Integrand size = 18, antiderivative size = 879 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=-\frac {4 b e \left (1+c^2 x^2\right )}{3 c d \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b e \sqrt {d+e x} \left (1+c^2 x^2\right )}{3 d \left (c^2 d^2+e^2\right )^{3/2} \sqrt {1+\frac {1}{c^2 x^2}} x \left (1+\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}\right )}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{c \sqrt {d} \sqrt {\frac {1}{c^2}+x^2}}\right )}{3 d^{3/2} e \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {4 b \sqrt {\frac {e^2 \left (1+c^2 x^2\right )}{\left (c^2 d^2+e^2\right ) \left (1+\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}\right )^2}} \left (1+\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}\right ) 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 (1+\frac {c d}{\sqrt {c^2 d^2+e^2}}\right )\right )}{3 \sqrt {c} d e \sqrt [4]{c^2 d^2+e^2} \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 b \left (c^2 d^2+2 e^2-c d \sqrt {c^2 d^2+e^2}\right ) \sqrt {\frac {e^2 \left (1+c^2 x^2\right )}{\left (c^2 d^2+e^2\right ) \left (1+\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}\right )^2}} \left (1+\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}\right ) \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 (1+\frac {c d}{\sqrt {c^2 d^2+e^2}}\right )\right )}{3 \sqrt {c} d e^3 \sqrt [4]{c^2 d^2+e^2} \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {b \left (c d-\sqrt {c^2 d^2+e^2}\right )^2 \sqrt {\frac {e^2 \left (1+c^2 x^2\right )}{\left (\sqrt {c^2 d^2+e^2}+c (d+e x)\right )^2}} \left (\sqrt {c^2 d^2+e^2}+c (d+e x)\right ) \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 (1+\frac {c d}{\sqrt {c^2 d^2+e^2}}\right )\right )}{3 c^{3/2} d^2 e^3 \sqrt [4]{c^2 d^2+e^2} \sqrt {1+\frac {1}{c^2 x^2}} x} \] Output:
-4/3*b*e*(c^2*x^2+1)/c/d/(c^2*d^2+e^2)/(1+1/c^2/x^2)^(1/2)/x/(e*x+d)^(1/2) +4/3*b*e*(e*x+d)^(1/2)*(c^2*x^2+1)/d/(c^2*d^2+e^2)^(3/2)/(1+1/c^2/x^2)^(1/ 2)/x/(1+c*(e*x+d)/(c^2*d^2+e^2)^(1/2))-2/3*(a+b*arccsch(c*x))/e/(e*x+d)^(3 /2)+2/3*b*(1/c^2+x^2)^(1/2)*arctanh((e*x+d)^(1/2)/c/d^(1/2)/(1/c^2+x^2)^(1 /2))/d^(3/2)/e/(1+1/c^2/x^2)^(1/2)/x-4/3*b*(e^2*(c^2*x^2+1)/(c^2*d^2+e^2)/ (1+c*(e*x+d)/(c^2*d^2+e^2)^(1/2))^2)^(1/2)*(1+c*(e*x+d)/(c^2*d^2+e^2)^(1/2 ))*EllipticE(sin(2*arctan(c^(1/2)*(e*x+d)^(1/2)/(c^2*d^2+e^2)^(1/4))),1/2* (2+2*c*d/(c^2*d^2+e^2)^(1/2))^(1/2))/c^(1/2)/d/e/(c^2*d^2+e^2)^(1/4)/(1+1/ c^2/x^2)^(1/2)/x+2/3*b*(c^2*d^2+2*e^2-c*d*(c^2*d^2+e^2)^(1/2))*(e^2*(c^2*x ^2+1)/(c^2*d^2+e^2)/(1+c*(e*x+d)/(c^2*d^2+e^2)^(1/2))^2)^(1/2)*(1+c*(e*x+d )/(c^2*d^2+e^2)^(1/2))*InverseJacobiAM(2*arctan(c^(1/2)*(e*x+d)^(1/2)/(c^2 *d^2+e^2)^(1/4)),1/2*(2+2*c*d/(c^2*d^2+e^2)^(1/2))^(1/2))/c^(1/2)/d/e^3/(c ^2*d^2+e^2)^(1/4)/(1+1/c^2/x^2)^(1/2)/x+1/3*b*(c*d-(c^2*d^2+e^2)^(1/2))^2* ((c^2*x^2+1)*e^2/(c*(e*x+d)+(c^2*d^2+e^2)^(1/2))^2)^(1/2)*(c*(e*x+d)+(c^2* d^2+e^2)^(1/2))*EllipticPi(sin(2*arctan(c^(1/2)*(e*x+d)^(1/2)/(c^2*d^2+e^2 )^(1/4))),1/4*(c*d+(c^2*d^2+e^2)^(1/2))^2/c/d/(c^2*d^2+e^2)^(1/2),1/2*(2+2 *c*d/(c^2*d^2+e^2)^(1/2))^(1/2))/c^(3/2)/d^2/e^3/(c^2*d^2+e^2)^(1/4)/(1+1/ c^2/x^2)^(1/2)/x
Result contains complex when optimal does not.
Time = 20.64 (sec) , antiderivative size = 784, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\frac {-\frac {2 a (d+e x)}{e}+\frac {4 b c \sqrt {1+\frac {1}{c^2 x^2}} (d+e x)^3}{d \left (c^2 d^2+e^2\right )}-\frac {2 b e x^2 (d+e x) \text {csch}^{-1}(c x)}{d^2}-\frac {2 b (d+e x)^3 \text {csch}^{-1}(c x)}{d^2 e}+\frac {4 b x (d+e x)^2 \left (-c d e \sqrt {1+\frac {1}{c^2 x^2}}+\left (c^2 d^2+e^2\right ) \text {csch}^{-1}(c x)\right )}{d^2 \left (c^2 d^2+e^2\right )}+\frac {2 i b c d \sqrt {2+2 i c x} (d+e x)^2 \sqrt {\frac {c e (i+c x) (d+e x)}{(i c d+e)^2}} \operatorname {EllipticPi}\left (1+\frac {i c d}{e},\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )}{(c d+i e) e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {4 b (d+e x)^2 \cosh \left (2 \text {csch}^{-1}(c x)\right ) \left (-c (d+e x) \left (1+c^2 x^2\right )+\frac {c x \left (c d \sqrt {2+2 i c x} (i+c x) \sqrt {\frac {c (d+e x)}{c d-i e}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )+2 \sqrt {\frac {e (1+i c x)}{-i c d+e}} (i+c x) \sqrt {\frac {c (d+e x)}{c d-i e}} \left ((c d+i e) E\left (\arcsin \left (\sqrt {\frac {c (d+e x)}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )-i e \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c (d+e x)}{c d-i e}}\right ),\frac {c d-i e}{c d+i e}\right )\right )+(i c d+e) \sqrt {\frac {e (1-i c x)}{i c d+e}} \sqrt {2+2 i c x} \sqrt {\frac {c e (i+c x) (d+e x)}{(i c d+e)^2}} \operatorname {EllipticPi}\left (1+\frac {i c d}{e},\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )\right )}{2 \sqrt {\frac {e (1-i c x)}{i c d+e}}}\right )}{d \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} \left (2+c^2 x^2\right )}}{3 (d+e x)^{5/2}} \] Input:
Integrate[(a + b*ArcCsch[c*x])/(d + e*x)^(5/2),x]
Output:
((-2*a*(d + e*x))/e + (4*b*c*Sqrt[1 + 1/(c^2*x^2)]*(d + e*x)^3)/(d*(c^2*d^ 2 + e^2)) - (2*b*e*x^2*(d + e*x)*ArcCsch[c*x])/d^2 - (2*b*(d + e*x)^3*ArcC sch[c*x])/(d^2*e) + (4*b*x*(d + e*x)^2*(-(c*d*e*Sqrt[1 + 1/(c^2*x^2)]) + ( c^2*d^2 + e^2)*ArcCsch[c*x]))/(d^2*(c^2*d^2 + e^2)) + ((2*I)*b*c*d*Sqrt[2 + (2*I)*c*x]*(d + e*x)^2*Sqrt[(c*e*(I + c*x)*(d + e*x))/(I*c*d + e)^2]*Ell ipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/((c*d + I*e)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x) + (4*b*(d + e*x)^2* Cosh[2*ArcCsch[c*x]]*(-(c*(d + e*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2 *I)*c*x]*(I + c*x)*Sqrt[(c*(d + e*x))/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[- ((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[(e*(1 + I*c*x)) /((-I)*c*d + e)]*(I + c*x)*Sqrt[(c*(d + e*x))/(c*d - I*e)]*((c*d + I*e)*El lipticE[ArcSin[Sqrt[(c*(d + e*x))/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*(d + e*x))/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[(e*(1 - I*c*x))/(I*c*d + e)]*Sqrt[2 + (2*I)*c* x]*Sqrt[(c*e*(I + c*x)*(d + e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[ (e*(1 - I*c*x))/(I*c*d + e)])))/(d*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*( 2 + c^2*x^2)))/(3*(d + e*x)^(5/2))
Time = 1.95 (sec) , antiderivative size = 1359, normalized size of antiderivative = 1.55, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6844, 1898, 635, 25, 27, 498, 27, 507, 631, 1459, 1416, 1509, 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.
| \(\displaystyle \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6844 |
| \(\displaystyle -\frac {2 b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}}dx}{3 c e}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1898 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \int \frac {1}{x (d+e x)^{3/2} \sqrt {x^2+\frac {1}{c^2}}}dx}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 635 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (\int -\frac {e}{d (d+e x)^{3/2} \sqrt {x^2+\frac {1}{c^2}}}dx+\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx}{d}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx}{d}-\int \frac {e}{d (d+e x)^{3/2} \sqrt {x^2+\frac {1}{c^2}}}dx\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx}{d}-\frac {e \int \frac {1}{(d+e x)^{3/2} \sqrt {x^2+\frac {1}{c^2}}}dx}{d}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx}{d}-\frac {e \left (-\frac {2 c^2 \int -\frac {\sqrt {d+e x}}{2 \sqrt {x^2+\frac {1}{c^2}}}dx}{c^2 d^2+e^2}-\frac {2 c^2 e \sqrt {\frac {1}{c^2}+x^2}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx}{d}-\frac {e \left (\frac {c^2 \int \frac {\sqrt {d+e x}}{\sqrt {x^2+\frac {1}{c^2}}}dx}{c^2 d^2+e^2}-\frac {2 c^2 e \sqrt {\frac {1}{c^2}+x^2}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 507 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx}{d}-\frac {e \left (\frac {2 c^2 \int \frac {d+e x}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e \left (c^2 d^2+e^2\right )}-\frac {2 c^2 e \sqrt {\frac {1}{c^2}+x^2}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 631 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (-\frac {e \left (\frac {2 c^2 \int \frac {d+e x}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e \left (c^2 d^2+e^2\right )}-\frac {2 c^2 e \sqrt {\frac {1}{c^2}+x^2}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}-\frac {2 \int -\frac {1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{d}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1459 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (-\frac {e \left (\frac {2 c^2 \left (\frac {\sqrt {c^2 d^2+e^2} \int \frac {1}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{c}-\frac {\sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 c^2 e \sqrt {\frac {1}{c^2}+x^2}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}-\frac {2 \int -\frac {1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{d}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (-\frac {e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}-\frac {\sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 c^2 e \sqrt {\frac {1}{c^2}+x^2}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}-\frac {2 \int -\frac {1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{d}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (-\frac {2 \int -\frac {1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{d}-\frac {e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}-\frac {\sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 c^2 e \sqrt {\frac {1}{c^2}+x^2}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1540 |
| \(\displaystyle -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {2 b \sqrt {x^2+\frac {1}{c^2}} \left (-\frac {e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \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^{3/2} \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 {\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 )}{e \left (c^2 d^2+e^2\right )}-\frac {2 c^2 e \sqrt {x^2+\frac {1}{c^2}}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}-\frac {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 {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 )}{d}\right )}{3 c e \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {2 b \sqrt {x^2+\frac {1}{c^2}} \left (-\frac {e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \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^{3/2} \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 {\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 )}{e \left (c^2 d^2+e^2\right )}-\frac {2 c^2 e \sqrt {x^2+\frac {1}{c^2}}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}-\frac {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}\right )}{3 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 )}{3 e (d+e x)^{3/2}}-\frac {2 b \sqrt {x^2+\frac {1}{c^2}} \left (-\frac {e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \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^{3/2} \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 {\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 )}{e \left (c^2 d^2+e^2\right )}-\frac {2 c^2 e \sqrt {x^2+\frac {1}{c^2}}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}-\frac {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}\right )}{3 c e \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
Input:
Int[(a + b*ArcCsch[c*x])/(d + e*x)^(5/2),x]
Output:
(-2*(a + b*ArcCsch[c*x]))/(3*e*(d + e*x)^(3/2)) - (2*b*Sqrt[c^(-2) + x^2]* (-((e*((-2*c^2*e*Sqrt[c^(-2) + x^2])/((c^2*d^2 + e^2)*Sqrt[d + e*x]) + (2* c^2*(-((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)]*Ellip ticE[2*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*d^2 + e^2)^(1/4)], (1 + (c*d)/S qrt[c^2*d^2 + e^2])/2])/(Sqrt[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)^(3/4)*(1 + (c*(d + e*x))/Sq rt[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)]*E llipticF[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^(3/2)*Sqrt[c^(-2) + d^2/e^2 - (2*d*(d + e *x))/e^2 + (d + e*x)^2/e^2])))/(e*(c^2*d^2 + e^2))))/d) - (2*(-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))/Sqr t[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)]*El lipticF[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))...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[2/ d Subst[Int[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/((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[( (c + d*x)^n/Sqrt[a + b*x^2])*ExpandToSum[(1 - c^(n + 1/2)*(c + d*x)^(-n - 1 /2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[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[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[1/q Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], 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[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[((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 = 10.90 (sec) , antiderivative size = 2078, normalized size of antiderivative = 2.36
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2078\) |
| default | \(\text {Expression too large to display}\) | \(2078\) |
| parts | \(\text {Expression too large to display}\) | \(2081\) |
Input:
int((a+b*arccsch(c*x))/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/e*(-1/3*a/(e*x+d)^(3/2)+b*(-1/3/(e*x+d)^(3/2)*arccsch(c*x)+2/3/c*(-I*((I *e+c*d)*c/(c^2*d^2+e^2))^(1/2)*d*e^3-I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c ^2*d*e*(e*x+d)^2-(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2) )^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*El lipticF((e*x+d)^(1/2)*((I*e+c*d)*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*(e*x+d)^(1/2)+(-(I*c*(e*x+d)*e+c^2*d *(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-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)*((I*e+c*d)*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*( e*x+d)^(1/2)-(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1 /2)*((I*c*(e*x+d)*e-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*Ellipt icPi((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),1/(I*e+c*d)/c*(c^2*d^ 2+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((I*e+c*d)*c/(c^2*d^2+e^2))^(1 /2))*c^3*d^3*(e*x+d)^(1/2)+((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^3*d^2*(e*x+ d)^2+I*(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*(( I*c*(e*x+d)*e-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)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),1/(I*e+c*d)/c*(c^2*d^2+e^2) /d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2))*c ^2*d^2*e*(e*x+d)^(1/2)-2*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^3*d^3*(e*x+d) -I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^2*d^3*e+I*(-(I*c*(e*x+d)*e+c^2*d...
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*arccsch(c*x))/(e*x+d)^(5/2),x, algorithm="fricas")
Output:
integral(sqrt(e*x + d)*(b*arccsch(c*x) + a)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2 *e*x + d^3), x)
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((a+b*acsch(c*x))/(e*x+d)**(5/2),x)
Output:
Integral((a + b*acsch(c*x))/(d + e*x)**(5/2), x)
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*arccsch(c*x))/(e*x+d)^(5/2),x, algorithm="maxima")
Output:
-1/3*(6*c^2*integrate(1/3*x/((c^2*e^2*x^3 + c^2*d*e*x^2 + e^2*x + d*e)*sqr t(c^2*x^2 + 1)*sqrt(e*x + d) + (c^2*e^2*x^3 + c^2*d*e*x^2 + e^2*x + d*e)*s qrt(e*x + d)), x) + 2*log(sqrt(c^2*x^2 + 1) + 1)/((e^2*x + d*e)*sqrt(e*x + d)) + 3*integrate(1/3*((3*e*log(c) - 2*e)*c^2*x^2 - 2*c^2*d*x + 3*e*log(c ) + 3*(c^2*e*x^2 + e)*log(x))/((c^2*e^3*x^4 + 2*c^2*d*e^2*x^3 + 2*d*e^2*x + d^2*e + (c^2*d^2*e + e^3)*x^2)*sqrt(e*x + d)), x))*b - 2/3*a/((e*x + d)^ (3/2)*e)
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*arccsch(c*x))/(e*x+d)^(5/2),x, algorithm="giac")
Output:
integrate((b*arccsch(c*x) + a)/(e*x + d)^(5/2), x)
Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:
int((a + b*asinh(1/(c*x)))/(d + e*x)^(5/2),x)
Output:
int((a + b*asinh(1/(c*x)))/(d + e*x)^(5/2), x)
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\frac {3 \sqrt {e x +d}\, \left (\int \frac {\mathit {acsch} \left (c x \right )}{\sqrt {e x +d}\, d^{2}+2 \sqrt {e x +d}\, d e x +\sqrt {e x +d}\, e^{2} x^{2}}d x \right ) b d e +3 \sqrt {e x +d}\, \left (\int \frac {\mathit {acsch} \left (c x \right )}{\sqrt {e x +d}\, d^{2}+2 \sqrt {e x +d}\, d e x +\sqrt {e x +d}\, e^{2} x^{2}}d x \right ) b \,e^{2} x -2 a}{3 \sqrt {e x +d}\, e \left (e x +d \right )} \] Input:
int((a+b*acsch(c*x))/(e*x+d)^(5/2),x)
Output:
(3*sqrt(d + e*x)*int(acsch(c*x)/(sqrt(d + e*x)*d**2 + 2*sqrt(d + e*x)*d*e* x + sqrt(d + e*x)*e**2*x**2),x)*b*d*e + 3*sqrt(d + e*x)*int(acsch(c*x)/(sq rt(d + e*x)*d**2 + 2*sqrt(d + e*x)*d*e*x + sqrt(d + e*x)*e**2*x**2),x)*b*e **2*x - 2*a)/(3*sqrt(d + e*x)*e*(d + e*x))