Integrand size = 29, antiderivative size = 424 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b^2 x \left (1-c^2 x^2\right )}{4 c^4 d \sqrt {d-c^2 d x^2}}+\frac {b^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c^5 d \sqrt {d-c^2 d x^2}}-\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^4 d^2}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}} \]
-1/4*b^2*x*(-c^2*x^2+1)/c^4/d/(-c^2*d*x^2+d)^(1/2)+x^3*(a+b*arcsin(c*x))^2 /c^2/d/(-c^2*d*x^2+d)^(1/2)+1/4*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c^5/d/( -c^2*d*x^2+d)^(1/2)-1/2*b*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3/d/( -c^2*d*x^2+d)^(1/2)-I*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^5/d/(-c^2*d *x^2+d)^(1/2)-1/2*(a+b*arcsin(c*x))^3*(-c^2*x^2+1)^(1/2)/b/c^5/d/(-c^2*d*x ^2+d)^(1/2)+2*b*(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)/c^5/d/(-c^2*d*x^2+d)^(1/2)-I*b^2*polylog(2,-(I*c*x+(-c^2*x^2 +1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/c^5/d/(-c^2*d*x^2+d)^(1/2)+3/2*x*(a+b*arc sin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^4/d^2
Time = 2.20 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.74 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-4 a^2 c \sqrt {d} x \left (-3+c^2 x^2\right )+12 a^2 \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+2 a b \sqrt {d} \left (8 c x \arcsin (c x)+\sqrt {1-c^2 x^2} \left (-6 \arcsin (c x)^2+\cos (2 \arcsin (c x))+4 \log \left (1-c^2 x^2\right )+2 \arcsin (c x) \sin (2 \arcsin (c x))\right )\right )+b^2 \sqrt {d} \left (8 c x \arcsin (c x)^2-8 i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )+\sqrt {1-c^2 x^2} \left (-4 \arcsin (c x)^3+2 \arcsin (c x) \left (\cos (2 \arcsin (c x))+8 \log \left (1+e^{2 i \arcsin (c x)}\right )\right )-\sin (2 \arcsin (c x))+2 \arcsin (c x)^2 (-4 i+\sin (2 \arcsin (c x)))\right )\right )}{8 c^5 d^{3/2} \sqrt {d-c^2 d x^2}} \]
(-4*a^2*c*Sqrt[d]*x*(-3 + c^2*x^2) + 12*a^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))] + 2*a*b*Sqrt[d]*(8*c*x*Ar cSin[c*x] + Sqrt[1 - c^2*x^2]*(-6*ArcSin[c*x]^2 + Cos[2*ArcSin[c*x]] + 4*L og[1 - c^2*x^2] + 2*ArcSin[c*x]*Sin[2*ArcSin[c*x]])) + b^2*Sqrt[d]*(8*c*x* ArcSin[c*x]^2 - (8*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + Sqrt[1 - c^2*x^2]*(-4*ArcSin[c*x]^3 + 2*ArcSin[c*x]*(Cos[2*ArcSin[c*x]] + 8*Log[1 + E^((2*I)*ArcSin[c*x])]) - Sin[2*ArcSin[c*x]] + 2*ArcSin[c*x]^ 2*(-4*I + Sin[2*ArcSin[c*x]]))))/(8*c^5*d^(3/2)*Sqrt[d - c^2*d*x^2])
Time = 1.82 (sec) , antiderivative size = 394, normalized size of antiderivative = 0.93, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {5206, 5210, 262, 223, 5138, 262, 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.
\(\displaystyle \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 5206 |
\(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^3 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 5210 |
\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}+\frac {b \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{2 c}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {3 \left (\frac {b \sqrt {1-c^2 x^2} \int x (a+b \arcsin (c x))dx}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}\right )}{c^2 d}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}+\frac {b \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 c}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {3 \left (\frac {b \sqrt {1-c^2 x^2} \int x (a+b \arcsin (c x))dx}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}\right )}{c^2 d}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {3 \left (\frac {b \sqrt {1-c^2 x^2} \int x (a+b \arcsin (c x))dx}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}+\frac {b \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle -\frac {3 \left (\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx\right )}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}+\frac {b \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {3 \left (\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}+\frac {b \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}+\frac {b \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {3 \left (\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}+\frac {b \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
\(\Big \downarrow \) 5180 |
\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {c x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^4}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}+\frac {b \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int (a+b \arcsin (c x)) \tan (\arcsin (c x))d\arcsin (c x)}{c^4}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}+\frac {b \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1+e^{2 i \arcsin (c x)}}d\arcsin (c x)}{c^4}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}+\frac {b \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{2} i b \int \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{c^4}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}+\frac {b \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1+e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{c^4}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}+\frac {b \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )}{c^4}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}+\frac {b \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}\) |
(x^3*(a + b*ArcSin[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) - (3*(-1/2*(x*Sqrt [d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(c^2*d) + (Sqrt[1 - c^2*x^2]*(a + b *ArcSin[c*x])^3)/(6*b*c^3*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*((x^ 2*(a + b*ArcSin[c*x]))/2 - (b*c*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c *x]/(2*c^3)))/2))/(c*Sqrt[d - c^2*d*x^2])))/(c^2*d) - (2*b*Sqrt[1 - c^2*x^ 2]*(-1/2*(x^2*(a + b*ArcSin[c*x]))/c^2 + (b*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^ 2 + ArcSin[c*x]/(2*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))/(c*d*Sqrt[d - c^2*d*x^2])
3.3.45.3.1 Defintions of rubi rules used
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)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 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_.) + (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_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
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]
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/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S imp[b*f*(n/(c*(m + 2*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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 806 vs. \(2 (402 ) = 804\).
Time = 0.26 (sec) , antiderivative size = 807, normalized size of antiderivative = 1.90
method | result | size |
default | \(-\frac {a^{2} x^{3}}{2 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{4} d \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}}{2 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {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 )}{c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (-i+8 \arcsin \left (c x \right )\right )}{8 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (18 \arcsin \left (c x \right )^{2}-1\right ) x}{16 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{8 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arcsin \left (c x \right )^{2}-1\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{16 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}\right )+\frac {3 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{2 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}}{8 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {9 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x}{4 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (3 \arcsin \left (c x \right )\right )}{8 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{4 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}\) | \(807\) |
parts | \(-\frac {a^{2} x^{3}}{2 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{4} d \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}}{2 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {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 )}{c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (-i+8 \arcsin \left (c x \right )\right )}{8 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (18 \arcsin \left (c x \right )^{2}-1\right ) x}{16 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{8 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arcsin \left (c x \right )^{2}-1\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{16 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}\right )+\frac {3 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{2 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}}{8 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {9 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x}{4 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (3 \arcsin \left (c x \right )\right )}{8 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{4 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}\) | \(807\) |
-1/2*a^2*x^3/c^2/d/(-c^2*d*x^2+d)^(1/2)+3/2*a^2/c^4*x/d/(-c^2*d*x^2+d)^(1/ 2)-3/2*a^2/c^4/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2) )+b^2*(1/2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5/d^2/(c^2*x^2-1)*a rcsin(c*x)^3+I*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c^5/d^2/(c^2*x^2- 1)*(2*I*arcsin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)+2*arcsin(c*x)^2+pol ylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2))-1/8*I*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x ^2-1))^(1/2)/c^5/d^2/(c^2*x^2-1)*arcsin(c*x)*(-I+8*arcsin(c*x))-1/16*(-d*( c^2*x^2-1))^(1/2)/c^4/d^2/(c^2*x^2-1)*(18*arcsin(c*x)^2-1)*x-1/8*(-d*(c^2* x^2-1))^(1/2)/c^5/d^2/(c^2*x^2-1)*arcsin(c*x)*cos(3*arcsin(c*x))-1/16*(-d* (c^2*x^2-1))^(1/2)/c^5/d^2/(c^2*x^2-1)*(2*arcsin(c*x)^2-1)*sin(3*arcsin(c* x)))+3/2*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5/d^2/(c^2*x^2-1) *arcsin(c*x)^2+2*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c^5/d^2/( c^2*x^2-1)*arcsin(c*x)-2*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5 /d^2/(c^2*x^2-1)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/8*a*b*(-d*(c^2*x^2-1 ))^(1/2)/c^5/d^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-9/4*a*b*(-d*(c^2*x^2-1))^( 1/2)/c^4/d^2/(c^2*x^2-1)*arcsin(c*x)*x-1/8*a*b*(-d*(c^2*x^2-1))^(1/2)/c^5/ d^2/(c^2*x^2-1)*cos(3*arcsin(c*x))-1/4*a*b*(-d*(c^2*x^2-1))^(1/2)/c^5/d^2/ (c^2*x^2-1)*arcsin(c*x)*sin(3*arcsin(c*x))
\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
-1/2*a^2*(x^3/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 3*x/(sqrt(-c^2*d*x^2 + d)*c^4 *d) + 3*arcsin(c*x)/(c^5*d^(3/2))) + 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)*sqrt(-c*x + 1)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
Exception generated. \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]