Integrand size = 18, antiderivative size = 486 \[ \int (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {4 b e \sqrt {d+e x} \left (1+c^2 x^2\right )}{15 c^3 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}+\frac {28 b c d \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}}}-\frac {4 b c \left (2 c^2 d^2-e^2\right ) \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{5/2} \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b d^3 \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{5 c e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
2/5*(e*x+d)^(5/2)*(a+b*arccsch(c*x))/e+4/15*b*e*(c^2*x^2+1)*(e*x+d)^(1/2)/ c^3/x/(1+1/c^2/x^2)^(1/2)+28/15*b*c*d*EllipticE(1/2*(1-x*(-c^2)^(1/2))^(1/ 2)*2^(1/2),(-2*e*(-c^2)^(1/2)/(c^2*d-e*(-c^2)^(1/2)))^(1/2))*(e*x+d)^(1/2) *(c^2*x^2+1)^(1/2)/(-c^2)^(3/2)/x/(1+1/c^2/x^2)^(1/2)/((e*x+d)/(d+e/(-c^2) ^(1/2)))^(1/2)-4/15*b*c*(2*c^2*d^2-e^2)*EllipticF(1/2*(1-x*(-c^2)^(1/2))^( 1/2)*2^(1/2),(-2*e*(-c^2)^(1/2)/(c^2*d-e*(-c^2)^(1/2)))^(1/2))*(c^2*x^2+1) ^(1/2)*((e*x+d)/(d+e/(-c^2)^(1/2)))^(1/2)/(-c^2)^(5/2)/x/(1+1/c^2/x^2)^(1/ 2)/(e*x+d)^(1/2)-4/5*b*d^3*EllipticPi(1/2*(1-x*(-c^2)^(1/2))^(1/2)*2^(1/2) ,2,2^(1/2)*(e/(d*(-c^2)^(1/2)+e))^(1/2))*(c^2*x^2+1)^(1/2)*((e*x+d)*(-c^2) ^(1/2)/(d*(-c^2)^(1/2)+e))^(1/2)/c/e/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 13.22 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.78 \[ \int (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {2 \left (\frac {2 b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}{c}+3 a (d+e x)^{5/2}+3 b (d+e x)^{5/2} \text {csch}^{-1}(c x)+\frac {2 i b \sqrt {-\frac {e (-i+c x)}{c d+i e}} \sqrt {-\frac {e (i+c x)}{c d-i e}} \left (7 c d (c d+i e) E\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )+\left (-9 c^2 d^2-7 i c d e+e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right ),\frac {c d-i e}{c d+i e}\right )+3 c^2 d^2 \operatorname {EllipticPi}\left (1-\frac {i e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right ),\frac {c d-i e}{c d+i e}\right )\right )}{c^3 \sqrt {-\frac {c}{c d-i e}} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{15 e} \]
(2*((2*b*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])/c + 3*a*(d + e*x)^(5/2 ) + 3*b*(d + e*x)^(5/2)*ArcCsch[c*x] + ((2*I)*b*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*(7*c*d*(c*d + I*e)*EllipticE[ I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] + (-9*c^2*d^2 - (7*I)*c*d*e + e^2)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d - I*e ))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] + 3*c^2*d^2*EllipticPi[1 - (I *e)/(c*d), I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c *d + I*e)]))/(c^3*Sqrt[-(c/(c*d - I*e))]*Sqrt[1 + 1/(c^2*x^2)]*x)))/(15*e)
Leaf count is larger than twice the leaf count of optimal. \(1357\) vs. \(2(486)=972\).
Time = 2.42 (sec) , antiderivative size = 1357, normalized size of antiderivative = 2.79, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6844, 1898, 634, 631, 1540, 1416, 2185, 27, 599, 25, 27, 1511, 1416, 1509, 2222}
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.
| \(\displaystyle \int (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 6844 |
| \(\displaystyle \frac {2 b \int \frac {(d+e x)^{5/2}}{\sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{5 c e}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}\) |
| \(\Big \downarrow \) 1898 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \int \frac {(d+e x)^{5/2}}{x \sqrt {x^2+\frac {1}{c^2}}}dx}{5 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}\) |
| \(\Big \downarrow \) 634 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (d^3 \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx\right )}{5 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}\) |
| \(\Big \downarrow \) 631 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx-2 d^3 \int -\frac {1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}\right )}{5 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}\) |
| \(\Big \downarrow \) 1540 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx-2 d^3 \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}-\frac {c \left (c d-\sqrt {c^2 d^2+e^2}\right ) \int \frac {1}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}\right )\right )}{5 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (-2 d^3 \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}\right )-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx\right )}{5 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (-2 d^3 \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}\right )-\frac {2 \int -\frac {e^3 \left (9 d^2+7 e x d-\frac {e^2}{c^2}\right )}{2 \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx}{3 e^2}+\frac {2}{3} e^2 \sqrt {\frac {1}{c^2}+x^2} \sqrt {d+e x}\right )}{5 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (-2 d^3 \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}\right )+\frac {1}{3} e \int \frac {9 d^2+7 e x d-\frac {e^2}{c^2}}{\sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx+\frac {2}{3} e^2 \sqrt {\frac {1}{c^2}+x^2} \sqrt {d+e x}\right )}{5 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (-2 d^3 \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}\right )-\frac {2 \int -\frac {e \left (2 d^2+7 (d+e x) d-\frac {e^2}{c^2}\right )}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{3 e}+\frac {2}{3} e^2 \sqrt {\frac {1}{c^2}+x^2} \sqrt {d+e x}\right )}{5 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (-2 d^3 \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}\right )+\frac {2 \int \frac {e \left (2 d^2+7 (d+e x) d-\frac {e^2}{c^2}\right )}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{3 e}+\frac {2}{3} e^2 \sqrt {\frac {1}{c^2}+x^2} \sqrt {d+e x}\right )}{5 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (-2 d^3 \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}\right )+\frac {2}{3} \int \frac {2 d^2+7 (d+e x) d-\frac {e^2}{c^2}}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}+\frac {2}{3} e^2 \sqrt {\frac {1}{c^2}+x^2} \sqrt {d+e x}\right )}{5 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (-2 d^3 \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}\right )+\frac {2}{3} \left (\frac {\left (7 c d \sqrt {c^2 d^2+e^2}+2 c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{c^2}-\frac {7 d \sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{c}\right )+\frac {2}{3} e^2 \sqrt {\frac {1}{c^2}+x^2} \sqrt {d+e x}\right )}{5 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{5/2}}{5 e}+\frac {2 b \sqrt {x^2+\frac {1}{c^2}} \left (-2 \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}{\left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}\right ) d^3+\frac {2}{3} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (2 c^2 d^2+7 c \sqrt {c^2 d^2+e^2} d-e^2\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}{\left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{5/2} \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}-\frac {7 d \sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{c}\right )+\frac {2}{3} e^2 \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}\right )}{5 c e \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{5/2}}{5 e}+\frac {2 b \sqrt {x^2+\frac {1}{c^2}} \left (-2 \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}{\left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}\right ) d^3+\frac {2}{3} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (2 c^2 d^2+7 c \sqrt {c^2 d^2+e^2} d-e^2\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}{\left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{5/2} \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}-\frac {7 d \sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}{\left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}-\frac {\sqrt {d+e x} \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}{\left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )+\frac {2}{3} e^2 \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}\right )}{5 c e \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
| \(\Big \downarrow \) 2222 |
| \(\displaystyle \frac {2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{5/2}}{5 e}+\frac {2 b \sqrt {x^2+\frac {1}{c^2}} \left (-2 \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{c \sqrt {d} \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}\right )}{2 \sqrt {d}}+\frac {\sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}{\left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c d+\sqrt {c^2 d^2+e^2}\right )^2}{4 c d \sqrt {c^2 d^2+e^2}},2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {\sqrt {c^2 d^2+e^2} d}{2 c \left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) e^2}+\frac {1}{2}\right )}{4 \sqrt {c} d \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}\right )}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}{\left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}\right ) d^3+\frac {2}{3} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (2 c^2 d^2+7 c \sqrt {c^2 d^2+e^2} d-e^2\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}{\left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{5/2} \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}-\frac {7 d \sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}{\left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}-\frac {\sqrt {d+e x} \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}{\left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )+\frac {2}{3} e^2 \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}\right )}{5 c e \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
(2*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e) + (2*b*Sqrt[c^(-2) + x^2]*( (2*e^2*Sqrt[d + e*x]*Sqrt[c^(-2) + x^2])/3 + (2*((-7*d*Sqrt[c^2*d^2 + e^2] *(-((Sqrt[d + e*x]*Sqrt[c^(-2) + d^2/e^2 - (2*d*(d + e*x))/e^2 + (d + e*x) ^2/e^2])/((c^(-2) + d^2/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2]))) + ( (c^2*d^2 + e^2)^(1/4)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])*Sqrt[(c^(-2) + d^2/e^2 - (2*d*(d + e*x))/e^2 + (d + e*x)^2/e^2)/((c^(-2) + d^2/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])^2)]*EllipticE[2*ArcTan[(Sqrt[c]*Sqrt [d + e*x])/(c^2*d^2 + e^2)^(1/4)], (1 + (c*d)/Sqrt[c^2*d^2 + e^2])/2])/(Sq rt[c]*Sqrt[c^(-2) + d^2/e^2 - (2*d*(d + e*x))/e^2 + (d + e*x)^2/e^2])))/c + ((c^2*d^2 + e^2)^(1/4)*(2*c^2*d^2 - e^2 + 7*c*d*Sqrt[c^2*d^2 + e^2])*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])*Sqrt[(c^(-2) + d^2/e^2 - (2*d*(d + e* x))/e^2 + (d + e*x)^2/e^2)/((c^(-2) + d^2/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2 *d^2 + e^2])^2)]*EllipticF[2*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*d^2 + e^2 )^(1/4)], (1 + (c*d)/Sqrt[c^2*d^2 + e^2])/2])/(2*c^(5/2)*Sqrt[c^(-2) + d^2 /e^2 - (2*d*(d + e*x))/e^2 + (d + e*x)^2/e^2])))/3 - 2*d^3*(-1/2*(Sqrt[c]* (c^2*d^2 + e^2)^(1/4)*(c*d - Sqrt[c^2*d^2 + e^2])*(1 + (c*(d + e*x))/Sqrt[ c^2*d^2 + e^2])*Sqrt[(c^(-2) + d^2/e^2 - (2*d*(d + e*x))/e^2 + (d + e*x)^2 /e^2)/((c^(-2) + d^2/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])^2)]*Elli pticF[2*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*d^2 + e^2)^(1/4)], (1 + (c*d)/ Sqrt[c^2*d^2 + e^2])/2])/(e^2*Sqrt[c^(-2) + d^2/e^2 - (2*d*(d + e*x))/e...
3.1.56.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a *d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > Simp[-2 Subst[Int[1/((c - x^2)*Sqrt[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^2/d^ 2) + b*(x^4/d^2)]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && PosQ[b/a]
Int[((c_) + (d_.)*(x_))^(n_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[c^(n + 1/2) Int[1/(x*Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] - Int[( 1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[(c^(n + 1/2) - (c + d*x)^(n + 1/2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n - 1/2, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S ymbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2) Int[1 /Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2) I nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^ (q_.), x_Symbol] :> Simp[x^(2*n*FracPart[p])*((a + c/x^(2*n))^FracPart[p]/( c + a*x^(2*n))^FracPart[p]) Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n ))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] && !I ntegerQ[p] && !IntegerQ[q] && PosQ[n]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCsch[c*x])/(e*(m + 1))), x] + Simp[ b/(c*e*(m + 1)) Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Result contains complex when optimal does not.
Time = 9.23 (sec) , antiderivative size = 1939, normalized size of antiderivative = 3.99
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1939\) |
| default | \(\text {Expression too large to display}\) | \(1939\) |
| parts | \(\text {Expression too large to display}\) | \(1941\) |
2/e*(1/5*a*(e*x+d)^(5/2)+b*(1/5*arccsch(c*x)*(e*x+d)^(5/2)+2/15/c^3*(-2*I* ((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(3/2)*c^2*d*e-((c*d+I*e)*c/(c^2* d^2+e^2))^(1/2)*c^3*d*(e*x+d)^(5/2)+2*I*(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)-c^2 *d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-c^2*d*(e*x+d)+c^2*d^2+e^2)/ (c^2*d^2+e^2))^(1/2)*EllipticF((e*x+d)^(1/2)*((c*d+I*e)*c/(c^2*d^2+e^2))^( 1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c^2*d^2*e-3*I*(-(I*c* e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)- c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticPi((e*x+d)^(1/2)*( (c*d+I*e)*c/(c^2*d^2+e^2))^(1/2),1/(c*d+I*e)/c*(c^2*d^2+e^2)/d,(-(I*e-c*d) *c/(c^2*d^2+e^2))^(1/2)/((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2))*c^2*d^2*e+I*((c *d+I*e)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(1/2)*e^3+2*((c*d+I*e)*c/(c^2*d^2+e ^2))^(1/2)*c^3*d^2*(e*x+d)^(3/2)-9*(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2- e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2* d^2+e^2))^(1/2)*EllipticF((e*x+d)^(1/2)*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2), (-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c^3*d^3+7*(-(I*c*e*(e*x+d) +c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-c^2*d*(e* x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticE((e*x+d)^(1/2)*((c*d+I*e)* c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c^3 *d^3+3*(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*(( I*c*e*(e*x+d)-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticP...
\[ \int (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { {\left (e x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \,d x } \]
\[ \int (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x\right )^{\frac {3}{2}}\, dx \]
\[ \int (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { {\left (e x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \,d x } \]
2/5*(e*x + d)^(5/2)*a/e - 1/75*(375*c^2*e^2*integrate(1/5*sqrt(e*x + d)*x^ 3*log(x)/(c^2*e*x^2 + e), x) + 375*c^2*d*e*integrate(1/5*sqrt(e*x + d)*x^2 *log(x)/(c^2*e*x^2 + e), x) + 75*d*e^2*integrate(sqrt(e*x + d)/((e*x + d)^ 2*c^2 - 2*(e*x + d)*c^2*d + c^2*d^2 + e^2), x)*log(c) + 30*c^2*d^2*(e^2*in tegrate(((e*x + d)*c^2*d - c^2*d^2 - e^2)/(((e*x + d)^2*c^2 - 2*(e*x + d)* c^2*d + c^2*d^2 + e^2)*sqrt(e*x + d)), x)/c^2 + 2*sqrt(e*x + d)*e/c^2)/e^2 + 375*e^2*integrate(1/5*sqrt(e*x + d)*x*log(x)/(c^2*e*x^2 + e), x) + 375* d*e*integrate(1/5*sqrt(e*x + d)*log(x)/(c^2*e*x^2 + e), x) - 25*(3*e^4*int egrate(sqrt(e*x + d)/((e*x + d)^2*c^2 - 2*(e*x + d)*c^2*d + c^2*d^2 + e^2) , x)/c^2 - 2*(e*x + d)^(3/2)*e/c^2)*c^2*d*log(c)/e^2 - 20*(3*e^4*integrate (sqrt(e*x + d)/((e*x + d)^2*c^2 - 2*(e*x + d)*c^2*d + c^2*d^2 + e^2), x)/c ^2 - 2*(e*x + d)^(3/2)*e/c^2)*c^2*d/e^2 + 75*(e^2*integrate(((e*x + d)*c^2 *d - c^2*d^2 - e^2)/(((e*x + d)^2*c^2 - 2*(e*x + d)*c^2*d + c^2*d^2 + e^2) *sqrt(e*x + d)), x)/c^2 + 2*sqrt(e*x + d)*e/c^2)*log(c) - 5*c^2*(15*e^4*in tegrate(((e*x + d)*c^2*d - c^2*d^2 - e^2)/(((e*x + d)^2*c^2 - 2*(e*x + d)* c^2*d + c^2*d^2 + e^2)*sqrt(e*x + d)), x)/c^4 - 2*(3*(e*x + d)^(5/2)*c^2*e - 5*(e*x + d)^(3/2)*c^2*d*e - 15*sqrt(e*x + d)*e^3)/c^4)*log(c)/e^2 - 30* (e^2*x^2 + 2*d*e*x + d^2)*sqrt(e*x + d)*log(sqrt(c^2*x^2 + 1) + 1)/e - 2*c ^2*(15*e^4*integrate(((e*x + d)*c^2*d - c^2*d^2 - e^2)/(((e*x + d)^2*c^2 - 2*(e*x + d)*c^2*d + c^2*d^2 + e^2)*sqrt(e*x + d)), x)/c^4 - 2*(3*(e*x ...
\[ \int (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { {\left (e x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \,d x } \]
Timed out. \[ \int (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int \left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^{3/2} \,d x \]