Integrand size = 29, antiderivative size = 634 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}} \] Output:
-b*c*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/d/x/(-c^2*d*x^2+d)^(1/2)+3/2*c^2 *(a+b*arcsin(c*x))^2/d/(-c^2*d*x^2+d)^(1/2)-1/2*(a+b*arcsin(c*x))^2/d/x^2/ (-c^2*d*x^2+d)^(1/2)+4*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/(-c^2*d*x^2+d)^(1/2)-3*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/(-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/(-c^2*d*x^2+d )^(1/2)+3*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/(-c^2*d*x^2+d)^(1/2)-2*I*b^2*c^2*(-c^2*x^2+1)^(1/2)*po lylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d/(-c^2*d*x^2+d)^(1/2)+2*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/(-c^2*d*x^2+d )^(1/2)-3*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/(-c^2*d*x^2+d)^(1/2)-3*b^2*c^2*(-c^2*x^2+1)^(1/2)*polyl og(3,-I*c*x-(-c^2*x^2+1)^(1/2))/d/(-c^2*d*x^2+d)^(1/2)+3*b^2*c^2*(-c^2*x^2 +1)^(1/2)*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))/d/(-c^2*d*x^2+d)^(1/2)
Time = 7.75 (sec) , antiderivative size = 844, normalized size of antiderivative = 1.33 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^(3/2)),x]
Output:
Sqrt[-(d*(-1 + c^2*x^2))]*(-1/2*a^2/(d^2*x^2) - (a^2*c^2)/(d^2*(-1 + c^2*x ^2))) + (3*a^2*c^2*Log[x])/(2*d^(3/2)) - (3*a^2*c^2*Log[d + Sqrt[d]*Sqrt[- (d*(-1 + c^2*x^2))]])/(2*d^(3/2)) + (a*b*c*((6*I)*PolyLog[2, -E^(I*ArcSin[ c*x])]*Sin[2*ArcSin[c*x]] - (6*I)*PolyLog[2, E^(I*ArcSin[c*x])]*Sin[2*ArcS in[c*x]] - (-2*ArcSin[c*x] + 6*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + 3*ArcSin[c *x]*Cos[3*ArcSin[c*x]]*Log[1 - E^(I*ArcSin[c*x])] - 3*ArcSin[c*x]*Cos[3*Ar cSin[c*x]]*Log[1 + E^(I*ArcSin[c*x])] + 2*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSi n[c*x]/2] - Sin[ArcSin[c*x]/2]] - 2*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x] /2] + Sin[ArcSin[c*x]/2]] + Sqrt[1 - c^2*x^2]*(-3*ArcSin[c*x]*Log[1 - E^(I *ArcSin[c*x])] + 3*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - 2*Log[Cos[ArcS in[c*x]/2] - Sin[ArcSin[c*x]/2]] + 2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c *x]/2]]) + 2*Sin[2*ArcSin[c*x]])/(c*x)))/(4*d*x*Sqrt[d*(1 - c^2*x^2)]) + ( b^2*c^2*Sqrt[1 - c^2*x^2]*(8*ArcSin[c*x]^2 - 4*ArcSin[c*x]*Cot[ArcSin[c*x] /2] - ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^2 + 8*Log[Tan[ArcSin[c*x]/2]] - 16* (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*ArcSin[c*x])] - PolyLog[2, I*E^(I*ArcSin[c*x])] )) + 12*(ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c *x])]) + (2*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^ (I*ArcSin[c*x])]) + 2*(-PolyLog[3, -E^(I*ArcSin[c*x])] + PolyLog[3, E^(I*A rcSin[c*x])])) + ArcSin[c*x]^2*Sec[ArcSin[c*x]/2]^2 + (8*ArcSin[c*x]^2*...
Time = 4.57 (sec) , antiderivative size = 457, normalized size of antiderivative = 0.72, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.759, Rules used = {5204, 5204, 243, 73, 221, 5164, 3042, 4669, 2715, 2838, 5208, 5164, 3042, 4669, 2715, 2838, 5218, 3042, 4671, 3011, 2720, 7143}
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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {3}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx+\frac {b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {3}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx+\frac {b c \sqrt {1-c^2 x^2} \left (c^2 \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx+b c \int \frac {1}{x \sqrt {1-c^2 x^2}}dx-\frac {a+b \arcsin (c x)}{x}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {3}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx+\frac {b c \sqrt {1-c^2 x^2} \left (c^2 \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx^2-\frac {a+b \arcsin (c x)}{x}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {3}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx+\frac {b c \sqrt {1-c^2 x^2} \left (c^2 \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx-\frac {b \int \frac {1}{\frac {1}{c^2}-\frac {x^4}{c^2}}d\sqrt {1-c^2 x^2}}{c}-\frac {a+b \arcsin (c x)}{x}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (c^2 \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (c \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (c \int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (c \left (-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {b c \sqrt {1-c^2 x^2} \left (c \left (i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {3}{2} c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx+\frac {b c \sqrt {1-c^2 x^2} \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle \frac {3}{2} c^2 \left (-\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle \frac {3}{2} c^2 \left (-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {3}{2} c^2 \left (-\frac {2 b \sqrt {1-c^2 x^2} \left (-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {3}{2} c^2 \left (-\frac {2 b \sqrt {1-c^2 x^2} \left (i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {1-c^2 x^2} \int \frac {(a+b \arcsin (c x))^2}{c x}d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {1-c^2 x^2} \int (a+b \arcsin (c x))^2 \csc (\arcsin (c x))d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {1-c^2 x^2} \left (-2 b \int (a+b \arcsin (c x)) \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {1-c^2 x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {1-c^2 x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3}{2} c^2 \left (-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )+\frac {b c \sqrt {1-c^2 x^2} \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
Input:
Int[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^(3/2)),x]
Output:
-1/2*(a + b*ArcSin[c*x])^2/(d*x^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[1 - c^2 *x^2]*(-((a + b*ArcSin[c*x])/x) - b*c*ArcTanh[Sqrt[1 - c^2*x^2]] + c*((-2* I)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])] + I*b*PolyLog[2, (-I)*E^( I*ArcSin[c*x])] - I*b*PolyLog[2, I*E^(I*ArcSin[c*x])])))/(d*Sqrt[d - c^2*d *x^2]) + (3*c^2*((a + b*ArcSin[c*x])^2/(d*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt [1 - c^2*x^2]*((-2*I)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])] + I*b* PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - I*b*PolyLog[2, I*E^(I*ArcSin[c*x])])) /(d*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(-2*(a + b*ArcSin[c*x])^2*Ar cTanh[E^(I*ArcSin[c*x])] + 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*Arc Sin[c*x])] - b*PolyLog[3, -E^(I*ArcSin[c*x])]) - 2*b*(I*(a + b*ArcSin[c*x] )*PolyLog[2, E^(I*ArcSin[c*x])] - b*PolyLog[3, E^(I*ArcSin[c*x])])))/(d*Sq rt[d - c^2*d*x^2])))/2
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_)^(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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
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), 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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 0.96 (sec) , antiderivative size = 869, normalized size of antiderivative = 1.37
| method | result | size |
| default | \(a^{2} \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+4 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+4 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-4 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}\right )-\frac {i a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}-3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}+i x^{3} c^{3}-4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )-i c x \right )}{d^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}\) | \(869\) |
| parts | \(a^{2} \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+4 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+4 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-4 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}\right )-\frac {i a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}-3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}+i x^{3} c^{3}-4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )-i c x \right )}{d^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}\) | \(869\) |
Input:
int((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
Output:
a^2*(-1/2/d/x^2/(-c^2*d*x^2+d)^(1/2)+3/2*c^2*(1/d/(-c^2*d*x^2+d)^(1/2)-1/d ^(3/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)))+b^2*(-1/2*(-d*(c^2*x^2 -1))^(1/2)/d^2/(c^2*x^2-1)/x^2*arcsin(c*x)*(3*c^2*x^2*arcsin(c*x)-2*c*x*(- c^2*x^2+1)^(1/2)-arcsin(c*x))+1/2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2 )/(c^2*x^2-1)/d^2*(3*arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-3*arcsin (c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-6*I*arcsin(c*x)*polylog(2,-I*c*x-(- c^2*x^2+1)^(1/2))+6*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-4*ar csin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+4*arcsin(c*x)*ln(1-I*(I*c*x+( -c^2*x^2+1)^(1/2)))+4*I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-4*I*dilog(1- I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+6*polylog(3 ,-I*c*x-(-c^2*x^2+1)^(1/2))-6*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))-2*ln(I*c *x+(-c^2*x^2+1)^(1/2)-1))*c^2)-I*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^( 1/2)*(3*I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))*x^4*c^4+4*arctan(I*c* x+(-c^2*x^2+1)^(1/2))*c^4*x^4+3*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4+ 3*dilog(I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4+3*I*(-c^2*x^2+1)^(1/2)*arcsin(c* x)*c^2*x^2-3*I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))*x^2*c^2+I*x^3*c^ 3-4*arctan(I*c*x+(-c^2*x^2+1)^(1/2))*c^2*x^2-3*dilog(1+I*c*x+(-c^2*x^2+1)^ (1/2))*c^2*x^2-3*dilog(I*c*x+(-c^2*x^2+1)^(1/2))*c^2*x^2-I*(-c^2*x^2+1)^(1 /2)*arcsin(c*x)-I*c*x)/d^2/(c^4*x^4-2*c^2*x^2+1)/x^2
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^(3/2),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^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*asin(c*x))**2/x**3/(-c**2*d*x**2+d)**(3/2),x)
Output:
Integral((a + b*asin(c*x))**2/(x**3*(-d*(c*x - 1)*(c*x + 1))**(3/2)), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxim a")
Output:
-1/2*(3*c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(3/2 ) - 3*c^2/(sqrt(-c^2*d*x^2 + d)*d) + 1/(sqrt(-c^2*d*x^2 + d)*d*x^2))*a^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)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1 )/(c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3), x)
Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^(3/2),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. \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \] Input:
int((a + b*asin(c*x))^2/(x^3*(d - c^2*d*x^2)^(3/2)),x)
Output:
int((a + b*asin(c*x))^2/(x^3*(d - c^2*d*x^2)^(3/2)), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-16 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{5}-\sqrt {-c^{2} x^{2}+1}\, x^{3}}d x \right ) a b \,x^{2}-8 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{5}-\sqrt {-c^{2} x^{2}+1}\, x^{3}}d x \right ) b^{2} x^{2}+12 \sqrt {-c^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a^{2} c^{2} x^{2}-9 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}+12 a^{2} c^{2} x^{2}-4 a^{2}}{8 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d \,x^{2}} \] Input:
int((a+b*asin(c*x))^2/x^3/(-c^2*d*x^2+d)^(3/2),x)
Output:
( - 16*sqrt( - c**2*x**2 + 1)*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*x **5 - sqrt( - c**2*x**2 + 1)*x**3),x)*a*b*x**2 - 8*sqrt( - c**2*x**2 + 1)* int(asin(c*x)**2/(sqrt( - c**2*x**2 + 1)*c**2*x**5 - sqrt( - c**2*x**2 + 1 )*x**3),x)*b**2*x**2 + 12*sqrt( - c**2*x**2 + 1)*log(tan(asin(c*x)/2))*a** 2*c**2*x**2 - 9*sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 + 12*a**2*c**2*x**2 - 4*a**2)/(8*sqrt(d)*sqrt( - c**2*x**2 + 1)*d*x**2)