Integrand size = 18, antiderivative size = 998 \[ \int \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx =\text {Too large to display} \] Output:
4/3*b*e*(e*x+d)^(1/2)*(c^2*x^2+1)/c^2/(1+1/c^2/x^2)^(1/2)/x/(c*(e*x+d)+(c^ 2*d^2+e^2)^(1/2))+2/3*(e*x+d)^(3/2)*(a+b*arccsch(c*x))/e-2/3*b*d^(3/2)*(1/ c^2+x^2)^(1/2)*arctanh((e*x+d)^(1/2)/c/d^(1/2)/(1/c^2+x^2)^(1/2))/e/(1+1/c ^2/x^2)^(1/2)/x-4/3*b*(c^2*d^2+e^2)^(3/4)*(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^(5/2)/e/(1+1/c^2/x^2)^(1/2)/x+2/3*b* (c^2*d^2+e^2)^(1/4)*(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^(5/2)/e/(1+1/c^2/x^2)^(1/2)/x-2 /3*b*d^2*(c^2*d^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)/e^3/(c^2*d^2+e^2)^(1/4)/ (1+1/c^2/x^2)^(1/2)/x-1/3*b*d*(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)/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 = 32.51 (sec) , antiderivative size = 926, normalized size of antiderivative = 0.93 \[ \int \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {2 a (d+e x)^{3/2}}{3 e}+\frac {b \left (-\frac {(c d+c e x) \left (-\frac {4}{3} \sqrt {1+\frac {1}{c^2 x^2}}-\frac {2 c d \text {csch}^{-1}(c x)}{3 e}-\frac {2}{3} c x \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}}-\frac {2 (c d+c e x) \left (-\frac {\sqrt {2} c d e \sqrt {1+i c x} (i+c x) \sqrt {\frac {c d+c 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 )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2} \sqrt {\frac {e (1-i c x)}{i c d+e}}}+\frac {i \sqrt {2} (c d-i e) \left (c^2 d^2+e^2\right ) \sqrt {1+i c x} \sqrt {\frac {e (i+c x) (c d+c 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 )}{e \sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}-\frac {2 e \cosh \left (2 \text {csch}^{-1}(c x)\right ) \left (-\left ((c d+c e x) \left (1+c^2 x^2\right )\right )+\frac {c x \left (c d \sqrt {2+2 i c x} (i+c x) \sqrt {\frac {c d+c 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 (-i+c x)}{c d+i e}} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} \left ((c d+i e) E\left (\arcsin \left (\sqrt {\frac {c d+c 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+c e x}{c d-i e}}\right ),\frac {c d-i e}{c d+i e}\right )\right )+(i c d+e) \sqrt {2+2 i c x} \sqrt {-\frac {e (i+c x)}{c d-i e}} \sqrt {\frac {e (i+c x) (c d+c 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 (i+c x)}{c d-i e}}}\right )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (2+c^2 x^2\right )}\right )}{3 e \sqrt {e+\frac {d}{x}} \sqrt {c x} \sqrt {d+e x}}\right )}{c^2} \] Input:
Integrate[Sqrt[d + e*x]*(a + b*ArcCsch[c*x]),x]
Output:
(2*a*(d + e*x)^(3/2))/(3*e) + (b*(-(((c*d + c*e*x)*((-4*Sqrt[1 + 1/(c^2*x^ 2)])/3 - (2*c*d*ArcCsch[c*x])/(3*e) - (2*c*x*ArcCsch[c*x])/3))/Sqrt[d + e* x]) - (2*(c*d + c*e*x)*(-((Sqrt[2]*c*d*e*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c *d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e) )]], (I*c*d + e)/(2*e)])/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)* Sqrt[(e*(1 - I*c*x))/(I*c*d + e)])) + (I*Sqrt[2]*(c*d - I*e)*(c^2*d^2 + e^ 2)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*Ellipti cPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e) /(2*e)])/(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (2*e*Cosh[2 *ArcCsch[c*x]]*(-((c*d + c*e*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)* c*x]*(I + c*x)*Sqrt[(c*d + c*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*(-I + c*x))/(c *d + I*e))]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*((c*d + I*e)*Ellipti cE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*e *EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e )]) + (I*c*d + e)*Sqrt[2 + (2*I)*c*x]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*S qrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, A rcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[-( (e*(I + c*x))/(c*d - I*e))])))/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c *x]*(2 + c^2*x^2))))/(3*e*Sqrt[e + d/x]*Sqrt[c*x]*Sqrt[d + e*x])))/c^2
Time = 1.84 (sec) , antiderivative size = 1313, normalized size of antiderivative = 1.32, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6844, 1898, 634, 599, 27, 631, 1511, 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.
| \(\displaystyle \int \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 6844 |
| \(\displaystyle \frac {2 b \int \frac {(d+e x)^{3/2}}{\sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{3 c e}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 1898 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \int \frac {(d+e x)^{3/2}}{x \sqrt {x^2+\frac {1}{c^2}}}dx}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 634 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx-\int \frac {-x e^2-2 d e}{\sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (\frac {2 \int \frac {e^2 (2 d+e x)}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}+d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (2 \int \frac {2 d+e x}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}+d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 631 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (2 \int \frac {2 d+e x}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}-2 d^2 \int -\frac {1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (2 \left (\frac {\left (\sqrt {c^2 d^2+e^2}+c d\right ) \int \frac {1}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{c}-\frac {\sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{c}\right )-2 d^2 \int -\frac {1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (2 \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\sqrt {c^2 d^2+e^2}+c d\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}-\frac {\sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{c}\right )-2 d^2 \int -\frac {1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c^2}+x^2} \left (2 \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\sqrt {c^2 d^2+e^2}+c d\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}-\frac {\sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )-2 d^2 \int -\frac {1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 1540 |
| \(\displaystyle \frac {2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{3 e}+\frac {2 b \sqrt {x^2+\frac {1}{c^2}} \left (2 \left (\frac {\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 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 )-2 d^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 )\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 ) (d+e x)^{3/2}}{3 e}+\frac {2 b \sqrt {x^2+\frac {1}{c^2}} \left (2 \left (\frac {\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 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 )-2 d^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 )\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 ) (d+e x)^{3/2}}{3 e}+\frac {2 b \sqrt {x^2+\frac {1}{c^2}} \left (2 \left (\frac {\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 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 )-2 d^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 )\right )}{3 c e \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
Input:
Int[Sqrt[d + e*x]*(a + b*ArcCsch[c*x]),x]
Output:
(2*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e) + (2*b*Sqrt[c^(-2) + x^2]*( 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))/S qrt[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)]*Ellipti cE[2*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*d^2 + e^2)^(1/4)], (1 + (c*d)/Sqr t[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)^(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)]*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^(3/2)*Sqrt[c^(- 2) + d^2/e^2 - (2*d*(d + e*x))/e^2 + (d + e*x)^2/e^2])) - 2*d^2*(-1/2*(Sqr t[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)] *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[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)/Sqrt[c^2*d^2 + e...
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[((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 = 9.41 (sec) , antiderivative size = 840, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(\frac {\frac {2 a \left (e x +d \right )^{\frac {3}{2}}}{3}+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsch}\left (c x \right )}{3}+\frac {2 \sqrt {-\frac {i c \left (e x +d \right ) e +c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c \left (e x +d \right ) e -c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (i \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 ) c d e -i \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 ) c d e -2 \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 ) c^{2} d^{2}+\operatorname {EllipticE}\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 ) c^{2} d^{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 ) c^{2} d^{2}-\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 ) e^{2}+\operatorname {EllipticE}\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 ) e^{2}\right )}{3 c^{2} \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}}}\, \left (-c d +i e \right )}\right )}{e}\) | \(840\) |
| default | \(\frac {\frac {2 a \left (e x +d \right )^{\frac {3}{2}}}{3}+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsch}\left (c x \right )}{3}+\frac {2 \sqrt {-\frac {i c \left (e x +d \right ) e +c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c \left (e x +d \right ) e -c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (i \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 ) c d e -i \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 ) c d e -2 \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 ) c^{2} d^{2}+\operatorname {EllipticE}\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 ) c^{2} d^{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 ) c^{2} d^{2}-\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 ) e^{2}+\operatorname {EllipticE}\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 ) e^{2}\right )}{3 c^{2} \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}}}\, \left (-c d +i e \right )}\right )}{e}\) | \(840\) |
| parts | \(\frac {2 a \left (e x +d \right )^{\frac {3}{2}}}{3 e}+\frac {2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsch}\left (c x \right )}{3}+\frac {2 \sqrt {-\frac {i c \left (e x +d \right ) e +c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c \left (e x +d \right ) e -c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (i \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 ) c d e -i \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 ) c d e -2 \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 ) c^{2} d^{2}+\operatorname {EllipticE}\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 ) c^{2} d^{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 ) c^{2} d^{2}-\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 ) e^{2}+\operatorname {EllipticE}\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 ) e^{2}\right )}{3 c^{2} \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}}}\, \left (-c d +i e \right )}\right )}{e}\) | \(842\) |
Input:
int((e*x+d)^(1/2)*(a+b*arccsch(c*x)),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^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)*(I*EllipticF((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*d*e-I*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*d*e-2*EllipticF((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^2*d^ 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^2*d^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-EllipticF((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))*e^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))*e^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/((I*e+c*d)*c/(c^2*d^2+ e^2))^(1/2)/(I*e-c*d)))
Timed out. \[ \int \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\text {Timed out} \] Input:
integrate((e*x+d)^(1/2)*(a+b*arccsch(c*x)),x, algorithm="fricas")
Output:
Timed out
\[ \int \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \sqrt {d + e x}\, dx \] Input:
integrate((e*x+d)**(1/2)*(a+b*acsch(c*x)),x)
Output:
Integral((a + b*acsch(c*x))*sqrt(d + e*x), x)
\[ \int \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \,d x } \] Input:
integrate((e*x+d)^(1/2)*(a+b*arccsch(c*x)),x, algorithm="maxima")
Output:
-1/9*(27*c^2*e*integrate(1/3*sqrt(e*x + d)*x^2*log(x)/(c^2*e*x^2 + e), x) + 9*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) + 6*c^2*d*(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)/e^2 + 27*e*integrate(1/3*sqrt(e*x + d )*log(x)/(c^2*e*x^2 + e), x) - 3*(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*log(c)/e^2 - 6*(e*x + d)^(3/2)*log(sqrt(c^2*x^2 + 1) + 1)/e - 2*( 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/e^2 - 9*integrate(2/3*(c^ 2*e*x^2 + c^2*d*x)*sqrt(e*x + d)/(c^2*e*x^2 + (c^2*e*x^2 + e)*sqrt(c^2*x^2 + 1) + e), x))*b + 2/3*(e*x + d)^(3/2)*a/e
\[ \int \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \,d x } \] Input:
integrate((e*x+d)^(1/2)*(a+b*arccsch(c*x)),x, algorithm="giac")
Output:
integrate(sqrt(e*x + d)*(b*arccsch(c*x) + a), x)
Timed out. \[ \int \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int \left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x} \,d x \] Input:
int((a + b*asinh(1/(c*x)))*(d + e*x)^(1/2),x)
Output:
int((a + b*asinh(1/(c*x)))*(d + e*x)^(1/2), x)
\[ \int \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {2 \sqrt {e x +d}\, a d +2 \sqrt {e x +d}\, a e x +3 \left (\int \sqrt {e x +d}\, \mathit {acsch} \left (c x \right )d x \right ) b e}{3 e} \] Input:
int((e*x+d)^(1/2)*(a+b*acsch(c*x)),x)
Output:
(2*sqrt(d + e*x)*a*d + 2*sqrt(d + e*x)*a*e*x + 3*int(sqrt(d + e*x)*acsch(c *x),x)*b*e)/(3*e)