Integrand size = 19, antiderivative size = 318 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {4 b c \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{\left (-c^2\right )^{3/2} e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {8 b d \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
2*d*(a+b*arccsch(c*x))/e^2/(e*x+d)^(1/2)+2*(a+b*arccsch(c*x))*(e*x+d)^(1/2 )/e^2-8*b*d*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^2/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+4*b*c*Elliptic F(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)*(c^2*(e*x+d)/(c^2*d-e*(-c^2)^(1/2)))^(1/2 )/(-c^2)^(3/2)/e/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 13.03 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.83 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\frac {2 \left (\frac {a (2 d+e x)}{\sqrt {d+e x}}+\frac {b (2 d+e x) \text {csch}^{-1}(c x)}{\sqrt {d+e x}}-\frac {2 i b \sqrt {-\frac {e (-i+c x)}{c d+i e}} \sqrt {-\frac {e (i+c x)}{c d-i e}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right ),\frac {c d-i e}{c d+i e}\right )-2 \operatorname {EllipticPi}\left (1-\frac {i e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right ),\frac {c d-i e}{c d+i e}\right )\right )}{c \sqrt {-\frac {c}{c d-i e}} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{e^2} \]
(2*((a*(2*d + e*x))/Sqrt[d + e*x] + (b*(2*d + e*x)*ArcCsch[c*x])/Sqrt[d + e*x] - ((2*I)*b*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*Sqrt[-((e*(I + c*x))/( c*d - I*e))]*(EllipticF[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], ( c*d - I*e)/(c*d + I*e)] - 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*Sqrt[-(c/(c*d - I*e))]*Sqrt[1 + 1/(c^2*x^2)]*x)))/e^2
Leaf count is larger than twice the leaf count of optimal. \(1009\) vs. \(2(318)=636\).
Time = 2.37 (sec) , antiderivative size = 1009, normalized size of antiderivative = 3.17, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {6864, 27, 7272, 2351, 27, 510, 631, 1416, 1540, 1416, 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.
| \(\displaystyle \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6864 |
| \(\displaystyle \frac {b \int \frac {2 (2 d+e x)}{e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}}dx}{c}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \int \frac {2 d+e x}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}}dx}{c e^2}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \int \frac {2 d+e x}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx}{c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (\int \frac {e}{\sqrt {d+e x} \sqrt {c^2 x^2+1}}dx+2 d \int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx\right )}{c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (e \int \frac {1}{\sqrt {d+e x} \sqrt {c^2 x^2+1}}dx+2 d \int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx\right )}{c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 510 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (2 \int \frac {1}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}+2 d \int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx\right )}{c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 631 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (2 \int \frac {1}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-4 d \int -\frac {1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}-4 d \int -\frac {1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1540 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}-4 d \left (\left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {c \left (c d-\sqrt {c^2 d^2+e^2}\right ) \int \frac {1}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e^2}\right )\right )}{c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}-4 d \left (\left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}\right )\right )}{c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 2222 |
| \(\displaystyle \frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 b \sqrt {c^2 x^2+1} \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+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\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 )}{\sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-4 d \left (\left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {\left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}\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+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\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 {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{4 \sqrt {c} d \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}\right )-\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+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\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+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}\right )\right )}{c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
(2*d*(a + b*ArcCsch[c*x]))/(e^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*A rcCsch[c*x]))/e^2 + (2*b*Sqrt[1 + c^2*x^2]*(((c^2*d^2 + e^2)^(1/4)*(1 + (c *(d + e*x))/Sqrt[c^2*d^2 + e^2])*Sqrt[(1 + (c^2*d^2)/e^2 - (2*c^2*d*(d + e *x))/e^2 + (c^2*(d + e*x)^2)/e^2)/((1 + (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])/(Sqrt[c]*Sqrt[1 + (c ^2*d^2)/e^2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^2)/e^2]) - 4*d*(-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[(1 + (c^2*d^2)/e^2 - (2*c^2*d*(d + e*x))/e ^2 + (c^2*(d + e*x)^2)/e^2)/((1 + (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])/(e^2*Sqrt[1 + (c^2*d^2)/e^ 2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^2)/e^2]) + (1 + (c^2*d^2)/e^2 )*(1 - (c*d)/Sqrt[c^2*d^2 + e^2])*(((1 + (c*d)/Sqrt[c^2*d^2 + e^2])*ArcTan h[Sqrt[d + e*x]/(Sqrt[d]*Sqrt[1 + (c^2*d^2)/e^2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^2)/e^2])])/(2*Sqrt[d]) + ((c^2*d^2 + e^2)^(1/4)*(1 - (c*d )/Sqrt[c^2*d^2 + e^2])*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])*Sqrt[(1 + ( c^2*d^2)/e^2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^2)/e^2)/((1 + (c^2 *d^2)/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])^2)]*EllipticPi[(c*d + S qrt[c^2*d^2 + e^2])^2/(4*c*d*Sqrt[c^2*d^2 + e^2]), 2*ArcTan[(Sqrt[c]*Sq...
3.1.65.3.1 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[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[2/d Subst[Int[1/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[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[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[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[((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[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, x, x] Int[(c + d*x)^n*((a + b*x^2)^p/x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHid e[u, x]}, Simp[(a + b*ArcCsch[c*x]) v, x] + Simp[b/c Int[SimplifyIntegr and[v/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x] ] /; FreeQ[{a, b, c}, x]
Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[b^IntPart[p]*(( a + b*x^n)^FracPart[p]/(x^(n*FracPart[p])*(1 + a*(1/(x^n*b)))^FracPart[p])) Int[u*x^(n*p)*(1 + a*(1/(x^n*b)))^p, x], x] /; FreeQ[{a, b, p}, x] && ! IntegerQ[p] && ILtQ[n, 0] && !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]
Result contains complex when optimal does not.
Time = 6.85 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.32
| method | result | size |
| parts | \(\frac {2 a \left (\sqrt {e x +d}+\frac {d}{\sqrt {e x +d}}\right )}{e^{2}}+\frac {2 b \left (\sqrt {e x +d}\, \operatorname {arccsch}\left (c x \right )+\frac {\operatorname {arccsch}\left (c x \right ) d}{\sqrt {e x +d}}+\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right )-2 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e^{2}}\) | \(421\) |
| derivativedivides | \(\frac {-2 a \left (-\sqrt {e x +d}-\frac {d}{\sqrt {e x +d}}\right )-2 b \left (-\sqrt {e x +d}\, \operatorname {arccsch}\left (c x \right )-\frac {\operatorname {arccsch}\left (c x \right ) d}{\sqrt {e x +d}}-\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right )-2 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e^{2}}\) | \(425\) |
| default | \(\frac {-2 a \left (-\sqrt {e x +d}-\frac {d}{\sqrt {e x +d}}\right )-2 b \left (-\sqrt {e x +d}\, \operatorname {arccsch}\left (c x \right )-\frac {\operatorname {arccsch}\left (c x \right ) d}{\sqrt {e x +d}}-\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right )-2 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e^{2}}\) | \(425\) |
2*a/e^2*((e*x+d)^(1/2)+d/(e*x+d)^(1/2))+2*b/e^2*((e*x+d)^(1/2)*arccsch(c*x )+arccsch(c*x)*d/(e*x+d)^(1/2)+2/c*(-(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))-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*(e*x+ d)^2-2*c^2*d*(e*x+d)+c^2*d^2+e^2)/c^2/e^2/x^2)^(1/2)/x/((c*d+I*e)*c/(c^2*d ^2+e^2))^(1/2))
Timed out. \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
b*(2*(e*x + 2*d)*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(e*x + d)*e^2) + integrat e(2*(c^2*e*x^2 + 2*c^2*d*x)/((c^2*e^2*x^2 + e^2)*sqrt(c^2*x^2 + 1)*sqrt(e* x + d) + (c^2*e^2*x^2 + e^2)*sqrt(e*x + d)), x) - integrate((6*c^2*d*e*x^2 + (e^2*log(c) + 2*e^2)*c^2*x^3 + (4*c^2*d^2 + e^2*log(c))*x + (c^2*e^2*x^ 3 + e^2*x)*log(x))/((c^2*e^3*x^3 + c^2*d*e^2*x^2 + e^3*x + d*e^2)*sqrt(e*x + d)), x)) + 2*a*(sqrt(e*x + d)/e^2 + d/(sqrt(e*x + d)*e^2))
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]