Integrand size = 27, antiderivative size = 439 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b^2 c^2}{3 d^2 x}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {5 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {26 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {b^2 c^3 \text {arctanh}(c x)}{d^2}+\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^2}-\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d^2} \] Output:
-1/3*b^2*c^2/d^2/x-2/3*b*c^3*(a+b*arcsin(c*x))/d^2/(-c^2*x^2+1)^(1/2)-1/3* b*c*(a+b*arcsin(c*x))/d^2/x^2/(-c^2*x^2+1)^(1/2)-1/3*(a+b*arcsin(c*x))^2/d ^2/x^3/(-c^2*x^2+1)-5/3*c^2*(a+b*arcsin(c*x))^2/d^2/x/(-c^2*x^2+1)+5/2*c^4 *x*(a+b*arcsin(c*x))^2/d^2/(-c^2*x^2+1)-5*I*c^3*(a+b*arcsin(c*x))^2*arctan (I*c*x+(-c^2*x^2+1)^(1/2))/d^2-26/3*b*c^3*(a+b*arcsin(c*x))*arctanh(I*c*x+ (-c^2*x^2+1)^(1/2))/d^2+b^2*c^3*arctanh(c*x)/d^2+13/3*I*b^2*c^3*polylog(2, -I*c*x-(-c^2*x^2+1)^(1/2))/d^2+5*I*b*c^3*(a+b*arcsin(c*x))*polylog(2,-I*(I *c*x+(-c^2*x^2+1)^(1/2)))/d^2-5*I*b*c^3*(a+b*arcsin(c*x))*polylog(2,I*(I*c *x+(-c^2*x^2+1)^(1/2)))/d^2-13/3*I*b^2*c^3*polylog(2,I*c*x+(-c^2*x^2+1)^(1 /2))/d^2-5*b^2*c^3*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^2+5*b^2*c^3* polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1390\) vs. \(2(439)=878\).
Time = 10.73 (sec) , antiderivative size = 1390, normalized size of antiderivative = 3.17 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^2),x]
Output:
-1/3*a^2/(d^2*x^3) - (2*a^2*c^2)/(d^2*x) - (a^2*c^4*x)/(2*d^2*(-1 + c^2*x^ 2)) - (5*a^2*c^3*Log[1 - c*x])/(4*d^2) + (5*a^2*c^3*Log[1 + c*x])/(4*d^2) + (2*a*b*((c^3*(Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(4*(-1 + c*x)) - (c^4*(S qrt[1 - c^2*x^2] + ArcSin[c*x]))/(4*(c + c^2*x)) + 2*c^2*(-(ArcSin[c*x]/x) - c*ArcTanh[Sqrt[1 - c^2*x^2]]) - (c*x*Sqrt[1 - c^2*x^2] + 2*ArcSin[c*x] + c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*x^3) - (5*c^4*((((3*I)/2)*Pi*ArcS in[c*x])/c - ((I/2)*ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])] )/c - (Pi*Log[1 + I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[c*x]*Log[1 + I*E^(I* ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c + (Pi*Log[-Cos[(Pi + 2 *ArcSin[c*x])/4]])/c - ((2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/c))/4 + (5*c^4*(((I/2)*Pi*ArcSin[c*x])/c - ((I/2)*ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c + (Pi*Log[1 - I*E^(I*ArcSin[c*x])])/c + (2*ArcSi n[c*x]*Log[1 - I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c - (Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])/c - ((2*I)*PolyLog[2, I*E^(I*ArcSi n[c*x])])/c))/4))/d^2 + (b^2*c^3*(-24*ArcSin[c*x] - (6*ArcSin[c*x]^2)/(-1 + c*x) - 4*Cot[ArcSin[c*x]/2] - 26*ArcSin[c*x]^2*Cot[ArcSin[c*x]/2] - 2*Ar cSin[c*x]*Csc[ArcSin[c*x]/2]^2 - (c*x*ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^4)/ 2 + 104*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 60*ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] + 60*Pi*ArcSin[c*x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcSi n[c*x])))/(2*E^((I/2)*ArcSin[c*x]))] - 60*ArcSin[c*x]^2*Log[1 + I*E^(I*...
Time = 4.91 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.19, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {5204, 27, 5204, 264, 219, 5162, 5164, 3042, 4669, 3011, 2720, 5182, 219, 5208, 219, 5218, 3042, 4671, 2715, 2838, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {5}{3} c^2 \int \frac {(a+b \arcsin (c x))^2}{d^2 x^2 \left (1-c^2 x^2\right )^2}dx+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x^3 \left (1-c^2 x^2\right )^{3/2}}dx}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 c^2 \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (1-c^2 x^2\right )^2}dx}{3 d^2}+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x^3 \left (1-c^2 x^2\right )^{3/2}}dx}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^2}dx+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (1-c^2 x^2\right )}dx-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^2}dx+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x^2}dx-\frac {1}{x}\right )-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 b c \left (\frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}+\frac {5 c^2 \left (3 c^2 \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^2}dx+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle \frac {2 b c \left (\frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}+\frac {5 c^2 \left (3 c^2 \left (-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{1-c^2 x^2}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle \frac {2 b c \left (\frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}+\frac {5 c^2 \left (3 c^2 \left (-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c \left (\frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}+\frac {5 c^2 \left (2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx+3 c^2 \left (-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {\int (a+b \arcsin (c x))^2 \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {-2 b \int (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-b c \int \frac {1}{1-c^2 x^2}dx+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}\right )-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-b c \int \frac {1}{1-c^2 x^2}dx+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}-b \text {arctanh}(c x)\right )-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}-b \text {arctanh}(c x)\right )-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \left (\int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}-b \text {arctanh}(c x)\right )-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \left (\int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}-b \text {arctanh}(c x)\right )-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \left (\int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}-b \text {arctanh}(c x)\right )-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \left (\int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}-b \text {arctanh}(c x)\right )-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}-b \text {arctanh}(c x)\right )-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}-b \text {arctanh}(c x)\right )-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}-b \text {arctanh}(c x)\right )-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}-b \text {arctanh}(c x)\right )-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-b \text {arctanh}(c x)\right )-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-b \text {arctanh}(c x)\right )-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+2 b c \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-b \text {arctanh}(c x)\right )-\frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}+\frac {2 b c \left (\frac {3}{2} c^2 \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-b \text {arctanh}(c x)\right )-\frac {a+b \arcsin (c x)}{2 x^2 \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
Input:
Int[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^2),x]
Output:
-1/3*(a + b*ArcSin[c*x])^2/(d^2*x^3*(1 - c^2*x^2)) + (2*b*c*(-1/2*(a + b*A rcSin[c*x])/(x^2*Sqrt[1 - c^2*x^2]) + (b*c*(-x^(-1) + c*ArcTanh[c*x]))/2 + (3*c^2*((a + b*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - 2*(a + b*ArcSin[c*x])*Arc Tanh[E^(I*ArcSin[c*x])] - b*ArcTanh[c*x] + I*b*PolyLog[2, -E^(I*ArcSin[c*x ])] - I*b*PolyLog[2, E^(I*ArcSin[c*x])]))/2))/(3*d^2) + (5*c^2*(-((a + b*A rcSin[c*x])^2/(x*(1 - c^2*x^2))) + 2*b*c*((a + b*ArcSin[c*x])/Sqrt[1 - c^2 *x^2] - 2*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] - b*ArcTanh[c*x] + I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*PolyLog[2, E^(I*ArcSin[c*x])]) + 3*c^2*((x*(a + b*ArcSin[c*x])^2)/(2*(1 - c^2*x^2)) - b*c*((a + b*ArcSin[ c*x])/(c^2*Sqrt[1 - c^2*x^2]) - (b*ArcTanh[c*x])/c^2) + ((-2*I)*(a + b*Arc Sin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])] + 2*b*(I*(a + b*ArcSin[c*x])*PolyLog [2, (-I)*E^(I*ArcSin[c*x])] - b*PolyLog[3, (-I)*E^(I*ArcSin[c*x])]) - 2*b* (I*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])] - b*PolyLog[3, I*E^ (I*ArcSin[c*x])]))/(2*c))))/(3*d^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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)^(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_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -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/(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 = 0.89 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.61
| method | result | size |
| derivativedivides | \(c^{3} \left (\frac {a^{2} \left (-\frac {1}{4 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{4}-\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {5 \ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {15 \arcsin \left (c x \right )^{2} x^{4} c^{4}-4 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{3} c^{3}-10 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}-2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -2 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+5 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\frac {13 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {13 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {13 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-5 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {15 c^{4} x^{4} \arcsin \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-10 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}-\frac {13 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {5 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\) | \(707\) |
| default | \(c^{3} \left (\frac {a^{2} \left (-\frac {1}{4 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{4}-\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {5 \ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {15 \arcsin \left (c x \right )^{2} x^{4} c^{4}-4 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{3} c^{3}-10 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}-2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -2 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+5 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\frac {13 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {13 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {13 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-5 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {15 c^{4} x^{4} \arcsin \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-10 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}-\frac {13 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {5 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\) | \(707\) |
| parts | \(\frac {a^{2} \left (-\frac {c^{3}}{4 \left (c x -1\right )}-\frac {5 c^{3} \ln \left (c x -1\right )}{4}-\frac {c^{3}}{4 \left (c x +1\right )}+\frac {5 c^{3} \ln \left (c x +1\right )}{4}-\frac {1}{3 x^{3}}-\frac {2 c^{2}}{x}\right )}{d^{2}}+\frac {b^{2} c^{3} \left (-\frac {15 \arcsin \left (c x \right )^{2} x^{4} c^{4}-4 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{3} c^{3}-10 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}-2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -2 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+5 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\frac {13 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {13 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {13 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-5 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d^{2}}+\frac {2 a b \,c^{3} \left (-\frac {15 c^{4} x^{4} \arcsin \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-10 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}-\frac {13 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {5 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\) | \(718\) |
Input:
int((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
c^3*(a^2/d^2*(-1/4/(c*x+1)+5/4*ln(c*x+1)-1/3/c^3/x^3-2/c/x-1/4/(c*x-1)-5/4 *ln(c*x-1))+b^2/d^2*(-1/6*(15*arcsin(c*x)^2*x^4*c^4-4*(-c^2*x^2+1)^(1/2)*a rcsin(c*x)*x^3*c^3-10*arcsin(c*x)^2*x^2*c^2+2*c^4*x^4-2*arcsin(c*x)*(-c^2* x^2+1)^(1/2)*c*x-2*arcsin(c*x)^2-2*c^2*x^2)/(c^2*x^2-1)/c^3/x^3+5/2*arcsin (c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-5*I*arcsin(c*x)*polylog(2,I*(I* c*x+(-c^2*x^2+1)^(1/2)))+5*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))-13/3*ar csin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+13/3*I*dilog(I*c*x+(-c^2*x^2+1)^( 1/2))+13/3*I*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*I*arctan(I*c*x+(-c^2*x^2+ 1)^(1/2))-5/2*arcsin(c*x)^2*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+5*I*arcsin( c*x)*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-5*polylog(3,-I*(I*c*x+(-c^2* x^2+1)^(1/2))))+2*a*b/d^2*(-1/6*(15*c^4*x^4*arcsin(c*x)-2*c^3*x^3*(-c^2*x^ 2+1)^(1/2)-10*c^2*x^2*arcsin(c*x)-c*x*(-c^2*x^2+1)^(1/2)-2*arcsin(c*x))/(c ^2*x^2-1)/c^3/x^3+13/6*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)-13/6*ln(1+I*c*x+(-c^ 2*x^2+1)^(1/2))-5/2*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+5/2*arc sin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+5/2*I*dilog(1+I*(I*c*x+(-c^2*x ^2+1)^(1/2)))-5/2*I*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))))
\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^2,x, algorithm="fricas")
Output:
integral((b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/(c^4*d^2*x^8 - 2*c^ 2*d^2*x^6 + d^2*x^4), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx}{d^{2}} \] Input:
integrate((a+b*asin(c*x))**2/x**4/(-c**2*d*x**2+d)**2,x)
Output:
(Integral(a**2/(c**4*x**8 - 2*c**2*x**6 + x**4), x) + Integral(b**2*asin(c *x)**2/(c**4*x**8 - 2*c**2*x**6 + x**4), x) + Integral(2*a*b*asin(c*x)/(c* *4*x**8 - 2*c**2*x**6 + x**4), x))/d**2
\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^2,x, algorithm="maxima")
Output:
1/12*(15*c^3*log(c*x + 1)/d^2 - 15*c^3*log(c*x - 1)/d^2 - 2*(15*c^4*x^4 - 10*c^2*x^2 - 2)/(c^2*d^2*x^5 - d^2*x^3))*a^2 + 1/12*(15*(b^2*c^5*x^5 - b^2 *c^3*x^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(c*x + 1) - 15*( b^2*c^5*x^5 - b^2*c^3*x^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*lo g(-c*x + 1) - 2*(15*b^2*c^4*x^4 - 10*b^2*c^2*x^2 - 2*b^2)*arctan2(c*x, sqr t(c*x + 1)*sqrt(-c*x + 1))^2 + 12*(c^2*d^2*x^5 - d^2*x^3)*integrate(1/6*(1 2*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + (15*(b^2*c^6*x^6 - b^2* c^4*x^4)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - 15*(b^2 *c^6*x^6 - b^2*c^4*x^4)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c* x + 1) - 2*(15*b^2*c^5*x^5 - 10*b^2*c^3*x^3 - 2*b^2*c*x)*arctan2(c*x, sqrt (c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^4*d^2*x^8 - 2* c^2*d^2*x^6 + d^2*x^4), x))/(c^2*d^2*x^5 - d^2*x^3)
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^2,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^4\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \] Input:
int((a + b*asin(c*x))^2/(x^4*(d - c^2*d*x^2)^2),x)
Output:
int((a + b*asin(c*x))^2/(x^4*(d - c^2*d*x^2)^2), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\frac {24 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{4} x^{8}-2 c^{2} x^{6}+x^{4}}d x \right ) a b \,c^{2} x^{5}-24 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{4} x^{8}-2 c^{2} x^{6}+x^{4}}d x \right ) a b \,x^{3}+12 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{c^{4} x^{8}-2 c^{2} x^{6}+x^{4}}d x \right ) b^{2} c^{2} x^{5}-12 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{c^{4} x^{8}-2 c^{2} x^{6}+x^{4}}d x \right ) b^{2} x^{3}-15 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{5} x^{5}+15 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{3} x^{3}+15 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{5} x^{5}-15 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{3} x^{3}-30 a^{2} c^{4} x^{4}+20 a^{2} c^{2} x^{2}+4 a^{2}}{12 d^{2} x^{3} \left (c^{2} x^{2}-1\right )} \] Input:
int((a+b*asin(c*x))^2/x^4/(-c^2*d*x^2+d)^2,x)
Output:
(24*int(asin(c*x)/(c**4*x**8 - 2*c**2*x**6 + x**4),x)*a*b*c**2*x**5 - 24*i nt(asin(c*x)/(c**4*x**8 - 2*c**2*x**6 + x**4),x)*a*b*x**3 + 12*int(asin(c* x)**2/(c**4*x**8 - 2*c**2*x**6 + x**4),x)*b**2*c**2*x**5 - 12*int(asin(c*x )**2/(c**4*x**8 - 2*c**2*x**6 + x**4),x)*b**2*x**3 - 15*log(c**2*x - c)*a* *2*c**5*x**5 + 15*log(c**2*x - c)*a**2*c**3*x**3 + 15*log(c**2*x + c)*a**2 *c**5*x**5 - 15*log(c**2*x + c)*a**2*c**3*x**3 - 30*a**2*c**4*x**4 + 20*a* *2*c**2*x**2 + 4*a**2)/(12*d**2*x**3*(c**2*x**2 - 1))