Integrand size = 29, antiderivative size = 400 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 d \sqrt {d-c^2 d x^2} \arcsin (c x)}{3 \sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 x^2}+\frac {c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}+\frac {4 i c^3 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{3 b \sqrt {1-c^2 x^2}}-\frac {8 b c^3 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {4 i b^2 c^3 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{3 \sqrt {1-c^2 x^2}} \] Output:
-1/3*b^2*c^2*d*(-c^2*d*x^2+d)^(1/2)/x-1/3*b^2*c^3*d*(-c^2*d*x^2+d)^(1/2)*a rcsin(c*x)/(-c^2*x^2+1)^(1/2)-1/3*b*c*d*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^ (1/2)*(a+b*arcsin(c*x))/x^2+c^2*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2 /x+4/3*I*c^3*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/(-c^2*x^2+1)^(1/2) -1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^3+1/3*c^3*d*(-c^2*d*x^2+d) ^(1/2)*(a+b*arcsin(c*x))^3/b/(-c^2*x^2+1)^(1/2)-8/3*b*c^3*d*(-c^2*d*x^2+d) ^(1/2)*(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/(-c^2*x^2+1)^( 1/2)+4/3*I*b^2*c^3*d*(-c^2*d*x^2+d)^(1/2)*polylog(2,(I*c*x+(-c^2*x^2+1)^(1 /2))^2)/(-c^2*x^2+1)^(1/2)
Time = 1.45 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.23 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\frac {-a b c d x \sqrt {d-c^2 d x^2}-a^2 d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}+4 a^2 c^2 d x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}-b^2 c^2 d x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}+b d \sqrt {d-c^2 d x^2} \left (3 a c^3 x^3+b \left (4 i c^3 x^3-\sqrt {1-c^2 x^2}+4 c^2 x^2 \sqrt {1-c^2 x^2}\right )\right ) \arcsin (c x)^2+b^2 c^3 d x^3 \sqrt {d-c^2 d x^2} \arcsin (c x)^3-3 a^2 c^3 d^{3/2} x^3 \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-b d \sqrt {d-c^2 d x^2} \arcsin (c x) \left (b c x+2 a \left (1-4 c^2 x^2\right ) \sqrt {1-c^2 x^2}+8 b c^3 x^3 \log \left (1-e^{2 i \arcsin (c x)}\right )\right )-8 a b c^3 d x^3 \sqrt {d-c^2 d x^2} \log (c x)+4 i b^2 c^3 d x^3 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{3 x^3 \sqrt {1-c^2 x^2}} \] Input:
Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/x^4,x]
Output:
(-(a*b*c*d*x*Sqrt[d - c^2*d*x^2]) - a^2*d*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d *x^2] + 4*a^2*c^2*d*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] - b^2*c^2*d* x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + b*d*Sqrt[d - c^2*d*x^2]*(3*a*c ^3*x^3 + b*((4*I)*c^3*x^3 - Sqrt[1 - c^2*x^2] + 4*c^2*x^2*Sqrt[1 - c^2*x^2 ]))*ArcSin[c*x]^2 + b^2*c^3*d*x^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^3 - 3*a^ 2*c^3*d^(3/2)*x^3*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt [d]*(-1 + c^2*x^2))] - b*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*(b*c*x + 2*a*(1 - 4*c^2*x^2)*Sqrt[1 - c^2*x^2] + 8*b*c^3*x^3*Log[1 - E^((2*I)*ArcSin[c*x] )]) - 8*a*b*c^3*d*x^3*Sqrt[d - c^2*d*x^2]*Log[c*x] + (4*I)*b^2*c^3*d*x^3*S qrt[d - c^2*d*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(3*x^3*Sqrt[1 - c^2* x^2])
Time = 3.30 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.98, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.759, Rules used = {5200, 5190, 247, 223, 5136, 3042, 25, 4200, 25, 2620, 2715, 2838, 5196, 5136, 3042, 25, 4200, 25, 2620, 2715, 2838, 5152}
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.
\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^4} \, dx\) |
\(\Big \downarrow \) 5200 |
\(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x^3}dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 5190 |
\(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (c^2 \left (-\int \frac {a+b \arcsin (c x)}{x}dx\right )+\frac {1}{2} b c \int \frac {\sqrt {1-c^2 x^2}}{x^2}dx-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (c^2 \left (-\int \frac {a+b \arcsin (c x)}{x}dx\right )+\frac {1}{2} b c \left (c^2 \left (-\int \frac {1}{\sqrt {1-c^2 x^2}}dx\right )-\frac {\sqrt {1-c^2 x^2}}{x}\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (c^2 \left (-\int \frac {a+b \arcsin (c x)}{x}dx\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 5136 |
\(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (c^2 \left (-\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c x}d\arcsin (c x)\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (c^2 \left (-\int -\left ((a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )\right )d\arcsin (c x)\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (c^2 \int (a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 4200 |
\(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 5196 |
\(\displaystyle c^2 (-d) \left (-\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{x}dx}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 5136 |
\(\displaystyle c^2 (-d) \left (-\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d-c^2 d x^2} \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c x}d\arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle c^2 (-d) \left (-\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d-c^2 d x^2} \int -\left ((a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )\right )d\arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c^2 (-d) \left (-\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}-\frac {2 b c \sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 4200 |
\(\displaystyle c^2 (-d) \left (-\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c^2 (-d) \left (-\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle c^2 (-d) \left (-\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle c^2 (-d) \left (-\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle c^2 (-d) \left (-\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle c^2 (-d) \left (\frac {2 b c \sqrt {d-c^2 d x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {1-c^2 x^2}}-\frac {c \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{3 b \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\left (c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
Input:
Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/x^4,x]
Output:
-1/3*((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/x^3 + (2*b*c*d*Sqrt[d - c^2*d*x^2]*(-1/2*((1 - c^2*x^2)*(a + b*ArcSin[c*x]))/x^2 + (b*c*(-(Sqrt[1 - c^2*x^2]/x) - c*ArcSin[c*x]))/2 - c^2*(((-1/2*I)*(a + b*ArcSin[c*x])^2) /b - (2*I)*((I/2)*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])] + (b* PolyLog[2, E^((2*I)*ArcSin[c*x])])/4))))/(3*Sqrt[1 - c^2*x^2]) - c^2*d*(-( (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/x) - (c*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*Sqrt[1 - c^2*x^2]) + (2*b*c*Sqrt[d - c^2*d*x^2]* (((-1/2*I)*(a + b*ArcSin[c*x])^2)/b - (2*I)*((I/2)*(a + b*ArcSin[c*x])*Log [1 - E^((2*I)*ArcSin[c*x])] + (b*PolyLog[2, E^((2*I)*ArcSin[c*x])])/4)))/S qrt[1 - c^2*x^2])
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*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[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_.) + Pi*(k_.) + (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*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] , x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 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_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcSin[c*x ])/(f*(m + 1))), x] + (-Simp[b*c*(d^p/(f*(m + 1))) Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2), x], 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]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]
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 + 1))), x] + (-Simp[b*c*(n/(f*(m + 1)))*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] + Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int [(f*x)^(m + 2)*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[ {a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2249 vs. \(2 (372 ) = 744\).
Time = 0.83 (sec) , antiderivative size = 2250, normalized size of antiderivative = 5.62
method | result | size |
default | \(\text {Expression too large to display}\) | \(2250\) |
parts | \(\text {Expression too large to display}\) | \(2250\) |
Input:
int((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^4,x,method=_RETURNVERBOSE)
Output:
-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arcsin(c*x) ^3*d*c^3-20/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c ^2*x^2-1)*c^8+29/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x ^3/(c^2*x^2-1)*c^6-10/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2 +1)*x/(c^2*x^2-1)*c^4+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x ^2+1)/x/(c^2*x^2-1)*c^2+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2 *x^2+1)/x^3/(c^2*x^2-1)*arcsin(c*x)^2+8*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c ^4*x^4-9*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^5+1/3 *b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x^2/(c^2*x^2-1)*arc sin(c*x)*(-c^2*x^2+1)^(1/2)*c+4/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x ^4-9*c^2*x^2+1)/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2*c^3-16/3*I*b^ 2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c^2*x^2-1)*arcsin (c*x)*c^8-8*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^4/(c ^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^7+20/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c ^4*x^4-9*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)*c^6+3*I*b^2*(-d*(c^2*x^2-1 ))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^5 -4/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x/(c^2*x^2-1) *arcsin(c*x)*c^4-1/3*a^2/d/x^3*(-c^2*d*x^2+d)^(5/2)+2/3*a^2*c^4*x*(-c^2*d* x^2+d)^(3/2)+a^2*c^4*d*x*(-c^2*d*x^2+d)^(1/2)+a^2*c^4*d^2/(c^2*d)^(1/2)*ar ctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+2/3*a^2*c^2/d/x*(-c^2*d*x^2+...
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^4,x, algorithm="frica s")
Output:
integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arcsin(c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)/x^4, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \] Input:
integrate((-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x))**2/x**4,x)
Output:
Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x))**2/x**4, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^4,x, algorithm="maxim a")
Output:
1/3*(3*sqrt(-c^2*d*x^2 + d)*c^4*d*x + 3*c^3*d^(3/2)*arcsin(c*x) + 2*(-c^2* d*x^2 + d)^(3/2)*c^2/x - (-c^2*d*x^2 + d)^(5/2)/(d*x^3))*a^2 - sqrt(d)*int egrate(((b^2*c^2*d*x^2 - b^2*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) ^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))) *sqrt(c*x + 1)*sqrt(-c*x + 1)/x^4, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^4,x, algorithm="giac" )
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^4} \,d x \] Input:
int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(3/2))/x^4,x)
Output:
int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(3/2))/x^4, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\frac {\sqrt {d}\, d \left (\mathit {asin} \left (c x \right )^{3} b^{2} c^{3} x^{3}+3 \mathit {asin} \left (c x \right )^{2} a b \,c^{3} x^{3}+3 \mathit {asin} \left (c x \right ) a^{2} c^{3} x^{3}+4 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}\, a^{2}-6 \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) a b \,c^{2} x^{3}-3 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) b^{2} c^{2} x^{3}+6 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{x^{4}}d x \right ) a b \,x^{3}+3 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2}}{x^{4}}d x \right ) b^{2} x^{3}\right )}{3 x^{3}} \] Input:
int((-c^2*d*x^2+d)^(3/2)*(a+b*asin(c*x))^2/x^4,x)
Output:
(sqrt(d)*d*(asin(c*x)**3*b**2*c**3*x**3 + 3*asin(c*x)**2*a*b*c**3*x**3 + 3 *asin(c*x)*a**2*c**3*x**3 + 4*sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - sqrt ( - c**2*x**2 + 1)*a**2 - 6*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*x**2),x) *a*b*c**2*x**3 - 3*int(asin(c*x)**2/(sqrt( - c**2*x**2 + 1)*x**2),x)*b**2* c**2*x**3 + 6*int((sqrt( - c**2*x**2 + 1)*asin(c*x))/x**4,x)*a*b*x**3 + 3* int((sqrt( - c**2*x**2 + 1)*asin(c*x)**2)/x**4,x)*b**2*x**3))/(3*x**3)