Integrand size = 29, antiderivative size = 752 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c^3 x (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d^2 x \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \arcsin (c x))^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \arcsin (c x))^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \] Output:
1/3*b^2*c^2/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*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)-b*c*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c* x))/d^2/x/(-c^2*d*x^2+d)^(1/2)+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)+5/2*c^2*(a+b*ar csin(c*x))^2/d^2/(-c^2*d*x^2+d)^(1/2)+26/3*I*b*c^2*(-c^2*x^2+1)^(1/2)*(a+b *arcsin(c*x))*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/d^2/(-c^2*d*x^2+d)^(1/2)-5* c^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2*arctanh(I*c*x+(-c^2*x^2+1)^(1/2 ))/d^2/(-c^2*d*x^2+d)^(1/2)-b^2*c^2*(-c^2*x^2+1)^(1/2)*arctanh((-c^2*x^2+1 )^(1/2))/d^2/(-c^2*d*x^2+d)^(1/2)+5*I*b*c^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin (c*x))*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))/d^2/(-c^2*d*x^2+d)^(1/2)-13/3* I*b^2*c^2*(-c^2*x^2+1)^(1/2)*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^2/ (-c^2*d*x^2+d)^(1/2)+13/3*I*b^2*c^2*(-c^2*x^2+1)^(1/2)*polylog(2,I*(I*c*x+ (-c^2*x^2+1)^(1/2)))/d^2/(-c^2*d*x^2+d)^(1/2)-5*I*b*c^2*(-c^2*x^2+1)^(1/2) *(a+b*arcsin(c*x))*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))/d^2/(-c^2*d*x^2+d)^ (1/2)-5*b^2*c^2*(-c^2*x^2+1)^(1/2)*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))/d^ 2/(-c^2*d*x^2+d)^(1/2)+5*b^2*c^2*(-c^2*x^2+1)^(1/2)*polylog(3,I*c*x+(-c^2* x^2+1)^(1/2))/d^2/(-c^2*d*x^2+d)^(1/2)
Time = 9.73 (sec) , antiderivative size = 1090, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^(5/2)),x]
Output:
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 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {1-c^2 x^2} \left (-2 b \int (a+b \arcsin (c x)) \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}-\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
Input:
Int[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^(5/2)),x]
Output:
$Aborted
Time = 1.08 (sec) , antiderivative size = 1121, normalized size of antiderivative = 1.49
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1121\) |
| parts | \(\text {Expression too large to display}\) | \(1121\) |
Input:
int((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
-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)*x^3*c^3+2*c^4*x^4-20*arcsin(c*x)^ 2*x^2*c^2+6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x-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 )/d^3/(c^2*x^2-1)*(15*arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-15*arcs in(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)*ln(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(1+I*c*x+(-c^2*x^2+1)^(1/2))+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(I*c*x+(-c^2*x^2+1)^(1/2)-1))*c^2)-1/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)* (-c^2*x^2+1)^(1/2)*(-30*I*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))*arcsin(c*x)*x^4*c ^4+15*dilog(I*c*x+(-c^2*x^2+1)^(1/2))*c^6*x^6+15*dilog(1+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^6-5*I*x^3*c^3+2 *I*x^5*c^5+15*I*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))*arcsin(c*x)*x^2*c^2-30*dilo g(I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4-30*dilog(1+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+3*I*(-c^2*x^2+1)^(1/...
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="frica s")
Output:
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)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((a+b*asin(c*x))**2/x**3/(-c**2*d*x**2+d)**(5/2),x)
Output:
Integral((a + b*asin(c*x))**2/(x**3*(-d*(c*x - 1)*(c*x + 1))**(5/2)), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxim a")
Output:
-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)
Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac" )
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:
int((a + b*asin(c*x))^2/(x^3*(d - c^2*d*x^2)^(5/2)),x)
Output:
int((a + b*asin(c*x))^2/(x^3*(d - c^2*d*x^2)^(5/2)), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {48 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{7}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {-c^{2} x^{2}+1}\, x^{3}}d x \right ) a b \,c^{2} x^{4}-48 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{7}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {-c^{2} x^{2}+1}\, x^{3}}d x \right ) a b \,x^{2}+24 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{7}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {-c^{2} x^{2}+1}\, x^{3}}d x \right ) b^{2} c^{2} x^{4}-24 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{7}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {-c^{2} x^{2}+1}\, x^{3}}d x \right ) b^{2} x^{2}+60 \sqrt {-c^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a^{2} c^{4} x^{4}-60 \sqrt {-c^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a^{2} c^{2} x^{2}-65 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}+65 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}+60 a^{2} c^{4} x^{4}-80 a^{2} c^{2} x^{2}+12 a^{2}}{24 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} x^{2} \left (c^{2} x^{2}-1\right )} \] Input:
int((a+b*asin(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),x)
Output:
(48*sqrt( - c**2*x**2 + 1)*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*c**4*x**7 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**5 + sqrt( - c**2*x**2 + 1)*x**3),x)*a* b*c**2*x**4 - 48*sqrt( - c**2*x**2 + 1)*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*c**4*x**7 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**5 + sqrt( - c**2*x**2 + 1 )*x**3),x)*a*b*x**2 + 24*sqrt( - c**2*x**2 + 1)*int(asin(c*x)**2/(sqrt( - c**2*x**2 + 1)*c**4*x**7 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**5 + sqrt( - c* *2*x**2 + 1)*x**3),x)*b**2*c**2*x**4 - 24*sqrt( - c**2*x**2 + 1)*int(asin( c*x)**2/(sqrt( - c**2*x**2 + 1)*c**4*x**7 - 2*sqrt( - c**2*x**2 + 1)*c**2* x**5 + sqrt( - c**2*x**2 + 1)*x**3),x)*b**2*x**2 + 60*sqrt( - c**2*x**2 + 1)*log(tan(asin(c*x)/2))*a**2*c**4*x**4 - 60*sqrt( - c**2*x**2 + 1)*log(ta n(asin(c*x)/2))*a**2*c**2*x**2 - 65*sqrt( - c**2*x**2 + 1)*a**2*c**4*x**4 + 65*sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 + 60*a**2*c**4*x**4 - 80*a**2*c **2*x**2 + 12*a**2)/(24*sqrt(d)*sqrt( - c**2*x**2 + 1)*d**2*x**2*(c**2*x** 2 - 1))