Integrand size = 29, antiderivative size = 752 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c (a+b \arcsin (c x))}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \arcsin (c x))^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \arcsin (c x))^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
5/6*c^2*(a+b*arcsin(c*x))^2/d/(-c^2*d*x^2+d)^(3/2)-1/2*(a+b*arcsin(c*x))^2 /d/x^2/(-c^2*d*x^2+d)^(3/2)+1/3*b^2*c^2/d^2/(-c^2*d*x^2+d)^(1/2)+5/2*c^2*( a+b*arcsin(c*x))^2/d^2/(-c^2*d*x^2+d)^(1/2)-b*c*(a+b*arcsin(c*x))/d^2/x/(- c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+2/3*b*c^3*x*(a+b*arcsin(c*x))/d^2/(- c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+26/3*I*b*c^2*(a+b*arcsin(c*x))*arcta n(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-5* c^2*(a+b*arcsin(c*x))^2*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/ 2)/d^2/(-c^2*d*x^2+d)^(1/2)-b^2*c^2*arctanh((-c^2*x^2+1)^(1/2))*(-c^2*x^2+ 1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)+5*I*b*c^2*(a+b*arcsin(c*x))*polylog(2,-I *c*x-(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-13/3* I*b^2*c^2*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/d^2/ (-c^2*d*x^2+d)^(1/2)+13/3*I*b^2*c^2*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)) )*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-5*I*b*c^2*(a+b*arcsin(c*x))* polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^ (1/2)-5*b^2*c^2*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^ 2/(-c^2*d*x^2+d)^(1/2)+5*b^2*c^2*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2 *x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)
Time = 9.81 (sec) , antiderivative size = 1090, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx =\text {Too large to display} \]
Sqrt[-(d*(-1 + c^2*x^2))]*(-1/2*a^2/(d^3*x^2) + (a^2*c^2)/(3*d^3*(-1 + c^2 *x^2)^2) - (2*a^2*c^2)/(d^3*(-1 + c^2*x^2))) + (5*a^2*c^2*Log[x])/(2*d^(5/ 2)) - (5*a^2*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(5/2)) + (a*b*c^2*Sqrt[1 - c^2*x^2]*((-2*(-1 + ArcSin[c*x]))/(-1 + c*x) + 52*ArcSi n[c*x] - 6*Cot[ArcSin[c*x]/2] - 3*ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 + 60*Ar cSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + 52*L og[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 52*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + (60*I)*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 3*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 + (4*ArcSin[c*x]* Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (52*ArcS in[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - (4 *ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) ^3 + (2*(1 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - ( 52*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2 ]) - 6*Tan[ArcSin[c*x]/2]))/(12*d^2*Sqrt[d*(1 - c^2*x^2)]) + (b^2*c^2*Sqrt [1 - c^2*x^2]*(8 - (2*(-2 + ArcSin[c*x])*ArcSin[c*x])/(-1 + c*x) + 52*ArcS in[c*x]^2 - 12*ArcSin[c*x]*Cot[ArcSin[c*x]/2] - 3*ArcSin[c*x]^2*Csc[ArcSin [c*x]/2]^2 + 24*Log[Tan[ArcSin[c*x]/2]] - 104*(ArcSin[c*x]*(Log[1 - I*E^(I *ArcSin[c*x])] - Log[1 + I*E^(I*ArcSin[c*x])]) + I*(PolyLog[2, (-I)*E^(I*A rcSin[c*x])] - PolyLog[2, I*E^(I*ArcSin[c*x])])) + 60*(ArcSin[c*x]^2*(L...
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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx+b c \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}dx^2-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx+\frac {1}{2} b c \left (\int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx^2+\frac {2}{\sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-\frac {2 \int \frac {1}{\frac {1}{c^2}-\frac {x^4}{c^2}}d\sqrt {1-c^2 x^2}}{c^2}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx-\frac {1}{2} b c \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {\int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {5}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle \frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle \frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx-\frac {1}{2} b c \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle \frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {\int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {-\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {-\frac {2 b \sqrt {1-c^2 x^2} \left (-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {-\frac {2 b \sqrt {1-c^2 x^2} \left (i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {1-c^2 x^2} \int \frac {(a+b \arcsin (c x))^2}{c x}d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {1-c^2 x^2} \int (a+b \arcsin (c x))^2 \csc (\arcsin (c x))d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {1-c^2 x^2} \left (-2 b \int (a+b \arcsin (c x)) \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
3.3.62.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1 ))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b*Ar cSin[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] && LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[1/(c*d) Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) ) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp[b*c *(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)* (1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b , c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && !G tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* x^2]] Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a , b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.42 (sec) , antiderivative size = 1121, normalized size of antiderivative = 1.49
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1121\) |
| parts | \(\text {Expression too large to display}\) | \(1121\) |
-1/2*a^2/d/x^2/(-c^2*d*x^2+d)^(3/2)+5/6*a^2*c^2/d/(-c^2*d*x^2+d)^(3/2)+5/2 *a^2*c^2/d^2/(-c^2*d*x^2+d)^(1/2)-5/2*a^2*c^2/d^(5/2)*ln((2*d+2*d^(1/2)*(- c^2*d*x^2+d)^(1/2))/x)+b^2*(-1/6*(-d*(c^2*x^2-1))^(1/2)*(15*arcsin(c*x)^2* x^4*c^4-4*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^3*x^3+2*c^4*x^4-20*arcsin(c*x)^ 2*x^2*c^2+6*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x*c-2*c^2*x^2+3*arcsin(c*x)^2)/ d^3/(c^4*x^4-2*c^2*x^2+1)/x^2+1/6*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2 )*(15*arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-15*arcsin(c*x)^2*ln(1-I *c*x-(-c^2*x^2+1)^(1/2))-30*I*arcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1 /2))+30*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-26*arcsin(c*x)*l n(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+26*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1 )^(1/2)))+26*I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-26*I*dilog(1-I*(I*c*x +(-c^2*x^2+1)^(1/2)))-6*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)-30*polylog(3,I*c*x+ (-c^2*x^2+1)^(1/2))+30*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+6*ln(1+I*c*x+( -c^2*x^2+1)^(1/2)))*c^2/d^3/(c^2*x^2-1))-1/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)* (-c^2*x^2+1)^(1/2)*(-5*I*x^3*c^3+15*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^6* x^6+15*dilog(I*c*x+(-c^2*x^2+1)^(1/2))*c^6*x^6+26*arctan(I*c*x+(-c^2*x^2+1 )^(1/2))*c^6*x^6+3*I*(-c^2*x^2+1)^(1/2)*arcsin(c*x)+15*I*arcsin(c*x)*ln(1+ I*c*x+(-c^2*x^2+1)^(1/2))*x^2*c^2-20*I*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2* c^2-30*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4-30*dilog(I*c*x+(-c^2*x^2+ 1)^(1/2))*c^4*x^4-52*arctan(I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4+2*I*x^5*c...
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^ 2)/(c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 - d^3*x^3), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
-1/6*a^2*(15*c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d ^(5/2) - 15*c^2/(sqrt(-c^2*d*x^2 + d)*d^2) - 5*c^2/((-c^2*d*x^2 + d)^(3/2) *d) + 3/((-c^2*d*x^2 + d)^(3/2)*d*x^2)) - sqrt(d)*integrate((b^2*arctan2(c *x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*arctan2(c*x, sqrt(c*x + 1)*sqr t(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 - d^3*x^3), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]