Integrand size = 23, antiderivative size = 413 \[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=-\frac {b \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{1680 c^5 e^2 \sqrt {-c^2 x^2}}-\frac {b \left (29 c^2 d+25 e\right ) x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}+\frac {b \left (105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right ) x \arctan \left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{1680 c^6 e^{5/2} \sqrt {-c^2 x^2}}+\frac {8 b c d^{7/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{105 e^3 \sqrt {-c^2 x^2}} \]
1/3*d^2*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/e^3-2/5*d*(e*x^2+d)^(5/2)*(a+b* arccsch(c*x))/e^3+1/7*(e*x^2+d)^(7/2)*(a+b*arccsch(c*x))/e^3+1/1680*b*(105 *c^6*d^3+35*c^4*d^2*e+63*c^2*d*e^2-75*e^3)*x*arctan(e^(1/2)*(-c^2*x^2-1)^( 1/2)/c/(e*x^2+d)^(1/2))/c^6/e^(5/2)/(-c^2*x^2)^(1/2)+8/105*b*c*d^(7/2)*x*a rctan((e*x^2+d)^(1/2)/d^(1/2)/(-c^2*x^2-1)^(1/2))/e^3/(-c^2*x^2)^(1/2)-1/8 40*b*(29*c^2*d+25*e)*x*(e*x^2+d)^(3/2)*(-c^2*x^2-1)^(1/2)/c^3/e^2/(-c^2*x^ 2)^(1/2)+1/42*b*x*(e*x^2+d)^(5/2)*(-c^2*x^2-1)^(1/2)/c/e^2/(-c^2*x^2)^(1/2 )-1/1680*b*(23*c^4*d^2-12*c^2*d*e-75*e^2)*x*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^( 1/2)/c^5/e^2/(-c^2*x^2)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 2.20 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.78 \[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {32 a \left (d+e x^2\right )^2 \left (8 d^2-12 d e x^2+15 e^2 x^4\right )+\frac {2 b e \sqrt {1+\frac {1}{c^2 x^2}} x \left (d+e x^2\right ) \left (75 e^2-2 c^2 e \left (19 d+25 e x^2\right )+c^4 \left (-41 d^2+22 d e x^2+40 e^2 x^4\right )\right )}{c^5}+\frac {b \left (-128 c^4 d^4 \sqrt {1+\frac {d}{e x^2}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )+\frac {e \left (105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right ) \sqrt {1+\frac {1}{c^2 x^2}} x^4 \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-c^2 x^2,-\frac {e x^2}{d}\right )}{\sqrt {1+c^2 x^2}}\right )}{c^5 x}+32 b \left (d+e x^2\right )^2 \left (8 d^2-12 d e x^2+15 e^2 x^4\right ) \text {csch}^{-1}(c x)}{3360 e^3 \sqrt {d+e x^2}} \]
(32*a*(d + e*x^2)^2*(8*d^2 - 12*d*e*x^2 + 15*e^2*x^4) + (2*b*e*Sqrt[1 + 1/ (c^2*x^2)]*x*(d + e*x^2)*(75*e^2 - 2*c^2*e*(19*d + 25*e*x^2) + c^4*(-41*d^ 2 + 22*d*e*x^2 + 40*e^2*x^4)))/c^5 + (b*(-128*c^4*d^4*Sqrt[1 + d/(e*x^2)]* AppellF1[1, 1/2, 1/2, 2, -(1/(c^2*x^2)), -(d/(e*x^2))] + (e*(105*c^6*d^3 + 35*c^4*d^2*e + 63*c^2*d*e^2 - 75*e^3)*Sqrt[1 + 1/(c^2*x^2)]*x^4*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, -(c^2*x^2), -((e*x^2)/d)])/Sqrt[1 + c^ 2*x^2]))/(c^5*x) + 32*b*(d + e*x^2)^2*(8*d^2 - 12*d*e*x^2 + 15*e^2*x^4)*Ar cCsch[c*x])/(3360*e^3*Sqrt[d + e*x^2])
Time = 1.60 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.91, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {6856, 27, 7282, 2118, 27, 171, 27, 171, 27, 175, 66, 104, 217, 218}
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.
| \(\displaystyle \int x^5 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 6856 |
| \(\displaystyle -\frac {b c x \int \frac {\left (e x^2+d\right )^{3/2} \left (15 e^2 x^4-12 d e x^2+8 d^2\right )}{105 e^3 x \sqrt {-c^2 x^2-1}}dx}{\sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \int \frac {\left (e x^2+d\right )^{3/2} \left (15 e^2 x^4-12 d e x^2+8 d^2\right )}{x \sqrt {-c^2 x^2-1}}dx}{105 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 7282 |
| \(\displaystyle -\frac {b c x \int \frac {\left (e x^2+d\right )^{3/2} \left (15 e^2 x^4-12 d e x^2+8 d^2\right )}{x^2 \sqrt {-c^2 x^2-1}}dx^2}{210 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 2118 |
| \(\displaystyle -\frac {b c x \left (-\frac {\int -\frac {3 e \left (e x^2+d\right )^{3/2} \left (16 c^2 d^2-e \left (29 d c^2+25 e\right ) x^2\right )}{2 x^2 \sqrt {-c^2 x^2-1}}dx^2}{3 c^2 e}-\frac {5 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{c^2}\right )}{210 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \left (\frac {\int \frac {\left (e x^2+d\right )^{3/2} \left (16 c^2 d^2-e \left (29 d c^2+25 e\right ) x^2\right )}{x^2 \sqrt {-c^2 x^2-1}}dx^2}{2 c^2}-\frac {5 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{c^2}\right )}{210 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {b c x \left (\frac {\frac {e \sqrt {-c^2 x^2-1} \left (29 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}-\frac {\int -\frac {\sqrt {e x^2+d} \left (64 c^4 d^3-e \left (23 d^2 c^4-12 d e c^2-75 e^2\right ) x^2\right )}{2 x^2 \sqrt {-c^2 x^2-1}}dx^2}{2 c^2}}{2 c^2}-\frac {5 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{c^2}\right )}{210 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \left (\frac {\frac {\int \frac {\sqrt {e x^2+d} \left (64 c^4 d^3-e \left (23 d^2 c^4-12 d e c^2-75 e^2\right ) x^2\right )}{x^2 \sqrt {-c^2 x^2-1}}dx^2}{4 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (29 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{2 c^2}-\frac {5 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{c^2}\right )}{210 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {b c x \left (\frac {\frac {\frac {e \sqrt {-c^2 x^2-1} \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) \sqrt {d+e x^2}}{c^2}-\frac {\int -\frac {128 d^4 c^6+e \left (105 d^3 c^6+35 d^2 e c^4+63 d e^2 c^2-75 e^3\right ) x^2}{2 x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{c^2}}{4 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (29 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{2 c^2}-\frac {5 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{c^2}\right )}{210 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \left (\frac {\frac {\frac {\int \frac {128 d^4 c^6+e \left (105 d^3 c^6+35 d^2 e c^4+63 d e^2 c^2-75 e^3\right ) x^2}{x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (29 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{2 c^2}-\frac {5 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{c^2}\right )}{210 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {b c x \left (\frac {\frac {\frac {128 c^6 d^4 \int \frac {1}{x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2+e \left (105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right ) \int \frac {1}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (29 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{2 c^2}-\frac {5 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{c^2}\right )}{210 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle -\frac {b c x \left (\frac {\frac {\frac {128 c^6 d^4 \int \frac {1}{x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2+2 e \left (105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {-c^2 x^2-1}}{\sqrt {e x^2+d}}}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (29 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{2 c^2}-\frac {5 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{c^2}\right )}{210 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {b c x \left (\frac {\frac {\frac {256 c^6 d^4 \int \frac {1}{-x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {-c^2 x^2-1}}+2 e \left (105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {-c^2 x^2-1}}{\sqrt {e x^2+d}}}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (29 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{2 c^2}-\frac {5 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{c^2}\right )}{210 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {b c x \left (\frac {\frac {\frac {2 e \left (105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {-c^2 x^2-1}}{\sqrt {e x^2+d}}-256 c^6 d^{7/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (29 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{2 c^2}-\frac {5 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{c^2}\right )}{210 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {b c x \left (\frac {\frac {\frac {-256 c^6 d^{7/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )-\frac {2 \sqrt {e} \left (105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right ) \arctan \left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{c}}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (29 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{2 c^2}-\frac {5 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{c^2}\right )}{210 e^3 \sqrt {-c^2 x^2}}\) |
(d^2*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) - (2*d*(d + e*x^2)^(5 /2)*(a + b*ArcCsch[c*x]))/(5*e^3) + ((d + e*x^2)^(7/2)*(a + b*ArcCsch[c*x] ))/(7*e^3) - (b*c*x*((-5*e*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(5/2))/c^2 + ((e *(29*c^2*d + 25*e)*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(2*c^2) + ((e*(23 *c^4*d^2 - 12*c^2*d*e - 75*e^2)*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/c^2 + ((-2*Sqrt[e]*(105*c^6*d^3 + 35*c^4*d^2*e + 63*c^2*d*e^2 - 75*e^3)*ArcTan[( Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/c - 256*c^6*d^(7/2)*ArcT an[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(2*c^2))/(4*c^2))/(2*c^2 )))/(210*e^3*Sqrt[-(c^2*x^2)])
3.2.18.3.1 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[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + q + 1)) Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x]
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[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 /lst[[2]] Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( lst[[3]]*x)^lst[[2]]], x] /; !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ u] && !RationalFunctionQ[u, x]
\[\int x^{5} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}d x\]
Time = 1.49 (sec) , antiderivative size = 1951, normalized size of antiderivative = 4.72 \[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
[1/6720*(128*b*c^7*d^(7/2)*log(((c^4*d^2 + 6*c^2*d*e + e^2)*x^4 + 8*(c^2*d ^2 + d*e)*x^2 - 4*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sq rt((c^2*x^2 + 1)/(c^2*x^2)) + 8*d^2)/x^4) - (105*b*c^6*d^3 + 35*b*c^4*d^2* e + 63*b*c^2*d*e^2 - 75*b*e^3)*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 + 6*c^2 *d*e + 8*(c^4*d*e + c^2*e^2)*x^2 - 4*(2*c^4*e*x^3 + (c^4*d + c^2*e)*x)*sqr t(e*x^2 + d)*sqrt(e)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + e^2) + 64*(15*b*c^7*e ^3*x^6 + 3*b*c^7*d*e^2*x^4 - 4*b*c^7*d^2*e*x^2 + 8*b*c^7*d^3)*sqrt(e*x^2 + d)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 4*(240*a*c^7*e^3* x^6 + 48*a*c^7*d*e^2*x^4 - 64*a*c^7*d^2*e*x^2 + 128*a*c^7*d^3 + (40*b*c^6* e^3*x^5 + 2*(11*b*c^6*d*e^2 - 25*b*c^4*e^3)*x^3 - (41*b*c^6*d^2*e + 38*b*c ^4*d*e^2 - 75*b*c^2*e^3)*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(e*x^2 + d) )/(c^7*e^3), 1/3360*(64*b*c^7*d^(7/2)*log(((c^4*d^2 + 6*c^2*d*e + e^2)*x^4 + 8*(c^2*d^2 + d*e)*x^2 - 4*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d) *sqrt(d)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 8*d^2)/x^4) - (105*b*c^6*d^3 + 35 *b*c^4*d^2*e + 63*b*c^2*d*e^2 - 75*b*e^3)*sqrt(-e)*arctan(1/2*(2*c^2*e*x^3 + (c^2*d + e)*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/( c^2*e^2*x^4 + (c^2*d*e + e^2)*x^2 + d*e)) + 32*(15*b*c^7*e^3*x^6 + 3*b*c^7 *d*e^2*x^4 - 4*b*c^7*d^2*e*x^2 + 8*b*c^7*d^3)*sqrt(e*x^2 + d)*log((c*x*sqr t((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 2*(240*a*c^7*e^3*x^6 + 48*a*c^7*d *e^2*x^4 - 64*a*c^7*d^2*e*x^2 + 128*a*c^7*d^3 + (40*b*c^6*e^3*x^5 + 2*(...
\[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int x^{5} \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]
Exception generated. \[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]
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
\[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{5} \,d x } \]
Timed out. \[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int x^5\,\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]