Integrand size = 29, antiderivative size = 591 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=-\frac {7}{12} b^2 c^4 d^2 x \sqrt {d-c^2 d x^2}-\frac {b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{3 x}+\frac {23 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{12 \sqrt {1-c^2 x^2}}-\frac {5 b c^5 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}-\frac {7}{3} b c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {7 i c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 \sqrt {1-c^2 x^2}}+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b \sqrt {1-c^2 x^2}}-\frac {14 b c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {7 i b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{3 \sqrt {1-c^2 x^2}} \]
5/3*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x-1/3*(-c^2*d*x^2+d)^(5 /2)*(a+b*arcsin(c*x))^2/x^3-7/12*b^2*c^4*d^2*x*(-c^2*d*x^2+d)^(1/2)-1/3*b^ 2*c^2*d^2*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/x-1/3*b*c*d^2*(-c^2*x^2+1)^(3/ 2)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/x^2+5/2*c^4*d^2*x*(a+b*arcsin(c* x))^2*(-c^2*d*x^2+d)^(1/2)+23/12*b^2*c^3*d^2*arcsin(c*x)*(-c^2*d*x^2+d)^(1 /2)/(-c^2*x^2+1)^(1/2)-5/2*b*c^5*d^2*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^ (1/2)/(-c^2*x^2+1)^(1/2)+7/3*I*c^3*d^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^ (1/2)/(-c^2*x^2+1)^(1/2)+5/6*c^3*d^2*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1 /2)/b/(-c^2*x^2+1)^(1/2)-14/3*b*c^3*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)+7/3*I*b^2*c^3*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)-7/3*b*c^3*d^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d) ^(1/2)
Time = 3.11 (sec) , antiderivative size = 690, normalized size of antiderivative = 1.17 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\frac {d^2 \left (-4 a b c x \sqrt {d-c^2 d x^2}+3 a b c^3 x^3 \sqrt {d-c^2 d x^2}-6 a b c^5 x^5 \sqrt {d-c^2 d x^2}-4 a^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}+28 a^2 c^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}-4 b^2 c^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}+6 a^2 c^4 x^4 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}-3 b^2 c^4 x^4 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}+10 b^2 c^3 x^3 \sqrt {d-c^2 d x^2} \arcsin (c x)^3-30 a^2 c^3 \sqrt {d} x^3 \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-56 a b c^3 x^3 \sqrt {d-c^2 d x^2} \log (c x)+28 i b^2 c^3 x^3 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+b \sqrt {d-c^2 d x^2} \arcsin (c x) \left (-4 b c x-6 a \sqrt {1-c^2 x^2}+48 a c^2 x^2 \sqrt {1-c^2 x^2}+3 b c^3 x^3 \cos (2 \arcsin (c x))-2 a \cos (3 \arcsin (c x))-56 b c^3 x^3 \log \left (1-e^{2 i \arcsin (c x)}\right )+6 a c^3 x^3 \sin (2 \arcsin (c x))\right )+b \sqrt {d-c^2 d x^2} \arcsin (c x)^2 \left (30 a c^3 x^3+4 b \left (7 i c^3 x^3-\sqrt {1-c^2 x^2}+7 c^2 x^2 \sqrt {1-c^2 x^2}\right )+3 b c^3 x^3 \sin (2 \arcsin (c x))\right )\right )}{12 x^3 \sqrt {1-c^2 x^2}} \]
(d^2*(-4*a*b*c*x*Sqrt[d - c^2*d*x^2] + 3*a*b*c^3*x^3*Sqrt[d - c^2*d*x^2] - 6*a*b*c^5*x^5*Sqrt[d - c^2*d*x^2] - 4*a^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2* d*x^2] + 28*a^2*c^2*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] - 4*b^2*c^2* x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + 6*a^2*c^4*x^4*Sqrt[1 - c^2*x^2 ]*Sqrt[d - c^2*d*x^2] - 3*b^2*c^4*x^4*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2 ] + 10*b^2*c^3*x^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^3 - 30*a^2*c^3*Sqrt[d]* 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))] - 56*a*b*c^3*x^3*Sqrt[d - c^2*d*x^2]*Log[c*x] + (28*I)*b^2*c^3*x^3* Sqrt[d - c^2*d*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])] + b*Sqrt[d - c^2*d*x ^2]*ArcSin[c*x]*(-4*b*c*x - 6*a*Sqrt[1 - c^2*x^2] + 48*a*c^2*x^2*Sqrt[1 - c^2*x^2] + 3*b*c^3*x^3*Cos[2*ArcSin[c*x]] - 2*a*Cos[3*ArcSin[c*x]] - 56*b* c^3*x^3*Log[1 - E^((2*I)*ArcSin[c*x])] + 6*a*c^3*x^3*Sin[2*ArcSin[c*x]]) + b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*(30*a*c^3*x^3 + 4*b*((7*I)*c^3*x^3 - Sqrt[1 - c^2*x^2] + 7*c^2*x^2*Sqrt[1 - c^2*x^2]) + 3*b*c^3*x^3*Sin[2*ArcSi n[c*x]])))/(12*x^3*Sqrt[1 - c^2*x^2])
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{x^3}dx}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5190 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx+\frac {1}{2} b c \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2}dx-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx+\frac {1}{2} b c \left (-3 c^2 \int \sqrt {1-c^2 x^2}dx-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx+\frac {1}{2} b c \left (-3 c^2 \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5188 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\int \frac {a+b \arcsin (c x)}{x}dx-\frac {1}{2} b c \int \sqrt {1-c^2 x^2}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\int \frac {a+b \arcsin (c x)}{x}dx-\frac {1}{2} b c \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\int \frac {a+b \arcsin (c x)}{x}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c x}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\int -\left ((a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )\right )d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (-\int (a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \left (-\frac {b c \sqrt {d-c^2 d x^2} \int x (a+b \arcsin (c x))dx}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx\right )}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-3 c^2 d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5188 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \arcsin (c x)}{x}dx-\frac {1}{2} b c \int \sqrt {1-c^2 x^2}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \arcsin (c x)}{x}dx-\frac {1}{2} b c \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \arcsin (c x)}{x}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c x}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\int -\left ((a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )\right )d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\int (a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
3.3.33.3.1 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*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[((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_.))*((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_.))*((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_)*((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. 2649 vs. \(2 (541 ) = 1082\).
Time = 0.35 (sec) , antiderivative size = 2650, normalized size of antiderivative = 4.48
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2650\) |
| parts | \(\text {Expression too large to display}\) | \(2650\) |
-35*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^2/(c^2*x^ 2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2*c^5+7/3*I*b^2*(-d*(c^2*x^2-1))^(1/2) *d^2/(63*c^4*x^4-15*c^2*x^2+1)*x/(c^2*x^2-1)*(-c^2*x^2+1)*arcsin(c*x)*c^4+ 147*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^4/(c^2*x^ 2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2*c^7-49/3*I*b^2*(-d*(c^2*x^2-1))^(1/2 )*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*arcsin(c*x)*c ^6-49/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^5/(c^ 2*x^2-1)*arcsin(c*x)*c^8-21*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-1 5*c^2*x^2+1)*x^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^7+56/3*I*b^2*(-d*(c^2*x^ 2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)*c^6+ 5*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^2/(c^2*x^2- 1)*(-c^2*x^2+1)^(1/2)*c^5+7/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4 -15*c^2*x^2+1)/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2*c^3+21*b^2*(-d *(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*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^2/(63*c^4*x^4- 15*c^2*x^2+1)/x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c-7/3*I*b^2*( -d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*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)^(7/2)+4/3*a^2*c^4*x*(-c^2*d*x^2+d)^(5 /2)-56/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^5/(c^2 *x^2-1)*c^8+71/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+...
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsin(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b *c^2*d^2*x^2 + a*b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)/x^4, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
1/6*(10*(-c^2*d*x^2 + d)^(3/2)*c^4*d*x + 15*sqrt(-c^2*d*x^2 + d)*c^4*d^2*x + 15*c^3*d^(5/2)*arcsin(c*x) + 8*(-c^2*d*x^2 + d)^(5/2)*c^2/x - 2*(-c^2*d *x^2 + d)^(7/2)/(d*x^3))*a^2 + sqrt(d)*integrate(((b^2*c^4*d^2*x^4 - 2*b^2 *c^2*d^2*x^2 + b^2*d^2)*arctan2(c*x, sqrt(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^4, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^4} \,d x \]