Integrand size = 24, antiderivative size = 332 \[ \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b^2 x}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b (a+b \arcsin (c x))}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b (a+b \arcsin (c x))}{4 c d^3 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {3 i (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 c d^3}+\frac {5 b^2 \text {arctanh}(c x)}{6 c d^3}+\frac {3 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 c d^3}-\frac {3 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 c d^3}-\frac {3 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{4 c d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 c d^3} \] Output:
1/12*b^2*x/d^3/(-c^2*x^2+1)-1/6*b*(a+b*arcsin(c*x))/c/d^3/(-c^2*x^2+1)^(3/ 2)-3/4*b*(a+b*arcsin(c*x))/c/d^3/(-c^2*x^2+1)^(1/2)+1/4*x*(a+b*arcsin(c*x) )^2/d^3/(-c^2*x^2+1)^2+3/8*x*(a+b*arcsin(c*x))^2/d^3/(-c^2*x^2+1)-3/4*I*(a +b*arcsin(c*x))^2*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/c/d^3+5/6*b^2*arctanh(c *x)/c/d^3+3/4*I*b*(a+b*arcsin(c*x))*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2) ))/c/d^3-3/4*I*b*(a+b*arcsin(c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))) /c/d^3-3/4*b^2*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c/d^3+3/4*b^2*poly log(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c/d^3
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(666\) vs. \(2(332)=664\).
Time = 3.57 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.01 \[ \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=-\frac {-\frac {12 a^2 x}{\left (-1+c^2 x^2\right )^2}+\frac {18 a^2 x}{-1+c^2 x^2}+\frac {9 a^2 \log (1-c x)}{c}-\frac {9 a^2 \log (1+c x)}{c}+\frac {2 a b \left (\frac {2 \sqrt {1-c^2 x^2}}{(-1+c x)^2}-\frac {c x \sqrt {1-c^2 x^2}}{(-1+c x)^2}-\frac {9 \sqrt {1-c^2 x^2}}{-1+c x}+\frac {2 \sqrt {1-c^2 x^2}}{(1+c x)^2}+\frac {c x \sqrt {1-c^2 x^2}}{(1+c x)^2}+\frac {9 \sqrt {1-c^2 x^2}}{1+c x}+9 i \pi \arcsin (c x)-\frac {3 \arcsin (c x)}{(-1+c x)^2}+\frac {9 \arcsin (c x)}{-1+c x}+\frac {3 \arcsin (c x)}{(1+c x)^2}+\frac {9 \arcsin (c x)}{1+c x}-9 \pi \log \left (1-i e^{i \arcsin (c x)}\right )-18 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-9 \pi \log \left (1+i e^{i \arcsin (c x)}\right )+18 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+9 \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+9 \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-18 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+18 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{c}+\frac {2 b^2 \left (\frac {2 c x}{-1+c^2 x^2}-18 \coth ^{-1}(c x)+\frac {4 \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}+\frac {18 \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {6 c x \arcsin (c x)^2}{\left (-1+c^2 x^2\right )^2}+\frac {9 c x \arcsin (c x)^2}{-1+c^2 x^2}+18 i \arcsin (c x)^2 \arctan \left (e^{i \arcsin (c x)}\right )-2 \text {arctanh}(c x)-18 i \arcsin (c x) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+18 i \arcsin (c x) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+18 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )-18 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )}{c}}{48 d^3} \] Input:
Integrate[(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^3,x]
Output:
-1/48*((-12*a^2*x)/(-1 + c^2*x^2)^2 + (18*a^2*x)/(-1 + c^2*x^2) + (9*a^2*L og[1 - c*x])/c - (9*a^2*Log[1 + c*x])/c + (2*a*b*((2*Sqrt[1 - c^2*x^2])/(- 1 + c*x)^2 - (c*x*Sqrt[1 - c^2*x^2])/(-1 + c*x)^2 - (9*Sqrt[1 - c^2*x^2])/ (-1 + c*x) + (2*Sqrt[1 - c^2*x^2])/(1 + c*x)^2 + (c*x*Sqrt[1 - c^2*x^2])/( 1 + c*x)^2 + (9*Sqrt[1 - c^2*x^2])/(1 + c*x) + (9*I)*Pi*ArcSin[c*x] - (3*A rcSin[c*x])/(-1 + c*x)^2 + (9*ArcSin[c*x])/(-1 + c*x) + (3*ArcSin[c*x])/(1 + c*x)^2 + (9*ArcSin[c*x])/(1 + c*x) - 9*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 18*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 9*Pi*Log[1 + I*E^(I*ArcSin [c*x])] + 18*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 9*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 9*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (18*I)*Poly Log[2, (-I)*E^(I*ArcSin[c*x])] + (18*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]))/ c + (2*b^2*((2*c*x)/(-1 + c^2*x^2) - 18*ArcCoth[c*x] + (4*ArcSin[c*x])/(1 - c^2*x^2)^(3/2) + (18*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (6*c*x*ArcSin[c*x] ^2)/(-1 + c^2*x^2)^2 + (9*c*x*ArcSin[c*x]^2)/(-1 + c^2*x^2) + (18*I)*ArcSi n[c*x]^2*ArcTan[E^(I*ArcSin[c*x])] - 2*ArcTanh[c*x] - (18*I)*ArcSin[c*x]*P olyLog[2, (-I)*E^(I*ArcSin[c*x])] + (18*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*A rcSin[c*x])] + 18*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 18*PolyLog[3, I*E^( I*ArcSin[c*x])]))/c)/d^3
Time = 1.90 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5162, 27, 5162, 5164, 3042, 4669, 3011, 2720, 5182, 215, 219, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle -\frac {b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {3 \int \frac {(a+b \arcsin (c x))^2}{d^2 \left (1-c^2 x^2\right )^2}dx}{4 d}+\frac {x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {3 \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^2}dx}{4 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle -\frac {b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {3 \left (-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{1-c^2 x^2}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle -\frac {b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {3 \left (-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {3 \left (-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {\int (a+b \arcsin (c x))^2 \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {3 \left (\frac {-2 b \int (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {3 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {3 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {3 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \int \frac {1}{\left (1-c^2 x^2\right )^2}dx}{3 c}\right )}{2 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {3 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {1}{2} \int \frac {1}{1-c^2 x^2}dx+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )}{2 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )}{2 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )}{2 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
Input:
Int[(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^3,x]
Output:
(x*(a + b*ArcSin[c*x])^2)/(4*d^3*(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*d^3) + (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*A rcSin[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)))/(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[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[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[((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[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 = 0.64 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.72
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2} \left (\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {9 \arcsin \left (c x \right )^{2} c^{3} x^{3}-18 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-15 \arcsin \left (c x \right )^{2} c x +2 c^{3} x^{3}+22 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-2 c x}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {5 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3}}-\frac {2 a b \left (\frac {9 c^{3} x^{3} \arcsin \left (c x \right )-9 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-15 c x \arcsin \left (c x \right )+11 \sqrt {-c^{2} x^{2}+1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}}{c}\) | \(570\) |
| default | \(\frac {-\frac {a^{2} \left (\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {9 \arcsin \left (c x \right )^{2} c^{3} x^{3}-18 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-15 \arcsin \left (c x \right )^{2} c x +2 c^{3} x^{3}+22 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-2 c x}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {5 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3}}-\frac {2 a b \left (\frac {9 c^{3} x^{3} \arcsin \left (c x \right )-9 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-15 c x \arcsin \left (c x \right )+11 \sqrt {-c^{2} x^{2}+1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}}{c}\) | \(570\) |
| parts | \(-\frac {a^{2} \left (-\frac {1}{16 c \left (c x -1\right )^{2}}+\frac {3}{16 c \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16 c}+\frac {1}{16 c \left (c x +1\right )^{2}}+\frac {3}{16 c \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16 c}\right )}{d^{3}}-\frac {b^{2} \left (\frac {9 \arcsin \left (c x \right )^{2} c^{3} x^{3}-18 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-15 \arcsin \left (c x \right )^{2} c x +2 c^{3} x^{3}+22 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-2 c x}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {5 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3} c}-\frac {2 a b \left (\frac {9 c^{3} x^{3} \arcsin \left (c x \right )-9 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-15 c x \arcsin \left (c x \right )+11 \sqrt {-c^{2} x^{2}+1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3} c}\) | \(590\) |
Input:
int((a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
1/c*(-a^2/d^3*(1/16/(c*x+1)^2+3/16/(c*x+1)-3/16*ln(c*x+1)-1/16/(c*x-1)^2+3 /16/(c*x-1)+3/16*ln(c*x-1))-b^2/d^3*(1/24*(9*arcsin(c*x)^2*c^3*x^3-18*arcs in(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-15*arcsin(c*x)^2*c*x+2*c^3*x^3+22*arcsi n(c*x)*(-c^2*x^2+1)^(1/2)-2*c*x)/(c^4*x^4-2*c^2*x^2+1)+3/8*arcsin(c*x)^2*l n(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/4*I*arcsin(c*x)*polylog(2,-I*(I*c*x+(- c^2*x^2+1)^(1/2)))+3/4*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/8*arcsin (c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3/4*I*arcsin(c*x)*polylog(2,I*( I*c*x+(-c^2*x^2+1)^(1/2)))-3/4*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+5/3 *I*arctan(I*c*x+(-c^2*x^2+1)^(1/2)))-2*a*b/d^3*(1/24*(9*c^3*x^3*arcsin(c*x )-9*c^2*x^2*(-c^2*x^2+1)^(1/2)-15*c*x*arcsin(c*x)+11*(-c^2*x^2+1)^(1/2))/( c^4*x^4-2*c^2*x^2+1)+3/8*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/ 8*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/8*I*dilog(1+I*(I*c*x+(- c^2*x^2+1)^(1/2)))+3/8*I*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))))
\[ \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \] Input:
integrate((a+b*arcsin(c*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^6 - 3*c ^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((a+b*asin(c*x))**2/(-c**2*d*x**2+d)**3,x)
Output:
Timed out
\[ \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="maxima")
Output:
-1/16*a^2*(2*(3*c^2*x^3 - 5*x)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3) - 3*log (c*x + 1)/(c*d^3) + 3*log(c*x - 1)/(c*d^3)) + 1/16*(3*(b^2*c^4*x^4 - 2*b^2 *c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(c*x + 1) - 3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c *x + 1))^2*log(-c*x + 1) - 2*(3*b^2*c^3*x^3 - 5*b^2*c*x)*arctan2(c*x, sqrt (c*x + 1)*sqrt(-c*x + 1))^2 + 16*(c^5*d^3*x^4 - 2*c^3*d^3*x^2 + c*d^3)*int egrate(-1/8*(16*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - (3*(b^2*c ^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*l og(c*x + 1) - 3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1) - 2*(3*b^2*c^3*x^3 - 5*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^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x))/(c^5*d^3*x^4 - 2*c^3*d^3 *x^2 + c*d^3)
Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arcsin(c*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. \[ \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \] Input:
int((a + b*asin(c*x))^2/(d - c^2*d*x^2)^3,x)
Output:
int((a + b*asin(c*x))^2/(d - c^2*d*x^2)^3, x)
\[ \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-32 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b \,c^{5} x^{4}+64 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b \,c^{3} x^{2}-32 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b c -16 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2} c^{5} x^{4}+32 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2} c^{3} x^{2}-16 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2} c -3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{4} x^{4}+6 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{2} x^{2}-3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2}+3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{4} x^{4}-6 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{2} x^{2}+3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2}-6 a^{2} c^{3} x^{3}+10 a^{2} c x}{16 c \,d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*asin(c*x))^2/(-c^2*d*x^2+d)^3,x)
Output:
( - 32*int(asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*a*b*c* *5*x**4 + 64*int(asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)* a*b*c**3*x**2 - 32*int(asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*a*b*c - 16*int(asin(c*x)**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*b**2*c**5*x**4 + 32*int(asin(c*x)**2/(c**6*x**6 - 3*c**4*x**4 + 3*c **2*x**2 - 1),x)*b**2*c**3*x**2 - 16*int(asin(c*x)**2/(c**6*x**6 - 3*c**4* x**4 + 3*c**2*x**2 - 1),x)*b**2*c - 3*log(c**2*x - c)*a**2*c**4*x**4 + 6*l og(c**2*x - c)*a**2*c**2*x**2 - 3*log(c**2*x - c)*a**2 + 3*log(c**2*x + c) *a**2*c**4*x**4 - 6*log(c**2*x + c)*a**2*c**2*x**2 + 3*log(c**2*x + c)*a** 2 - 6*a**2*c**3*x**3 + 10*a**2*c*x)/(16*c*d**3*(c**4*x**4 - 2*c**2*x**2 + 1))