Integrand size = 29, antiderivative size = 579 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=-\frac {7}{12} b^2 c^4 d^2 x \sqrt {d-c^2 d x^2}-\frac {b^2 c^2 d \left (d-c^2 d x^2\right )^{3/2}}{3 x}+\frac {23 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{12 \sqrt {1-c^2 x^2}}-\frac {5 b c^5 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}-\frac {7}{3} b c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {7 i c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 \sqrt {1-c^2 x^2}}+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b \sqrt {1-c^2 x^2}}-\frac {14 b c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {7 i b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{3 \sqrt {1-c^2 x^2}} \] Output:
-7/12*b^2*c^4*d^2*x*(-c^2*d*x^2+d)^(1/2)-1/3*b^2*c^2*d*(-c^2*d*x^2+d)^(3/2 )/x+23/12*b^2*c^3*d^2*(-c^2*d*x^2+d)^(1/2)*arcsin(c*x)/(-c^2*x^2+1)^(1/2)- 5/2*b*c^5*d^2*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2 )-7/3*b*c^3*d^2*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))- 1/3*b*c*d^2*(-c^2*x^2+1)^(3/2)*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^2+ 5/2*c^4*d^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2+7/3*I*c^3*d^2*(-c^2 *d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/(-c^2*x^2+1)^(1/2)+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+5/6*c^3*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^3/b/(-c^2*x^2+ 1)^(1/2)-14/3*b*c^3*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))*ln(1-(I*c*x +(-c^2*x^2+1)^(1/2))^2)/(-c^2*x^2+1)^(1/2)+7/3*I*b^2*c^3*d^2*(-c^2*d*x^2+d )^(1/2)*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/(-c^2*x^2+1)^(1/2)
Time = 2.63 (sec) , antiderivative size = 690, normalized size of antiderivative = 1.19 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\frac {d^2 \left (-4 a b c x \sqrt {d-c^2 d x^2}+3 a b c^3 x^3 \sqrt {d-c^2 d x^2}-6 a b c^5 x^5 \sqrt {d-c^2 d x^2}-4 a^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}+28 a^2 c^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}-4 b^2 c^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}+6 a^2 c^4 x^4 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}-3 b^2 c^4 x^4 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}+10 b^2 c^3 x^3 \sqrt {d-c^2 d x^2} \arcsin (c x)^3-30 a^2 c^3 \sqrt {d} x^3 \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-56 a b c^3 x^3 \sqrt {d-c^2 d x^2} \log (c x)+28 i b^2 c^3 x^3 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+b \sqrt {d-c^2 d x^2} \arcsin (c x) \left (-4 b c x-6 a \sqrt {1-c^2 x^2}+48 a c^2 x^2 \sqrt {1-c^2 x^2}+3 b c^3 x^3 \cos (2 \arcsin (c x))-2 a \cos (3 \arcsin (c x))-56 b c^3 x^3 \log \left (1-e^{2 i \arcsin (c x)}\right )+6 a c^3 x^3 \sin (2 \arcsin (c x))\right )+b \sqrt {d-c^2 d x^2} \arcsin (c x)^2 \left (30 a c^3 x^3+4 b \left (7 i c^3 x^3-\sqrt {1-c^2 x^2}+7 c^2 x^2 \sqrt {1-c^2 x^2}\right )+3 b c^3 x^3 \sin (2 \arcsin (c x))\right )\right )}{12 x^3 \sqrt {1-c^2 x^2}} \] Input:
Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/x^4,x]
Output:
(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 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (-3 c^2 \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {\left (1-c^2 x^2\right )^{3/2}}{x}\right )\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 x^3}\) |
Input:
Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/x^4,x]
Output:
$Aborted
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2649 vs. \(2 (529 ) = 1058\).
Time = 0.92 (sec) , antiderivative size = 2650, normalized size of antiderivative = 4.58
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2650\) |
| parts | \(\text {Expression too large to display}\) | \(2650\) |
Input:
int((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/x^4,x,method=_RETURNVERBOSE)
Output:
-1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^6/(c^2*x^2-1)*x^3+1/4*b^2*(-d*(c^2*x ^2-1))^(1/2)*d^2*c^4/(c^2*x^2-1)*x-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)+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)*x/(c^2*x^2-1)*arcsin(c*x)*c^ 4-49/3*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+4/3*a^2*c^2/d/x*(-c^2*d*x^2+d)^(7/2)-5*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)*arcsin(c*x)*(-c^ 2*x^2+1)^(1/2)*c^3+14*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*d^2*c^ 3/(3*c^2*x^2-3)*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-14*I*b^2*(-c^2* x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*d^2*c^3/(3*c^2*x^2-3)*arcsin(c*x)^2-14 *I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*d^2*c^3/(3*c^2*x^2-3)*pol ylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-14*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2 -1))^(1/2)*d^2*c^3/(3*c^2*x^2-3)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-1/3*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)*c^3+14*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*d^2*c^ 3/(3*c^2*x^2-3)*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+147*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)^ 2*c^8-203*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)^2*c^6+190/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4...
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/x^4,x, algorithm="frica s")
Output:
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)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \] Input:
integrate((-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**2/x**4,x)
Output:
Integral((-d*(c*x - 1)*(c*x + 1))**(5/2)*(a + b*asin(c*x))**2/x**4, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/x^4,x, algorithm="maxim a")
Output:
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/x^4,x, algorithm="giac" )
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^4} \,d x \] Input:
int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2))/x^4,x)
Output:
int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2))/x^4, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^4} \, dx=\frac {\sqrt {d}\, d^{2} \left (4 \mathit {asin} \left (c x \right )^{3} b^{2} c^{3} x^{3}+12 \mathit {asin} \left (c x \right )^{2} a b \,c^{3} x^{3}+15 \mathit {asin} \left (c x \right ) a^{2} c^{3} x^{3}+3 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}+14 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a^{2}-24 \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) a b \,c^{2} x^{3}-12 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) b^{2} c^{2} x^{3}+12 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{x^{4}}d x \right ) a b \,x^{3}+6 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2}}{x^{4}}d x \right ) b^{2} x^{3}+12 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )d x \right ) a b \,c^{4} x^{3}+6 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2}d x \right ) b^{2} c^{4} x^{3}\right )}{6 x^{3}} \] Input:
int((-c^2*d*x^2+d)^(5/2)*(a+b*asin(c*x))^2/x^4,x)
Output:
(sqrt(d)*d**2*(4*asin(c*x)**3*b**2*c**3*x**3 + 12*asin(c*x)**2*a*b*c**3*x* *3 + 15*asin(c*x)*a**2*c**3*x**3 + 3*sqrt( - c**2*x**2 + 1)*a**2*c**4*x**4 + 14*sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - 2*sqrt( - c**2*x**2 + 1)*a** 2 - 24*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*x**2),x)*a*b*c**2*x**3 - 12*i nt(asin(c*x)**2/(sqrt( - c**2*x**2 + 1)*x**2),x)*b**2*c**2*x**3 + 12*int(( sqrt( - c**2*x**2 + 1)*asin(c*x))/x**4,x)*a*b*x**3 + 6*int((sqrt( - c**2*x **2 + 1)*asin(c*x)**2)/x**4,x)*b**2*x**3 + 12*int(sqrt( - c**2*x**2 + 1)*a sin(c*x),x)*a*b*c**4*x**3 + 6*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2,x)*b **2*c**4*x**3))/(6*x**3)