Integrand size = 27, antiderivative size = 268 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d^2}{3 x}-2 b^2 c^4 d^2 x+\frac {5}{3} b c^3 d^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 x^2}+\frac {8}{3} c^4 d^2 x (a+b \arcsin (c x))^2+\frac {4 c^2 d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 x}-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}+\frac {22}{3} b c^3 d^2 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )-\frac {11}{3} i b^2 c^3 d^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+\frac {11}{3} i b^2 c^3 d^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \]
-1/3*b^2*c^2*d^2/x-2*b^2*c^4*d^2*x-1/3*b*c*d^2*(-c^2*x^2+1)^(3/2)*(a+b*arc sin(c*x))/x^2+8/3*c^4*d^2*x*(a+b*arcsin(c*x))^2+4/3*c^2*d^2*(-c^2*x^2+1)*( a+b*arcsin(c*x))^2/x-1/3*d^2*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2/x^3+22/3*b *c^3*d^2*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))-11/3*I*b^2*c^ 3*d^2*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+11/3*I*b^2*c^3*d^2*polylog(2,I* c*x+(-c^2*x^2+1)^(1/2))+5/3*b*c^3*d^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)
Time = 0.63 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.40 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^4} \, dx=\frac {d^2 \left (-a^2+6 a^2 c^2 x^2-b^2 c^2 x^2+3 a^2 c^4 x^4-6 b^2 c^4 x^4-a b c x \sqrt {1-c^2 x^2}+6 a b c^3 x^3 \sqrt {1-c^2 x^2}-2 a b \arcsin (c x)+12 a b c^2 x^2 \arcsin (c x)+6 a b c^4 x^4 \arcsin (c x)-b^2 c x \sqrt {1-c^2 x^2} \arcsin (c x)+6 b^2 c^3 x^3 \sqrt {1-c^2 x^2} \arcsin (c x)-b^2 \arcsin (c x)^2+6 b^2 c^2 x^2 \arcsin (c x)^2+3 b^2 c^4 x^4 \arcsin (c x)^2+11 a b c^3 x^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-11 b^2 c^3 x^3 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+11 b^2 c^3 x^3 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-11 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+11 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{3 x^3} \]
(d^2*(-a^2 + 6*a^2*c^2*x^2 - b^2*c^2*x^2 + 3*a^2*c^4*x^4 - 6*b^2*c^4*x^4 - a*b*c*x*Sqrt[1 - c^2*x^2] + 6*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] - 2*a*b*ArcSi n[c*x] + 12*a*b*c^2*x^2*ArcSin[c*x] + 6*a*b*c^4*x^4*ArcSin[c*x] - b^2*c*x* Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 6*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x ] - b^2*ArcSin[c*x]^2 + 6*b^2*c^2*x^2*ArcSin[c*x]^2 + 3*b^2*c^4*x^4*ArcSin [c*x]^2 + 11*a*b*c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]] - 11*b^2*c^3*x^3*ArcSi n[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 11*b^2*c^3*x^3*ArcSin[c*x]*Log[1 + E^( I*ArcSin[c*x])] - (11*I)*b^2*c^3*x^3*PolyLog[2, -E^(I*ArcSin[c*x])] + (11* I)*b^2*c^3*x^3*PolyLog[2, E^(I*ArcSin[c*x])]))/(3*x^3)
Time = 1.96 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.36, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5200, 27, 5200, 244, 2009, 5130, 5182, 24, 5198, 24, 5218, 3042, 4671, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle \frac {2}{3} b c d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x^3}dx-\frac {4}{3} c^2 d \int \frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x^2}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4}{3} c^2 d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x^2}dx+\frac {2}{3} b c d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x^3}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle \frac {2}{3} b c d^2 \left (-\frac {3}{2} c^2 \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx+\frac {1}{2} b c \int \frac {1-c^2 x^2}{x^2}dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}\right )-\frac {4}{3} c^2 d^2 \left (2 b c \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-2 c^2 \int (a+b \arcsin (c x))^2dx-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {2}{3} b c d^2 \left (-\frac {3}{2} c^2 \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx+\frac {1}{2} b c \int \left (\frac {1}{x^2}-c^2\right )dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}\right )-\frac {4}{3} c^2 d^2 \left (2 b c \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-2 c^2 \int (a+b \arcsin (c x))^2dx-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} b c d^2 \left (-\frac {3}{2} c^2 \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (c^2 (-x)-\frac {1}{x}\right )\right )-\frac {4}{3} c^2 d^2 \left (2 b c \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-2 c^2 \int (a+b \arcsin (c x))^2dx-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle -\frac {4}{3} c^2 d^2 \left (-2 c^2 \left (x (a+b \arcsin (c x))^2-2 b c \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )+2 b c \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^2 \left (-\frac {3}{2} c^2 \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (c^2 (-x)-\frac {1}{x}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle -\frac {4}{3} c^2 d^2 \left (-2 c^2 \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )+2 b c \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^2 \left (-\frac {3}{2} c^2 \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (c^2 (-x)-\frac {1}{x}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {4}{3} c^2 d^2 \left (2 b c \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-2 c^2 \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^2 \left (-\frac {3}{2} c^2 \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (c^2 (-x)-\frac {1}{x}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5198 |
| \(\displaystyle -\frac {4}{3} c^2 d^2 \left (2 b c \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-b c \int 1dx+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )-2 c^2 \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^2 \left (-\frac {3}{2} c^2 \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-b c \int 1dx+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (c^2 (-x)-\frac {1}{x}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {4}{3} c^2 d^2 \left (2 b c \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-b c x\right )-2 c^2 \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^2 \left (-\frac {3}{2} c^2 \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-b c x\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (c^2 (-x)-\frac {1}{x}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle -\frac {4}{3} c^2 d^2 \left (2 b c \left (\int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-b c x\right )-2 c^2 \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^2 \left (-\frac {3}{2} c^2 \left (\int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-b c x\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (c^2 (-x)-\frac {1}{x}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4}{3} c^2 d^2 \left (2 b c \left (\int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-b c x\right )-2 c^2 \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^2 \left (-\frac {3}{2} c^2 \left (\int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-b c x\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (c^2 (-x)-\frac {1}{x}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {4}{3} c^2 d^2 \left (2 b c \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-b c x\right )-2 c^2 \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^2 \left (-\frac {3}{2} c^2 \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-b c x\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (c^2 (-x)-\frac {1}{x}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {4}{3} c^2 d^2 \left (2 b c \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-b c x\right )-2 c^2 \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^2 \left (-\frac {3}{2} c^2 \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-b c x\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (c^2 (-x)-\frac {1}{x}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {4}{3} c^2 d^2 \left (2 b c \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-b c x\right )-2 c^2 \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^2 \left (-\frac {3}{2} c^2 \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-b c x\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (c^2 (-x)-\frac {1}{x}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 x^3}\) |
-1/3*(d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/x^3 - (4*c^2*d^2*(-(((1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/x) - 2*c^2*(x*(a + b*ArcSin[c*x])^2 - 2*b *c*((b*x)/c - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^2)) + 2*b*c*(-(b*c *x) + Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - 2*(a + b*ArcSin[c*x])*ArcTan h[E^(I*ArcSin[c*x])] + I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*PolyLog[2, E^(I*ArcSin[c*x])])))/3 + (2*b*c*d^2*((b*c*(-x^(-1) - c^2*x))/2 - ((1 - c ^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(2*x^2) - (3*c^2*(-(b*c*x) + Sqrt[1 - c ^2*x^2]*(a + b*ArcSin[c*x]) - 2*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c* x])] + I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*PolyLog[2, E^(I*ArcSin[c*x ])]))/2))/3
3.2.73.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 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[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cSin[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcS in[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x ] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[ (f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcS in[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 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] && GtQ[p, 0] && LtQ[m, -1]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* x^2]] Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a , b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.23 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.41
| method | result | size |
| derivativedivides | \(c^{3} \left (d^{2} a^{2} \left (c x -\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}\right )+2 d^{2} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+d^{2} b^{2} \arcsin \left (c x \right )^{2} c x -2 d^{2} b^{2} c x +\frac {2 d^{2} b^{2} \arcsin \left (c x \right )^{2}}{c x}-\frac {d^{2} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {d^{2} b^{2}}{3 c x}+\frac {11 d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {11 i d^{2} b^{2} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {11 d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {11 i d^{2} b^{2} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+2 d^{2} a b \left (c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}+\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(378\) |
| default | \(c^{3} \left (d^{2} a^{2} \left (c x -\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}\right )+2 d^{2} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+d^{2} b^{2} \arcsin \left (c x \right )^{2} c x -2 d^{2} b^{2} c x +\frac {2 d^{2} b^{2} \arcsin \left (c x \right )^{2}}{c x}-\frac {d^{2} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {d^{2} b^{2}}{3 c x}+\frac {11 d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {11 i d^{2} b^{2} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {11 d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {11 i d^{2} b^{2} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+2 d^{2} a b \left (c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}+\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(378\) |
| parts | \(d^{2} a^{2} \left (c^{4} x +\frac {2 c^{2}}{x}-\frac {1}{3 x^{3}}\right )+2 d^{2} b^{2} c^{3} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )+d^{2} b^{2} c^{4} \arcsin \left (c x \right )^{2} x -2 b^{2} c^{4} d^{2} x +\frac {2 d^{2} b^{2} c^{2} \arcsin \left (c x \right )^{2}}{x}-\frac {d^{2} b^{2} c \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{3 x^{2}}-\frac {b^{2} c^{2} d^{2}}{3 x}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2}}{3 x^{3}}+\frac {11 d^{2} b^{2} c^{3} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {11 i b^{2} c^{3} d^{2} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {11 d^{2} b^{2} c^{3} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {11 i b^{2} c^{3} d^{2} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+2 d^{2} a b \,c^{3} \left (c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}+\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\) | \(390\) |
c^3*(d^2*a^2*(c*x-1/3/c^3/x^3+2/c/x)+2*d^2*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1 /2)+d^2*b^2*arcsin(c*x)^2*c*x-2*d^2*b^2*c*x+2*d^2*b^2/c/x*arcsin(c*x)^2-1/ 3*d^2*b^2/c^2/x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-1/3*d^2*b^2/c^3/x^3*arcsi n(c*x)^2-1/3*d^2*b^2/c/x+11/3*d^2*b^2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^ (1/2))-11/3*I*d^2*b^2*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-11/3*d^2*b^2*ar csin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+11/3*I*d^2*b^2*polylog(2,I*c*x+(- c^2*x^2+1)^(1/2))+2*d^2*a*b*(c*x*arcsin(c*x)-1/3/c^3/x^3*arcsin(c*x)+2/c/x *arcsin(c*x)+(-c^2*x^2+1)^(1/2)-1/6/c^2/x^2*(-c^2*x^2+1)^(1/2)+11/6*arctan h(1/(-c^2*x^2+1)^(1/2))))
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^4} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsin(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b *c^2*d^2*x^2 + a*b*d^2)*arcsin(c*x))/x^4, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^4} \, dx=d^{2} \left (\int a^{2} c^{4}\, dx + \int \frac {a^{2}}{x^{4}}\, dx + \int \left (- \frac {2 a^{2} c^{2}}{x^{2}}\right )\, dx + \int b^{2} c^{4} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int 2 a b c^{4} \operatorname {asin}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{4}}\, dx + \int \left (- \frac {2 b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{2}}\right )\, dx + \int \left (- \frac {4 a b c^{2} \operatorname {asin}{\left (c x \right )}}{x^{2}}\right )\, dx\right ) \]
d**2*(Integral(a**2*c**4, x) + Integral(a**2/x**4, x) + Integral(-2*a**2*c **2/x**2, x) + Integral(b**2*c**4*asin(c*x)**2, x) + Integral(b**2*asin(c* x)**2/x**4, x) + Integral(2*a*b*c**4*asin(c*x), x) + Integral(2*a*b*asin(c *x)/x**4, x) + Integral(-2*b**2*c**2*asin(c*x)**2/x**2, x) + Integral(-4*a *b*c**2*asin(c*x)/x**2, x))
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^4} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
b^2*c^4*d^2*x*arcsin(c*x)^2 - 2*b^2*c^4*d^2*(x - sqrt(-c^2*x^2 + 1)*arcsin (c*x)/c) + a^2*c^4*d^2*x + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*c^ 3*d^2 + 4*(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)* a*b*c^2*d^2 - 1/3*((c^2*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + sqrt (-c^2*x^2 + 1)/x^2)*c + 2*arcsin(c*x)/x^3)*a*b*d^2 + 2*a^2*c^2*d^2/x - 1/3 *a^2*d^2/x^3 + 1/3*(3*x^3*integrate(2/3*(6*b^2*c^3*d^2*x^2 - b^2*c*d^2)*sq rt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2 *x^5 - x^3), x) + (6*b^2*c^2*d^2*x^2 - b^2*d^2)*arctan2(c*x, sqrt(c*x + 1) *sqrt(-c*x + 1))^2)/x^3
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2}{x^4} \,d x \]