Integrand size = 29, antiderivative size = 421 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 x}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x^2 (a+b \arcsin (c x))}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {x (a+b \arcsin (c x))^2}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}} \] Output:
1/3*b^2*x/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*b^2*(-c^2*x^2+1)^(1/2)*arcsin(c *x)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*b*x^2*(a+b*arcsin(c*x))/c^3/d^2/(-c^2 *x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/3*x^3*(a+b*arcsin(c*x))^2/c^2/d/(-c^2 *d*x^2+d)^(3/2)-x*(a+b*arcsin(c*x))^2/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+4/3*I*( -c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2/c^5/d^2/(-c^2*d*x^2+d)^(1/2)+1/3*(-c ^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^3/b/c^5/d^2/(-c^2*d*x^2+d)^(1/2)-8/3*b*( -c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)/c^5 /d^2/(-c^2*d*x^2+d)^(1/2)+4/3*I*b^2*(-c^2*x^2+1)^(1/2)*polylog(2,-(I*c*x+( -c^2*x^2+1)^(1/2))^2)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)
Time = 1.15 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.89 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {a^2 c \sqrt {d} x \left (-3+4 c^2 x^2\right )+3 a^2 \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+b^2 \sqrt {d} \left (c x-c^3 x^3-\sqrt {1-c^2 x^2} \arcsin (c x)-3 c x \arcsin (c x)^2+4 c^3 x^3 \arcsin (c x)^2+4 i \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)^2+\left (1-c^2 x^2\right )^{3/2} \arcsin (c x)^3-8 \left (1-c^2 x^2\right )^{3/2} \arcsin (c x) \log \left (1+e^{2 i \arcsin (c x)}\right )+4 i \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )-a b \sqrt {d} \left (\sqrt {1-c^2 x^2}+\left (1-c^2 x^2\right )^{3/2} \left (-3 \arcsin (c x)^2+4 \log \left (1-c^2 x^2\right )\right )+2 \arcsin (c x) \sin (3 \arcsin (c x))\right )}{3 c^5 d^{5/2} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \] Input:
Integrate[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]
Output:
(a^2*c*Sqrt[d]*x*(-3 + 4*c^2*x^2) + 3*a^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^ 2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + b^2*Sqrt[d ]*(c*x - c^3*x^3 - Sqrt[1 - c^2*x^2]*ArcSin[c*x] - 3*c*x*ArcSin[c*x]^2 + 4 *c^3*x^3*ArcSin[c*x]^2 + (4*I)*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]^2 + (1 - c^ 2*x^2)^(3/2)*ArcSin[c*x]^3 - 8*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]*Log[1 + E^( (2*I)*ArcSin[c*x])] + (4*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcSi n[c*x])]) - a*b*Sqrt[d]*(Sqrt[1 - c^2*x^2] + (1 - c^2*x^2)^(3/2)*(-3*ArcSi n[c*x]^2 + 4*Log[1 - c^2*x^2]) + 2*ArcSin[c*x]*Sin[3*ArcSin[c*x]]))/(3*c^5 *d^(5/2)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])
Time = 1.92 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {5206, 5206, 252, 223, 5152, 5180, 3042, 4202, 2620, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^3 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx}{c^2 d}+\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {b \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2}}dx}{2 c}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}}{c^2 d}+\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{c^2}\right )}{2 c}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}}{c^2 d}+\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}+\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5180 |
| \(\displaystyle -\frac {-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {c x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {c x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^4}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \tan (\arcsin (c x))d\arcsin (c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\int (a+b \arcsin (c x)) \tan (\arcsin (c x))d\arcsin (c x)}{c^4}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle -\frac {-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1+e^{2 i \arcsin (c x)}}d\arcsin (c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1+e^{2 i \arcsin (c x)}}d\arcsin (c x)}{c^4}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{2} i b \int \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{2} i b \int \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{c^4}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1+e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1+e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{c^4}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\frac {x (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )}{c^4}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\) |
Input:
Int[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]
Output:
(x^3*(a + b*ArcSin[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (2*b*Sqrt[1 - c^2*x^2]*((x^2*(a + b*ArcSin[c*x]))/(2*c^2*(1 - c^2*x^2)) - (b*(x/(c^2*S qrt[1 - c^2*x^2]) - ArcSin[c*x]/c^3))/(2*c) - (((I/2)*(a + b*ArcSin[c*x])^ 2)/b - (2*I)*((-1/2*I)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])] - (b*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/4))/c^4))/(3*c*d^2*Sqrt[d - c^2*d *x^2]) - ((x*(a + b*ArcSin[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c^3*d*Sqrt[d - c^2*d*x^2]) - (2*b*S qrt[1 - c^2*x^2]*(((I/2)*(a + b*ArcSin[c*x])^2)/b - (2*I)*((-1/2*I)*(a + b *ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])] - (b*PolyLog[2, -E^((2*I)*Arc Sin[c*x])])/4)))/(c^3*d*Sqrt[d - c^2*d*x^2]))/(c^2*d)
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-e^(-1) Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp [b*f*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{ a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[m, 1]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 800 vs. \(2 (393 ) = 786\).
Time = 0.84 (sec) , antiderivative size = 801, normalized size of antiderivative = 1.90
| method | result | size |
| default | \(\frac {a^{2} x^{3}}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {a^{2} x}{c^{4} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{4} d^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{3 d^{3} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \left (-1+6 \arcsin \left (c x \right )^{2}-2 i \arcsin \left (c x \right )\right )}{12 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{5}}-\frac {4 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right )}{3 d^{3} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (10 \arcsin \left (c x \right )^{2}-2 i \arcsin \left (c x \right )-3\right )}{12 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{5}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-1\right ) \left (2 \arcsin \left (c x \right )^{2}-1-2 i \arcsin \left (c x \right )\right ) x}{12 c^{4} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) d^{3}}\right )-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 \arcsin \left (c x \right )^{2} x^{4} c^{4}+8 i \arcsin \left (c x \right ) x^{4} c^{4}-8 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}+8 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{3} c^{3}-6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-16 i \arcsin \left (c x \right ) x^{2} c^{2}+16 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}+3 \arcsin \left (c x \right )^{2}+8 i \arcsin \left (c x \right )-8 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-1\right )}{3 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) c^{5}}\) | \(801\) |
| parts | \(\frac {a^{2} x^{3}}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {a^{2} x}{c^{4} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{4} d^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{3 d^{3} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \left (-1+6 \arcsin \left (c x \right )^{2}-2 i \arcsin \left (c x \right )\right )}{12 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{5}}-\frac {4 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right )}{3 d^{3} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (10 \arcsin \left (c x \right )^{2}-2 i \arcsin \left (c x \right )-3\right )}{12 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{5}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-1\right ) \left (2 \arcsin \left (c x \right )^{2}-1-2 i \arcsin \left (c x \right )\right ) x}{12 c^{4} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) d^{3}}\right )-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 \arcsin \left (c x \right )^{2} x^{4} c^{4}+8 i \arcsin \left (c x \right ) x^{4} c^{4}-8 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}+8 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{3} c^{3}-6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-16 i \arcsin \left (c x \right ) x^{2} c^{2}+16 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}+3 \arcsin \left (c x \right )^{2}+8 i \arcsin \left (c x \right )-8 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-1\right )}{3 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) c^{5}}\) | \(801\) |
Input:
int(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/3*a^2*x^3/c^2/d/(-c^2*d*x^2+d)^(3/2)-a^2/c^4/d^2*x/(-c^2*d*x^2+d)^(1/2)+ a^2/c^4/d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^2 *(-1/3*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^3/c^5/(c^2*x^2-1)*arcsi n(c*x)^3+1/12*(-d*(c^2*x^2-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)+c*x)*(-1+6*arc sin(c*x)^2-2*I*arcsin(c*x))/d^3/(c^4*x^4-2*c^2*x^2+1)/c^5-4/3*I*(-c^2*x^2+ 1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(2*I*arcsin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^ (1/2))^2)+2*arcsin(c*x)^2+polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2))/d^3/c^ 5/(c^2*x^2-1)+1/12*(-d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+ 2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*(10*arcsin(c*x)^2-2*I*arcsin(c*x)-3) /d^3/(c^4*x^4-2*c^2*x^2+1)/c^5-1/12*(-d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+ 1)^(1/2)*c*x+2*c^2*x^2-1)*(2*arcsin(c*x)^2-1-2*I*arcsin(c*x))*x/c^4/(c^4*x ^4-2*c^2*x^2+1)/d^3)-1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(3* arcsin(c*x)^2*x^4*c^4+8*I*arcsin(c*x)*x^4*c^4-8*ln(1+(I*c*x+(-c^2*x^2+1)^( 1/2))^2)*x^4*c^4+8*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^3*c^3-6*arcsin(c*x)^2* x^2*c^2-16*I*arcsin(c*x)*x^2*c^2+16*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)*x^2 *c^2-6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2+3*arcsin(c*x)^2+8*I*arcs in(c*x)-8*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1)/d^3/(c^6*x^6-3*c^4*x^4+3*c ^2*x^2-1)/c^5
\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="frica s")
Output:
integral(-(b^2*x^4*arcsin(c*x)^2 + 2*a*b*x^4*arcsin(c*x) + a^2*x^4)*sqrt(- c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(x**4*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(5/2),x)
Output:
Integral(x**4*(a + b*asin(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxim a")
Output:
1/3*(x*(3*x^2/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 2/((-c^2*d*x^2 + d)^(3/2)*c ^4*d)) - x/(sqrt(-c^2*d*x^2 + d)*c^4*d^2) + 3*arcsin(c*x)/(c^5*d^(5/2)))*a ^2 - sqrt(d)*integrate((b^2*x^4*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) ^2 + 2*a*b*x^4*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*s qrt(-c*x + 1)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
Exception generated. \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac" )
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:
int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(5/2),x)
Output:
int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(5/2), x)
\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a^{2} c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right ) x^{4}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{7} x^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right ) x^{4}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{5}+3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2} x^{4}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{7} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2} x^{4}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{5}-4 a^{2} c^{3} x^{3}+3 a^{2} c x}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} \left (c^{2} x^{2}-1\right )} \] Input:
int(x^4*(a+b*asin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x)
Output:
(3*sqrt( - c**2*x**2 + 1)*asin(c*x)*a**2*c**2*x**2 - 3*sqrt( - c**2*x**2 + 1)*asin(c*x)*a**2 + 6*sqrt( - c**2*x**2 + 1)*int((asin(c*x)*x**4)/(sqrt( - c**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*a*b*c**7*x**2 - 6*sqrt( - c**2*x**2 + 1)*int((asin(c*x) *x**4)/(sqrt( - c**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x **2 + sqrt( - c**2*x**2 + 1)),x)*a*b*c**5 + 3*sqrt( - c**2*x**2 + 1)*int(( asin(c*x)**2*x**4)/(sqrt( - c**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*b**2*c**7*x**2 - 3*sqrt( - c* *2*x**2 + 1)*int((asin(c*x)**2*x**4)/(sqrt( - c**2*x**2 + 1)*c**4*x**4 - 2 *sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*b**2*c**5 - 4*a**2*c**3*x**3 + 3*a**2*c*x)/(3*sqrt(d)*sqrt( - c**2*x**2 + 1)*c**5*d** 2*(c**2*x**2 - 1))