Integrand size = 29, antiderivative size = 561 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\frac {31}{64} b^2 c^2 d^2 x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 c^2 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {89 b^2 c d^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{64 \sqrt {1-c^2 x^2}}+\frac {15 b c^3 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}+b c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {1}{8} b c d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {i c d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-\frac {5 c d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{8 b \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b^2 c d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \]
-5/4*c^2*d*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2-(-c^2*d*x^2+d)^(5/2) *(a+b*arcsin(c*x))^2/x+31/64*b^2*c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+1/32*b^2*c ^2*d^2*x*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)-1/8*b*c*d^2*(-c^2*x^2+1)^(3/2)* (a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)-15/8*c^2*d^2*x*(a+b*arcsin(c*x))^2* (-c^2*d*x^2+d)^(1/2)-89/64*b^2*c*d^2*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/(-c^ 2*x^2+1)^(1/2)+15/8*b*c^3*d^2*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/( -c^2*x^2+1)^(1/2)-I*c*d^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x ^2+1)^(1/2)-5/8*c*d^2*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/(-c^2*x^2 +1)^(1/2)+2*b*c*d^2*(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)*( -c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-I*b^2*c*d^2*polylog(2,(I*c*x+(-c^2* x^2+1)^(1/2))^2)*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+b*c*d^2*(a+b*arcs in(c*x))*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)
Time = 2.31 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.04 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\frac {d^2 \left (-256 a^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}-288 a^2 c^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}+64 a^2 c^4 x^4 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}-160 b^2 c x \sqrt {d-c^2 d x^2} \arcsin (c x)^3+480 a^2 c \sqrt {d} x \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-128 a b c x \sqrt {d-c^2 d x^2} \cos (2 \arcsin (c x))-4 a b c x \sqrt {d-c^2 d x^2} \cos (4 \arcsin (c x))+512 a b c x \sqrt {d-c^2 d x^2} \log (c x)-256 i b^2 c x \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+64 b^2 c x \sqrt {d-c^2 d x^2} \sin (2 \arcsin (c x))+b^2 c x \sqrt {d-c^2 d x^2} \sin (4 \arcsin (c x))-4 b \sqrt {d-c^2 d x^2} \arcsin (c x) \left (128 a \sqrt {1-c^2 x^2}+32 b c x \cos (2 \arcsin (c x))+b c x \cos (4 \arcsin (c x))-128 b c x \log \left (1-e^{2 i \arcsin (c x)}\right )+64 a c x \sin (2 \arcsin (c x))+4 a c x \sin (4 \arcsin (c x))\right )-8 b \sqrt {d-c^2 d x^2} \arcsin (c x)^2 \left (60 a c x+32 i b c x+32 b \sqrt {1-c^2 x^2}+16 b c x \sin (2 \arcsin (c x))+b c x \sin (4 \arcsin (c x))\right )\right )}{256 x \sqrt {1-c^2 x^2}} \]
(d^2*(-256*a^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] - 288*a^2*c^2*x^2*Sqr t[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + 64*a^2*c^4*x^4*Sqrt[1 - c^2*x^2]*Sqrt [d - c^2*d*x^2] - 160*b^2*c*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^3 + 480*a^2* c*Sqrt[d]*x*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(- 1 + c^2*x^2))] - 128*a*b*c*x*Sqrt[d - c^2*d*x^2]*Cos[2*ArcSin[c*x]] - 4*a* b*c*x*Sqrt[d - c^2*d*x^2]*Cos[4*ArcSin[c*x]] + 512*a*b*c*x*Sqrt[d - c^2*d* x^2]*Log[c*x] - (256*I)*b^2*c*x*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^((2*I)*Ar cSin[c*x])] + 64*b^2*c*x*Sqrt[d - c^2*d*x^2]*Sin[2*ArcSin[c*x]] + b^2*c*x* Sqrt[d - c^2*d*x^2]*Sin[4*ArcSin[c*x]] - 4*b*Sqrt[d - c^2*d*x^2]*ArcSin[c* x]*(128*a*Sqrt[1 - c^2*x^2] + 32*b*c*x*Cos[2*ArcSin[c*x]] + b*c*x*Cos[4*Ar cSin[c*x]] - 128*b*c*x*Log[1 - E^((2*I)*ArcSin[c*x])] + 64*a*c*x*Sin[2*Arc Sin[c*x]] + 4*a*c*x*Sin[4*ArcSin[c*x]]) - 8*b*Sqrt[d - c^2*d*x^2]*ArcSin[c *x]^2*(60*a*c*x + (32*I)*b*c*x + 32*b*Sqrt[1 - c^2*x^2] + 16*b*c*x*Sin[2*A rcSin[c*x]] + b*c*x*Sin[4*ArcSin[c*x]])))/(256*x*Sqrt[1 - c^2*x^2])
Time = 2.91 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.11, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.897, Rules used = {5200, 5158, 5156, 5138, 262, 223, 5152, 5182, 211, 211, 223, 5188, 211, 211, 223, 5188, 211, 223, 5136, 3042, 25, 4200, 25, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-5 c^2 d \int \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \int x (a+b \arcsin (c x))dx}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx\right )}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \int \left (1-c^2 x^2\right )^{3/2}dx}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \int \sqrt {1-c^2 x^2}dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 5188 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx-\frac {1}{4} b c \int \left (1-c^2 x^2\right )^{3/2}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx-\frac {1}{4} b c \left (\frac {3}{4} \int \sqrt {1-c^2 x^2}dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 5188 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \arcsin (c x)}{x}dx-\frac {1}{2} b c \int \sqrt {1-c^2 x^2}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \arcsin (c x)}{x}dx-\frac {1}{2} b c \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \arcsin (c x)}{x}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c x}d\arcsin (c x)+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int -\left ((a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )\right )d\arcsin (c x)+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\int (a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
-(((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/x) - 5*c^2*d*((x*(d - c^2* d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/4 + (3*d*((x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/2 + (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c*S qrt[1 - c^2*x^2]) - (b*c*Sqrt[d - c^2*d*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))/Sqrt[1 - c^2*x^2]))/4 - (b*c*d*Sqrt[d - c^2*d*x^2]*(-1/4*((1 - c^2*x^2)^2*(a + b* ArcSin[c*x]))/c^2 + (b*((x*(1 - c^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 - c^2*x^ 2])/2 + ArcSin[c*x]/(2*c)))/4))/(4*c)))/(2*Sqrt[1 - c^2*x^2])) + (2*b*c*d^ 2*Sqrt[d - c^2*d*x^2]*(((1 - c^2*x^2)*(a + b*ArcSin[c*x]))/2 + ((1 - c^2*x ^2)^2*(a + b*ArcSin[c*x]))/4 - ((I/2)*(a + b*ArcSin[c*x])^2)/b - (b*c*((x* Sqrt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/2 - (b*c*((x*(1 - c^2*x^2)^(3/2 ))/4 + (3*((x*Sqrt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/4))/4 - (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)))/Sqrt[1 - c^2*x^2]
3.3.31.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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_.) + 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_.)*((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_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcSin[c*x])^n/Sqrt[ 1 - c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 ]] Int[x*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x ] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c , d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 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_.))*((d_) + (e_.)*(x_)^2)^(p_.))/(x_), x_Symbol] :> Simp[(d + e*x^2)^p*((a + b*ArcSin[c*x])/(2*p)), x] + (Simp[d Int[(d + e*x^2)^(p - 1)*((a + b*ArcSin[c*x])/x), x], x] - Simp[b*c*(d^p/(2 *p)) Int[(1 - c^2*x^2)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
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]
Time = 0.32 (sec) , antiderivative size = 972, normalized size of antiderivative = 1.73
| method | result | size |
| default | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}-a^{2} c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}-\frac {5 \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a^{2} c^{2} d x}{4}-\frac {15 a^{2} d^{2} \sqrt {-c^{2} d \,x^{2}+d}\, c^{2} x}{8}-\frac {15 a^{2} c^{2} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+b^{2} \left (\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} c \,d^{2}}{8 \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (4 i \arcsin \left (c x \right )+8 \arcsin \left (c x \right )^{2}-1\right ) c \,d^{2}}{512 c^{2} x^{2}-512}-\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 (-2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right ) c \,d^{2}}{8 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2} d^{2}}{\left (c^{2} x^{2}-1\right ) x}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c \,d^{2}}{c^{2} x^{2}-1}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (44 i \arcsin \left (c x \right )+40 \arcsin \left (c x \right )^{2}-21\right ) \cos \left (3 \arcsin \left (c x \right )\right ) c \,d^{2}}{512 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (124 i \arcsin \left (c x \right )+136 \arcsin \left (c x \right )^{2}-65\right ) \sin \left (3 \arcsin \left (c x \right )\right ) c \,d^{2}}{512 \left (c^{2} x^{2}-1\right )}\right )+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-32 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{4} c^{4}+8 c^{5} x^{5}+144 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-72 c^{3} x^{3}+120 c x \arcsin \left (c x \right )^{2}+128 i \arcsin \left (c x \right ) x c -128 \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x c +128 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+33 c x \right ) d^{2}}{64 \left (c^{2} x^{2}-1\right ) x}\) | \(972\) |
| parts | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}-a^{2} c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}-\frac {5 \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a^{2} c^{2} d x}{4}-\frac {15 a^{2} d^{2} \sqrt {-c^{2} d \,x^{2}+d}\, c^{2} x}{8}-\frac {15 a^{2} c^{2} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+b^{2} \left (\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} c \,d^{2}}{8 \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (4 i \arcsin \left (c x \right )+8 \arcsin \left (c x \right )^{2}-1\right ) c \,d^{2}}{512 c^{2} x^{2}-512}-\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 (-2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right ) c \,d^{2}}{8 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2} d^{2}}{\left (c^{2} x^{2}-1\right ) x}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c \,d^{2}}{c^{2} x^{2}-1}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (44 i \arcsin \left (c x \right )+40 \arcsin \left (c x \right )^{2}-21\right ) \cos \left (3 \arcsin \left (c x \right )\right ) c \,d^{2}}{512 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (124 i \arcsin \left (c x \right )+136 \arcsin \left (c x \right )^{2}-65\right ) \sin \left (3 \arcsin \left (c x \right )\right ) c \,d^{2}}{512 \left (c^{2} x^{2}-1\right )}\right )+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-32 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{4} c^{4}+8 c^{5} x^{5}+144 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-72 c^{3} x^{3}+120 c x \arcsin \left (c x \right )^{2}+128 i \arcsin \left (c x \right ) x c -128 \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x c +128 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+33 c x \right ) d^{2}}{64 \left (c^{2} x^{2}-1\right ) x}\) | \(972\) |
-a^2/d/x*(-c^2*d*x^2+d)^(7/2)-a^2*c^2*x*(-c^2*d*x^2+d)^(5/2)-5/4*(-c^2*d*x ^2+d)^(3/2)*a^2*c^2*d*x-15/8*a^2*d^2*(-c^2*d*x^2+d)^(1/2)*c^2*x-15/8*a^2*c ^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^2*(5/8 *(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arcsin(c*x)^3*c*d^2 +1/512*(-d*(c^2*x^2-1))^(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8 *I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*(4*I* arcsin(c*x)+8*arcsin(c*x)^2-1)*c*d^2/(c^2*x^2-1)-1/8*(-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)*(- 2*I*arcsin(c*x)+2*arcsin(c*x)^2-1)*c*d^2/(c^2*x^2-1)-(-d*(c^2*x^2-1))^(1/2 )*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*arcsin(c*x)^2*d^2/(c^2*x^2-1)/x+2*I *(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)*(I*arcsin(c*x)*ln(1 +I*c*x+(-c^2*x^2+1)^(1/2))+I*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+ar csin(c*x)^2+polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+polylog(2,I*c*x+(-c^2*x^2 +1)^(1/2)))*c*d^2+3/512*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1) ^(1/2)-I)*(44*I*arcsin(c*x)+40*arcsin(c*x)^2-21)*cos(3*arcsin(c*x))*c*d^2/ (c^2*x^2-1)-1/512*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2 -1)*(124*I*arcsin(c*x)+136*arcsin(c*x)^2-65)*sin(3*arcsin(c*x))*c*d^2/(c^2 *x^2-1))+1/64*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/x* (-32*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^4*c^4+8*c^5*x^5+144*(-c^2*x^2+1)^(1/ 2)*arcsin(c*x)*x^2*c^2-72*c^3*x^3+120*c*x*arcsin(c*x)^2+128*I*arcsin(c*...
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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))*sqrt(-c^2*d*x^2 + d)/x^2, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
-1/8*(10*(-c^2*d*x^2 + d)^(3/2)*c^2*d*x + 15*sqrt(-c^2*d*x^2 + d)*c^2*d^2* x + 15*c*d^(5/2)*arcsin(c*x) + 8*(-c^2*d*x^2 + d)^(5/2)/x)*a^2 + sqrt(d)*i ntegrate(((b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arctan2(c*x, sqr t(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2 + a* b*d^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/x^2, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^2} \,d x \]