 1)^(1/3)*(b*arcsinh(c*x) + a)/x^2, x)
Exception generated. \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c^2*x^2+1)^(1/3)*(a+b*arcsinh(c*x))/x^2,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 = 3.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (c^2\,x^2+1\right )}^{1/3}}{x^2} \,d x \] Input:
int(((a + b*asinh(c*x))*(c^2*x^2 + 1)^(1/3))/x^2,x)
Output:
int(((a + b*asinh(c*x))*(c^2*x^2 + 1)^(1/3))/x^2, x)
Not integrable
Time = 0.39 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.32 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {-3 \left (c^{2} x^{2}+1\right )^{\frac {1}{3}} a -2 \left (\int \frac {\left (c^{2} x^{2}+1\right )^{\frac {1}{3}}}{c^{2} x^{4}+x^{2}}d x \right ) a x +\left (\int \frac {\left (c^{2} x^{2}+1\right )^{\frac {1}{3}} \mathit {asinh} \left (c x \right )}{c^{2} x^{4}+x^{2}}d x \right ) b x +\left (\int \frac {\mathit {asinh} \left (c x \right )}{\left (c^{2} x^{2}+1\right )^{\frac {2}{3}}}d x \right ) b \,c^{2} x}{x} \] Input:
int((c^2*x^2+1)^(1/3)*(a+b*asinh(c*x))/x^2,x)
Output:
( - 3*(c**2*x**2 + 1)**(1/3)*a - 2*int((c**2*x**2 + 1)**(1/3)/(c**2*x**4 + x**2),x)*a*x + int(((c**2*x**2 + 1)**(1/3)*asinh(c*x))/(c**2*x**4 + x**2) ,x)*b*x + int(((c**2*x**2 + 1)**(1/3)*asinh(c*x))/(c**2*x**2 + 1),x)*b*c** 2*x)/x