Integrand size = 23, antiderivative size = 527 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\frac {b c^3 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) x^2 \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}+\frac {b c \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2}}-\frac {b c \left (12 c^2 d+e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {b c^2 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) x \sqrt {d+e x^2} E\left (\arctan (c x)\left |1-\frac {e}{c^2 d}\right .\right )}{225 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac {2 b e \left (6 c^4 d^2-4 c^2 d e-15 e^2\right ) x \sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{225 d^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}} \]
-1/5*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/d/x^5+2/15*e*(e*x^2+d)^(3/2)*(a+b* arccsch(c*x))/d^2/x^3-1/45*b*c^3*(2*c^2*d-e)*e*x^2*(e*x^2+d)^(1/2)/d^2/(-c ^2*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)-2/15*b*c^3*e^2*x^2*(e*x^2+d)^(1/2)/d^2/(- c^2*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)+1/75*b*c^3*(8*c^4*d^2-3*c^2*d*e-2*e^2)*x ^2*(e*x^2+d)^(1/2)/d^2/(-c^2*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)-1/45*b*c*(2*c^2 *d-e)*e*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(-c^2*x^2)^(1/2)-2/15*b*c*e ^2*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(-c^2*x^2)^(1/2)+1/75*b*c*(8*c^4 *d^2-3*c^2*d*e-2*e^2)*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(-c^2*x^2)^(1 /2)+1/25*b*c*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/x^4/(-c^2*x^2)^(1/2)-1/75* b*c*(4*c^2*d-e)*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/x^2/(-c^2*x^2)^(1/2)+ 1/45*b*c*e*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/x^2/(-c^2*x^2)^(1/2)+1/45* b*c^2*(2*c^2*d-e)*e*x*(1/(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)*EllipticE(c* x/(c^2*x^2+1)^(1/2),(1-e/c^2/d)^(1/2))*(e*x^2+d)^(1/2)/d^2/(-c^2*x^2)^(1/2 )/(-c^2*x^2-1)^(1/2)/((e*x^2+d)/d/(c^2*x^2+1))^(1/2)+2/15*b*c^2*e^2*x*(1/( c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)*EllipticE(c*x/(c^2*x^2+1)^(1/2),(1-e/c ^2/d)^(1/2))*(e*x^2+d)^(1/2)/d^2/(-c^2*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)/((e*x ^2+d)/d/(c^2*x^2+1))^(1/2)-1/75*b*c^2*(8*c^4*d^2-3*c^2*d*e-2*e^2)*x*(1/(c^ 2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)*EllipticE(c*x/(c^2*x^2+1)^(1/2),(1-e/c^2 /d)^(1/2))*(e*x^2+d)^(1/2)/d^2/(-c^2*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)/((e*x^2 +d)/d/(c^2*x^2+1))^(1/2)+1/75*b*c^2*(4*c^2*d-e)*e*x*(1/(c^2*x^2+1))^(1/...
Result contains complex when optimal does not.
Time = 11.99 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\frac {\sqrt {d+e x^2} \left (-15 a \left (3 d^2+d e x^2-2 e^2 x^4\right )+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (-31 e^2 x^4+d e x^2 \left (8-19 c^2 x^2\right )+3 d^2 \left (3-4 c^2 x^2+8 c^4 x^4\right )\right )-15 b \left (3 d^2+d e x^2-2 e^2 x^4\right ) \text {csch}^{-1}(c x)\right )}{225 d^2 x^5}+\frac {i b c \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) E\left (i \text {arcsinh}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )+\left (-24 c^6 d^3+31 c^4 d^2 e+23 c^2 d e^2-30 e^3\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {c^2} x\right ),\frac {e}{c^2 d}\right )\right )}{225 \sqrt {c^2} d^2 \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \]
(Sqrt[d + e*x^2]*(-15*a*(3*d^2 + d*e*x^2 - 2*e^2*x^4) + b*c*Sqrt[1 + 1/(c^ 2*x^2)]*x*(-31*e^2*x^4 + d*e*x^2*(8 - 19*c^2*x^2) + 3*d^2*(3 - 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(3*d^2 + d*e*x^2 - 2*e^2*x^4)*ArcCsch[c*x]))/(225*d^2* x^5) + ((I/225)*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(24 *c^4*d^2 - 19*c^2*d*e - 31*e^2)*EllipticE[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d )] + (-24*c^6*d^3 + 31*c^4*d^2*e + 23*c^2*d*e^2 - 30*e^3)*EllipticF[I*ArcS inh[Sqrt[c^2]*x], e/(c^2*d)]))/(Sqrt[c^2]*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + e *x^2])
Time = 0.86 (sec) , antiderivative size = 447, normalized size of antiderivative = 0.85, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6856, 27, 442, 442, 445, 27, 406, 320, 388, 313}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 6856 |
| \(\displaystyle -\frac {b c x \int -\frac {\left (3 d-2 e x^2\right ) \left (e x^2+d\right )^{3/2}}{15 d^2 x^6 \sqrt {-c^2 x^2-1}}dx}{\sqrt {-c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \int \frac {\left (3 d-2 e x^2\right ) \left (e x^2+d\right )^{3/2}}{x^6 \sqrt {-c^2 x^2-1}}dx}{15 d^2 \sqrt {-c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle \frac {b c x \left (\frac {3 d \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}-\frac {1}{5} \int \frac {\sqrt {e x^2+d} \left (e \left (3 d c^2+10 e\right ) x^2+d \left (12 d c^2+e\right )\right )}{x^4 \sqrt {-c^2 x^2-1}}dx\right )}{15 d^2 \sqrt {-c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle \frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {2 e \left (6 d^2 c^4-4 d e c^2-15 e^2\right ) x^2+d \left (24 d^2 c^4-19 d e c^2-31 e^2\right )}{x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx-\frac {d \sqrt {-c^2 x^2-1} \left (12 c^2 d+e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {-c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\int \frac {d e \left (c^2 \left (24 d^2 c^4-19 d e c^2-31 e^2\right ) x^2+2 \left (6 d^2 c^4-4 d e c^2-15 e^2\right )\right )}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx}{d}+\frac {\sqrt {-c^2 x^2-1} \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {-c^2 x^2-1} \left (12 c^2 d+e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {-c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (e \int \frac {c^2 \left (24 d^2 c^4-19 d e c^2-31 e^2\right ) x^2+2 \left (6 d^2 c^4-4 d e c^2-15 e^2\right )}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx+\frac {\sqrt {-c^2 x^2-1} \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {-c^2 x^2-1} \left (12 c^2 d+e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {-c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (c^2 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \int \frac {x^2}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx+2 \left (6 c^4 d^2-4 c^2 d e-15 e^2\right ) \int \frac {1}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx\right )+\frac {\sqrt {-c^2 x^2-1} \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {-c^2 x^2-1} \left (12 c^2 d+e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {-c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (c^2 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \int \frac {x^2}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx+\frac {2 \left (6 c^4 d^2-4 c^2 d e-15 e^2\right ) \sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{c d \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}\right )+\frac {\sqrt {-c^2 x^2-1} \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {-c^2 x^2-1} \left (12 c^2 d+e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {-c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (c^2 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \left (\frac {\int \frac {\sqrt {e x^2+d}}{\left (-c^2 x^2-1\right )^{3/2}}dx}{e}+\frac {x \sqrt {d+e x^2}}{e \sqrt {-c^2 x^2-1}}\right )+\frac {2 \left (6 c^4 d^2-4 c^2 d e-15 e^2\right ) \sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{c d \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}\right )+\frac {\sqrt {-c^2 x^2-1} \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {-c^2 x^2-1} \left (12 c^2 d+e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {-c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {2 \left (6 c^4 d^2-4 c^2 d e-15 e^2\right ) \sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{c d \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+c^2 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \left (\frac {x \sqrt {d+e x^2}}{e \sqrt {-c^2 x^2-1}}-\frac {\sqrt {d+e x^2} E\left (\arctan (c x)\left |1-\frac {e}{c^2 d}\right .\right )}{c e \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}\right )\right )+\frac {\sqrt {-c^2 x^2-1} \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {-c^2 x^2-1} \left (12 c^2 d+e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {-c^2 x^2}}\) |
-1/5*((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(d*x^5) + (2*e*(d + e*x^2)^( 3/2)*(a + b*ArcCsch[c*x]))/(15*d^2*x^3) + (b*c*x*((3*d*Sqrt[-1 - c^2*x^2]* (d + e*x^2)^(3/2))/(5*x^5) + (-1/3*(d*(12*c^2*d + e)*Sqrt[-1 - c^2*x^2]*Sq rt[d + e*x^2])/x^3 + (((24*c^4*d^2 - 19*c^2*d*e - 31*e^2)*Sqrt[-1 - c^2*x^ 2]*Sqrt[d + e*x^2])/x + e*(c^2*(24*c^4*d^2 - 19*c^2*d*e - 31*e^2)*((x*Sqrt [d + e*x^2])/(e*Sqrt[-1 - c^2*x^2]) - (Sqrt[d + e*x^2]*EllipticE[ArcTan[c* x], 1 - e/(c^2*d)])/(c*e*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x ^2))])) + (2*(6*c^4*d^2 - 4*c^2*d*e - 15*e^2)*Sqrt[d + e*x^2]*EllipticF[Ar cTan[c*x], 1 - e/(c^2*d)])/(c*d*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])))/3)/5))/(15*d^2*Sqrt[-(c^2*x^2)])
3.2.27.3.1 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[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^2*(m + 1)) Int[(g*x) ^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*2 *(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*2*(p + q + 1))*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && GtQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^2])
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*( x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Si mp[(a + b*ArcCsch[c*x]) u, x] - Simp[b*c*(x/Sqrt[(-c^2)*x^2]) Int[Simpl ifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I LtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}}{x^{6}}d x\]
Time = 0.11 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=-\frac {{\left (24 \, b c^{8} d^{3} - 19 \, b c^{6} d^{2} e - 31 \, b c^{4} d e^{2}\right )} \sqrt {-c^{2}} \sqrt {d} x^{5} E(\arcsin \left (\sqrt {-c^{2}} x\right )\,|\,\frac {e}{c^{2} d}) - {\left (24 \, b c^{8} d^{3} - {\left (19 \, b c^{6} - 12 \, b c^{4}\right )} d^{2} e - {\left (31 \, b c^{4} + 8 \, b c^{2}\right )} d e^{2} - 30 \, b e^{3}\right )} \sqrt {-c^{2}} \sqrt {d} x^{5} F(\arcsin \left (\sqrt {-c^{2}} x\right )\,|\,\frac {e}{c^{2} d}) - 15 \, {\left (2 \, b c^{2} d e^{2} x^{4} - b c^{2} d^{2} e x^{2} - 3 \, b c^{2} d^{3}\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (30 \, a c^{2} d e^{2} x^{4} - 15 \, a c^{2} d^{2} e x^{2} - 45 \, a c^{2} d^{3} + {\left (9 \, b c^{3} d^{3} x + {\left (24 \, b c^{7} d^{3} - 19 \, b c^{5} d^{2} e - 31 \, b c^{3} d e^{2}\right )} x^{5} - 4 \, {\left (3 \, b c^{5} d^{3} - 2 \, b c^{3} d^{2} e\right )} x^{3}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d}}{225 \, c^{2} d^{3} x^{5}} \]
-1/225*((24*b*c^8*d^3 - 19*b*c^6*d^2*e - 31*b*c^4*d*e^2)*sqrt(-c^2)*sqrt(d )*x^5*elliptic_e(arcsin(sqrt(-c^2)*x), e/(c^2*d)) - (24*b*c^8*d^3 - (19*b* c^6 - 12*b*c^4)*d^2*e - (31*b*c^4 + 8*b*c^2)*d*e^2 - 30*b*e^3)*sqrt(-c^2)* sqrt(d)*x^5*elliptic_f(arcsin(sqrt(-c^2)*x), e/(c^2*d)) - 15*(2*b*c^2*d*e^ 2*x^4 - b*c^2*d^2*e*x^2 - 3*b*c^2*d^3)*sqrt(e*x^2 + d)*log((c*x*sqrt((c^2* x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) - (30*a*c^2*d*e^2*x^4 - 15*a*c^2*d^2*e*x^2 - 45*a*c^2*d^3 + (9*b*c^3*d^3*x + (24*b*c^7*d^3 - 19*b*c^5*d^2*e - 31*b*c ^3*d*e^2)*x^5 - 4*(3*b*c^5*d^3 - 2*b*c^3*d^2*e)*x^3)*sqrt((c^2*x^2 + 1)/(c ^2*x^2)))*sqrt(e*x^2 + d))/(c^2*d^3*x^5)
\[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{6}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{6}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \]