Integrand size = 27, antiderivative size = 429 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\frac {b^2 c^2 x}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {7 b c (a+b \arcsin (c x))}{4 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {15 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^3}+\frac {11 b^2 c \text {arctanh}(c x)}{6 d^3}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^3}+\frac {15 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {15 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^3}-\frac {15 b^2 c \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {15 b^2 c \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 d^3} \] Output:
1/12*b^2*c^2*x/d^3/(-c^2*x^2+1)-1/6*b*c*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1) ^(3/2)-7/4*b*c*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)^(1/2)-(a+b*arcsin(c*x))^ 2/d^3/x/(-c^2*x^2+1)^2+5/4*c^2*x*(a+b*arcsin(c*x))^2/d^3/(-c^2*x^2+1)^2+15 /8*c^2*x*(a+b*arcsin(c*x))^2/d^3/(-c^2*x^2+1)-15/4*I*c*(a+b*arcsin(c*x))^2 *arctan(I*c*x+(-c^2*x^2+1)^(1/2))/d^3-4*b*c*(a+b*arcsin(c*x))*arctanh(I*c* x+(-c^2*x^2+1)^(1/2))/d^3+11/6*b^2*c*arctanh(c*x)/d^3+2*I*b^2*c*polylog(2, -I*c*x-(-c^2*x^2+1)^(1/2))/d^3+15/4*I*b*c*(a+b*arcsin(c*x))*polylog(2,-I*( I*c*x+(-c^2*x^2+1)^(1/2)))/d^3-15/4*I*b*c*(a+b*arcsin(c*x))*polylog(2,I*(I *c*x+(-c^2*x^2+1)^(1/2)))/d^3-2*I*b^2*c*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2) )/d^3-15/4*b^2*c*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^3+15/4*b^2*c*p olylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^3
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1400\) vs. \(2(429)=858\).
Time = 10.47 (sec) , antiderivative size = 1400, normalized size of antiderivative = 3.26 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)^3),x]
Output:
-(a^2/(d^3*x)) + (a^2*c^2*x)/(4*d^3*(-1 + c^2*x^2)^2) - (7*a^2*c^2*x)/(8*d ^3*(-1 + c^2*x^2)) - (15*a^2*c*Log[1 - c*x])/(16*d^3) + (15*a^2*c*Log[1 + c*x])/(16*d^3) - (2*a*b*c*((-7*(Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(16*(-1 + c*x)) + ArcSin[c*x]/(c*x) + (7*(Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/(16*(1 + c*x)) - ((-2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x])/(48*(-1 + c*x)^2 ) + ((2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x])/(48*(1 + c*x)^2) + ArcTa nh[Sqrt[1 - c^2*x^2]] + (15*(((3*I)/2)*Pi*ArcSin[c*x] - (I/2)*ArcSin[c*x]^ 2 + 2*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 2*Pi*Log[Cos[ArcSin[c*x]/2]] + Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*PolyLog[2, (-I)*E^(I*ArcSi n[c*x])]))/16 - (15*((I/2)*Pi*ArcSin[c*x] - (I/2)*ArcSin[c*x]^2 + 2*Pi*Log [1 + E^((-I)*ArcSin[c*x])] + Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 2*ArcSin[c* x]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*Pi*Log[Cos[ArcSin[c*x]/2]] - Pi*Log[Si n[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]))/16))/ d^3 - (b^2*c*((-2*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (44*ArcSin[c*x] + 15 *ArcSin[c*x]^3 - 45*ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] - 45*Pi*Arc Sin[c*x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcSin[c*x])))/(2*E^((I/2)*ArcSin[c*x ]))] + 45*ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] + 45*ArcSin[c*x]^2*Lo g[((1/2 + I/2)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] - 45*Pi*Ar cSin[c*x]*Log[-1/2*((-1)^(1/4)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSi...
Time = 4.26 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.17, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.852, Rules used = {5204, 27, 5162, 5162, 5164, 3042, 4669, 3011, 2720, 5182, 215, 219, 5208, 215, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle 5 c^2 \int \frac {(a+b \arcsin (c x))^2}{d^3 \left (1-c^2 x^2\right )^3}dx+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 c^2 \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle \frac {5 c^2 \left (-\frac {1}{2} b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {3}{4} \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^2}dx+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle \frac {5 c^2 \left (-\frac {1}{2} b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {3}{4} \left (-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{1-c^2 x^2}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle \frac {5 c^2 \left (-\frac {1}{2} b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {3}{4} \left (-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}+\frac {5 c^2 \left (-\frac {1}{2} b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {3}{4} \left (-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {\int (a+b \arcsin (c x))^2 \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {-2 b \int (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \int \frac {1}{\left (1-c^2 x^2\right )^2}dx}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {1}{2} \int \frac {1}{1-c^2 x^2}dx+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \left (\int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {1}{3} b c \int \frac {1}{\left (1-c^2 x^2\right )^2}dx+\frac {a+b \arcsin (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \left (\int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {1}{3} b c \left (\frac {1}{2} \int \frac {1}{1-c^2 x^2}dx+\frac {x}{2 \left (1-c^2 x^2\right )}\right )+\frac {a+b \arcsin (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \left (\int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx+\frac {a+b \arcsin (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-b c \int \frac {1}{1-c^2 x^2}dx+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )-b \text {arctanh}(c x)\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \left (\int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )-b \text {arctanh}(c x)\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \left (\int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )-b \text {arctanh}(c x)\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )-b \text {arctanh}(c x)\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )-b \text {arctanh}(c x)\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )-b \text {arctanh}(c x)\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )-\frac {1}{2} b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}+\frac {2 b c \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )-b \text {arctanh}(c x)\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}\) |
Input:
Int[(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)^3),x]
Output:
-((a + b*ArcSin[c*x])^2/(d^3*x*(1 - c^2*x^2)^2)) + (2*b*c*((a + b*ArcSin[c *x])/(3*(1 - c^2*x^2)^(3/2)) + (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] - (b*c*(x/( 2*(1 - c^2*x^2)) + ArcTanh[c*x]/(2*c)))/3 + I*b*PolyLog[2, -E^(I*ArcSin[c* x])] - I*b*PolyLog[2, E^(I*ArcSin[c*x])]))/d^3 + (5*c^2*((x*(a + b*ArcSin[ c*x])^2)/(4*(1 - c^2*x^2)^2) - (b*c*((a + b*ArcSin[c*x])/(3*c^2*(1 - c^2*x ^2)^(3/2)) - (b*(x/(2*(1 - c^2*x^2)) + ArcTanh[c*x]/(2*c)))/(3*c)))/2 + (3 *((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*ArcSin[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*ArcS in[c*x])]))/(2*c)))/4))/d^3
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)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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[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 = 0.86 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.70
| method | result | size |
| derivativedivides | \(c \left (-\frac {a^{2} \left (\frac {1}{16 \left (c x +1\right )^{2}}+\frac {7}{16 \left (c x +1\right )}-\frac {15 \ln \left (c x +1\right )}{16}+\frac {1}{c x}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {7}{16 \left (c x -1\right )}+\frac {15 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {45 \arcsin \left (c x \right )^{2} x^{4} c^{4}-42 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{3} c^{3}-75 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}+46 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +24 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{24 c x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {15 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {15 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {15 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {11 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {15 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {15 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}\right )}{d^{3}}-\frac {2 a b \left (\frac {45 c^{4} x^{4} \arcsin \left (c x \right )-21 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-75 c^{2} x^{2} \arcsin \left (c x \right )+23 c x \sqrt {-c^{2} x^{2}+1}+24 \arcsin \left (c x \right )}{24 c x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {15 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {15 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\right )\) | \(730\) |
| default | \(c \left (-\frac {a^{2} \left (\frac {1}{16 \left (c x +1\right )^{2}}+\frac {7}{16 \left (c x +1\right )}-\frac {15 \ln \left (c x +1\right )}{16}+\frac {1}{c x}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {7}{16 \left (c x -1\right )}+\frac {15 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {45 \arcsin \left (c x \right )^{2} x^{4} c^{4}-42 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{3} c^{3}-75 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}+46 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +24 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{24 c x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {15 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {15 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {15 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {11 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {15 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {15 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}\right )}{d^{3}}-\frac {2 a b \left (\frac {45 c^{4} x^{4} \arcsin \left (c x \right )-21 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-75 c^{2} x^{2} \arcsin \left (c x \right )+23 c x \sqrt {-c^{2} x^{2}+1}+24 \arcsin \left (c x \right )}{24 c x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {15 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {15 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\right )\) | \(730\) |
| parts | \(-\frac {a^{2} \left (-\frac {c}{16 \left (c x -1\right )^{2}}+\frac {7 c}{16 \left (c x -1\right )}+\frac {15 c \ln \left (c x -1\right )}{16}+\frac {c}{16 \left (c x +1\right )^{2}}+\frac {7 c}{16 \left (c x +1\right )}-\frac {15 c \ln \left (c x +1\right )}{16}+\frac {1}{x}\right )}{d^{3}}-\frac {b^{2} c \left (\frac {45 \arcsin \left (c x \right )^{2} x^{4} c^{4}-42 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{3} c^{3}-75 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}+46 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +24 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{24 c x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {15 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {15 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {15 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {11 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {15 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {15 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}\right )}{d^{3}}-\frac {2 a b c \left (\frac {45 c^{4} x^{4} \arcsin \left (c x \right )-21 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-75 c^{2} x^{2} \arcsin \left (c x \right )+23 c x \sqrt {-c^{2} x^{2}+1}+24 \arcsin \left (c x \right )}{24 c x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {15 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {15 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\) | \(732\) |
Input:
int((a+b*arcsin(c*x))^2/x^2/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
c*(-a^2/d^3*(1/16/(c*x+1)^2+7/16/(c*x+1)-15/16*ln(c*x+1)+1/c/x-1/16/(c*x-1 )^2+7/16/(c*x-1)+15/16*ln(c*x-1))-b^2/d^3*(1/24*(45*arcsin(c*x)^2*x^4*c^4- 42*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^3*c^3-75*arcsin(c*x)^2*x^2*c^2+2*c^4*x ^4+46*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x+24*arcsin(c*x)^2-2*c^2*x^2)/c/x/( c^4*x^4-2*c^2*x^2+1)-15/8*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))) +15/4*I*arcsin(c*x)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))-15/4*polylog(3 ,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2) )-2*I*dilog(I*c*x+(-c^2*x^2+1)^(1/2))-2*I*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2) )+11/3*I*arctan(I*c*x+(-c^2*x^2+1)^(1/2))+15/8*arcsin(c*x)^2*ln(1+I*(I*c*x +(-c^2*x^2+1)^(1/2)))-15/4*I*arcsin(c*x)*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^ (1/2)))+15/4*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2))))-2*a*b/d^3*(1/24*(45 *c^4*x^4*arcsin(c*x)-21*c^3*x^3*(-c^2*x^2+1)^(1/2)-75*c^2*x^2*arcsin(c*x)+ 23*c*x*(-c^2*x^2+1)^(1/2)+24*arcsin(c*x))/c/x/(c^4*x^4-2*c^2*x^2+1)-ln(I*c *x+(-c^2*x^2+1)^(1/2)-1)+ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+15/8*arcsin(c*x)*l n(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-15/8*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2 +1)^(1/2)))-15/8*I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+15/8*I*dilog(1-I* (I*c*x+(-c^2*x^2+1)^(1/2)))))
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{2}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/x^2/(-c^2*d*x^2+d)^3,x, algorithm="fricas")
Output:
integral(-(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/(c^6*d^3*x^8 - 3*c ^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2}}{c^{6} x^{8} - 3 c^{4} x^{6} + 3 c^{2} x^{4} - x^{2}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{8} - 3 c^{4} x^{6} + 3 c^{2} x^{4} - x^{2}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{8} - 3 c^{4} x^{6} + 3 c^{2} x^{4} - x^{2}}\, dx}{d^{3}} \] Input:
integrate((a+b*asin(c*x))**2/x**2/(-c**2*d*x**2+d)**3,x)
Output:
-(Integral(a**2/(c**6*x**8 - 3*c**4*x**6 + 3*c**2*x**4 - x**2), x) + Integ ral(b**2*asin(c*x)**2/(c**6*x**8 - 3*c**4*x**6 + 3*c**2*x**4 - x**2), x) + Integral(2*a*b*asin(c*x)/(c**6*x**8 - 3*c**4*x**6 + 3*c**2*x**4 - x**2), x))/d**3
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{2}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/x^2/(-c^2*d*x^2+d)^3,x, algorithm="maxima")
Output:
-1/16*a^2*(2*(15*c^4*x^4 - 25*c^2*x^2 + 8)/(c^4*d^3*x^5 - 2*c^2*d^3*x^3 + d^3*x) - 15*c*log(c*x + 1)/d^3 + 15*c*log(c*x - 1)/d^3) + 1/16*(15*(b^2*c^ 5*x^5 - 2*b^2*c^3*x^3 + b^2*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1) )^2*log(c*x + 1) - 15*(b^2*c^5*x^5 - 2*b^2*c^3*x^3 + b^2*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) - 2*(15*b^2*c^4*x^4 - 25*b^ 2*c^2*x^2 + 8*b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 16*(c^4* d^3*x^5 - 2*c^2*d^3*x^3 + d^3*x)*integrate(-1/8*(16*a*b*arctan2(c*x, sqrt( c*x + 1)*sqrt(-c*x + 1)) - (15*(b^2*c^6*x^6 - 2*b^2*c^4*x^4 + b^2*c^2*x^2) *arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - 15*(b^2*c^6*x^6 - 2*b^2*c^4*x^4 + b^2*c^2*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) *log(-c*x + 1) - 2*(15*b^2*c^5*x^5 - 25*b^2*c^3*x^3 + 8*b^2*c*x)*arctan2(c *x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^6*d^3* x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x))/(c^4*d^3*x^5 - 2*c^2*d ^3*x^3 + d^3*x)
Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arcsin(c*x))^2/x^2/(-c^2*d*x^2+d)^3,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. \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^2\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \] Input:
int((a + b*asin(c*x))^2/(x^2*(d - c^2*d*x^2)^3),x)
Output:
int((a + b*asin(c*x))^2/(x^2*(d - c^2*d*x^2)^3), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\frac {-32 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{8}-3 c^{4} x^{6}+3 c^{2} x^{4}-x^{2}}d x \right ) a b \,c^{4} x^{5}+64 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{8}-3 c^{4} x^{6}+3 c^{2} x^{4}-x^{2}}d x \right ) a b \,c^{2} x^{3}-32 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{8}-3 c^{4} x^{6}+3 c^{2} x^{4}-x^{2}}d x \right ) a b x -16 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{c^{6} x^{8}-3 c^{4} x^{6}+3 c^{2} x^{4}-x^{2}}d x \right ) b^{2} c^{4} x^{5}+32 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{c^{6} x^{8}-3 c^{4} x^{6}+3 c^{2} x^{4}-x^{2}}d x \right ) b^{2} c^{2} x^{3}-16 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{c^{6} x^{8}-3 c^{4} x^{6}+3 c^{2} x^{4}-x^{2}}d x \right ) b^{2} x -15 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{5} x^{5}+30 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{3} x^{3}-15 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c x +15 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{5} x^{5}-30 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{3} x^{3}+15 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c x -30 a^{2} c^{4} x^{4}+50 a^{2} c^{2} x^{2}-16 a^{2}}{16 d^{3} x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*asin(c*x))^2/x^2/(-c^2*d*x^2+d)^3,x)
Output:
( - 32*int(asin(c*x)/(c**6*x**8 - 3*c**4*x**6 + 3*c**2*x**4 - x**2),x)*a*b *c**4*x**5 + 64*int(asin(c*x)/(c**6*x**8 - 3*c**4*x**6 + 3*c**2*x**4 - x** 2),x)*a*b*c**2*x**3 - 32*int(asin(c*x)/(c**6*x**8 - 3*c**4*x**6 + 3*c**2*x **4 - x**2),x)*a*b*x - 16*int(asin(c*x)**2/(c**6*x**8 - 3*c**4*x**6 + 3*c* *2*x**4 - x**2),x)*b**2*c**4*x**5 + 32*int(asin(c*x)**2/(c**6*x**8 - 3*c** 4*x**6 + 3*c**2*x**4 - x**2),x)*b**2*c**2*x**3 - 16*int(asin(c*x)**2/(c**6 *x**8 - 3*c**4*x**6 + 3*c**2*x**4 - x**2),x)*b**2*x - 15*log(c**2*x - c)*a **2*c**5*x**5 + 30*log(c**2*x - c)*a**2*c**3*x**3 - 15*log(c**2*x - c)*a** 2*c*x + 15*log(c**2*x + c)*a**2*c**5*x**5 - 30*log(c**2*x + c)*a**2*c**3*x **3 + 15*log(c**2*x + c)*a**2*c*x - 30*a**2*c**4*x**4 + 50*a**2*c**2*x**2 - 16*a**2)/(16*d**3*x*(c**4*x**4 - 2*c**2*x**2 + 1))