Integrand size = 27, antiderivative size = 403 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\frac {b^2 c^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c (a+b \arcsin (c x))}{d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^3 x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {6 c^2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d^3}+\frac {b^2 c^2 \log (x)}{d^3}-\frac {7 b^2 c^2 \log \left (1-c^2 x^2\right )}{6 d^3}+\frac {3 i b c^2 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {3 i b c^2 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 d^3}+\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )}{2 d^3} \]
1/12*b^2*c^2/d^3/(-c^2*x^2+1)-b*c*(a+b*arcsin(c*x))/d^3/x/(-c^2*x^2+1)^(3/ 2)+5/6*b*c^3*x*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)^(3/2)+3/4*c^2*(a+b*arcsi n(c*x))^2/d^3/(-c^2*x^2+1)^2-1/2*(a+b*arcsin(c*x))^2/d^3/x^2/(-c^2*x^2+1)^ 2+3/2*c^2*(a+b*arcsin(c*x))^2/d^3/(-c^2*x^2+1)-6*c^2*(a+b*arcsin(c*x))^2*a rctanh((I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3+b^2*c^2*ln(x)/d^3-7/6*b^2*c^2*ln( -c^2*x^2+1)/d^3+3*I*b*c^2*(a+b*arcsin(c*x))*polylog(2,-(I*c*x+(-c^2*x^2+1) ^(1/2))^2)/d^3-3*I*b*c^2*(a+b*arcsin(c*x))*polylog(2,(I*c*x+(-c^2*x^2+1)^( 1/2))^2)/d^3-3/2*b^2*c^2*polylog(3,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3+3/2* b^2*c^2*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3-4/3*b*c^3*x*(a+b*arcsi n(c*x))/d^3/(-c^2*x^2+1)^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1003\) vs. \(2(403)=806\).
Time = 7.92 (sec) , antiderivative size = 1003, normalized size of antiderivative = 2.49 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {a^2}{2 d^3 x^2}+\frac {a^2 c^2}{4 d^3 \left (-1+c^2 x^2\right )^2}-\frac {a^2 c^2}{d^3 \left (-1+c^2 x^2\right )}+\frac {3 a^2 c^2 \log (x)}{d^3}-\frac {3 a^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^3}-\frac {2 a b \left (\frac {c^2 \left ((2-c x) \sqrt {1-c^2 x^2}-3 \arcsin (c x)\right )}{48 (-1+c x)^2}-\frac {9 c^2 \left (\sqrt {1-c^2 x^2}-\arcsin (c x)\right )}{16 (-1+c x)}-\frac {9 c^3 \left (\sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{16 \left (c+c^2 x\right )}+\frac {c x \sqrt {1-c^2 x^2}+\arcsin (c x)}{2 x^2}-\frac {c^2 \left ((2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)\right )}{48 (1+c x)^2}+\frac {3}{2} c^3 \left (\frac {3 i \pi \arcsin (c x)}{2 c}-\frac {i \arcsin (c x)^2}{2 c}+\frac {2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )}{c}-\frac {\pi \log \left (1+i e^{i \arcsin (c x)}\right )}{c}+\frac {2 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )}{c}-\frac {2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )}{c}+\frac {\pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )}{c}-\frac {2 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c}\right )+\frac {3}{2} c^3 \left (\frac {i \pi \arcsin (c x)}{2 c}-\frac {i \arcsin (c x)^2}{2 c}+\frac {2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )}{c}+\frac {\pi \log \left (1-i e^{i \arcsin (c x)}\right )}{c}+\frac {2 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )}{c}-\frac {2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )}{c}-\frac {\pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )}{c}-\frac {2 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c}\right )-3 c^2 \left (\arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-\frac {1}{2} i \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )\right )\right )}{d^3}-\frac {b^2 c^2 \left (\frac {i \pi ^3}{8}-\frac {1}{12 \left (1-c^2 x^2\right )}+\frac {c x \arcsin (c x)}{6 \left (1-c^2 x^2\right )^{3/2}}+\frac {7 c x \arcsin (c x)}{3 \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \arcsin (c x)}{c x}+\frac {\arcsin (c x)^2}{2 c^2 x^2}-\frac {\arcsin (c x)^2}{4 \left (1-c^2 x^2\right )^2}-\frac {\arcsin (c x)^2}{1-c^2 x^2}-2 i \arcsin (c x)^3-3 \arcsin (c x)^2 \log \left (1-e^{-2 i \arcsin (c x)}\right )+3 \arcsin (c x)^2 \log \left (1+e^{2 i \arcsin (c x)}\right )-\log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+\frac {4}{3} \log \left (\sqrt {1-c^2 x^2}\right )-3 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c x)}\right )-3 i \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )\right )}{d^3} \]
-1/2*a^2/(d^3*x^2) + (a^2*c^2)/(4*d^3*(-1 + c^2*x^2)^2) - (a^2*c^2)/(d^3*( -1 + c^2*x^2)) + (3*a^2*c^2*Log[x])/d^3 - (3*a^2*c^2*Log[1 - c^2*x^2])/(2* d^3) - (2*a*b*((c^2*((2 - c*x)*Sqrt[1 - c^2*x^2] - 3*ArcSin[c*x]))/(48*(-1 + c*x)^2) - (9*c^2*(Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(16*(-1 + c*x)) - ( 9*c^3*(Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/(16*(c + c^2*x)) + (c*x*Sqrt[1 - c^2*x^2] + ArcSin[c*x])/(2*x^2) - (c^2*((2 + c*x)*Sqrt[1 - c^2*x^2] + 3*Ar cSin[c*x]))/(48*(1 + c*x)^2) + (3*c^3*((((3*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*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))/2 + (3*c^3*(((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*ArcSin[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*ArcSin[c*x])])/c))/2 - 3*c^2*(ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - (I/2)*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c*x])]))))/d^3 - (b^2*c^2*((I/8)*Pi^3 - 1/(12 *(1 - c^2*x^2)) + (c*x*ArcSin[c*x])/(6*(1 - c^2*x^2)^(3/2)) + (7*c*x*ArcSi n[c*x])/(3*Sqrt[1 - c^2*x^2]) + (Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*x) + Ar cSin[c*x]^2/(2*c^2*x^2) - ArcSin[c*x]^2/(4*(1 - c^2*x^2)^2) - ArcSin[c*...
Time = 3.03 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.24, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.815, Rules used = {5204, 27, 5194, 27, 1578, 1195, 2009, 5208, 5162, 241, 5160, 240, 5208, 5160, 240, 5184, 4919, 3042, 4671, 3011, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle 3 c^2 \int \frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^3}dx+\frac {b c \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^{5/2}}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5194 |
| \(\displaystyle \frac {3 c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \left (-b c \int -\frac {8 c^4 x^4-12 c^2 x^2+3}{3 x \left (1-c^2 x^2\right )^2}dx+\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \left (\frac {1}{3} b c \int \frac {8 c^4 x^4-12 c^2 x^2+3}{x \left (1-c^2 x^2\right )^2}dx+\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {3 c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \left (\frac {1}{6} b c \int \frac {8 c^4 x^4-12 c^2 x^2+3}{x^2 \left (1-c^2 x^2\right )^2}dx^2+\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {3 c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \left (\frac {1}{6} b c \int \left (\frac {5 c^2}{c^2 x^2-1}-\frac {c^2}{\left (c^2 x^2-1\right )^2}+\frac {3}{x^2}\right )dx^2+\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^3}dx}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle \frac {3 c^2 \left (-\frac {1}{2} b c \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{5/2}}dx+\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle \frac {3 c^2 \left (-\frac {1}{2} b c \left (\frac {2}{3} \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}dx-\frac {1}{3} b c \int \frac {x}{\left (1-c^2 x^2\right )^2}dx+\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}\right )+\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {3 c^2 \left (-\frac {1}{2} b c \left (\frac {2}{3} \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )+\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 5160 |
| \(\displaystyle \frac {3 c^2 \left (-\frac {1}{2} b c \left (\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-b c \int \frac {x}{1-c^2 x^2}dx\right )+\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )+\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {3 c^2 \left (\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}-\frac {1}{2} b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle \frac {3 c^2 \left (-b c \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}dx+\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}dx+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}-\frac {1}{2} b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 5160 |
| \(\displaystyle \frac {3 c^2 \left (-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-b c \int \frac {x}{1-c^2 x^2}dx\right )+\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}dx+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}-\frac {1}{2} b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {3 c^2 \left (\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}dx+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 5184 |
| \(\displaystyle \frac {3 c^2 \left (\int \frac {(a+b \arcsin (c x))^2}{c x \sqrt {1-c^2 x^2}}d\arcsin (c x)+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 4919 |
| \(\displaystyle \frac {3 c^2 \left (2 \int (a+b \arcsin (c x))^2 \csc (2 \arcsin (c x))d\arcsin (c x)+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 c^2 \left (2 \int (a+b \arcsin (c x))^2 \csc (2 \arcsin (c x))d\arcsin (c x)+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {3 c^2 \left (2 \left (-b \int (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)+b \int (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {3 c^2 \left (2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {3 c^2 \left (2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3 c^2 \left (2 \left (-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )\right )\right )+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}+\frac {(a+b \arcsin (c x))^2}{4 \left (1-c^2 x^2\right )^2}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )\right )}{d^3}-\frac {(a+b \arcsin (c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {8 c^2 x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 c^2 x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
-1/2*(a + b*ArcSin[c*x])^2/(d^3*x^2*(1 - c^2*x^2)^2) + (b*c*(-((a + b*ArcS in[c*x])/(x*(1 - c^2*x^2)^(3/2))) + (4*c^2*x*(a + b*ArcSin[c*x]))/(3*(1 - c^2*x^2)^(3/2)) + (8*c^2*x*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) + (b *c*((-1 + c^2*x^2)^(-1) + 3*Log[x^2] + 5*Log[1 - c^2*x^2]))/6))/d^3 + (3*c ^2*((a + b*ArcSin[c*x])^2/(4*(1 - c^2*x^2)^2) + (a + b*ArcSin[c*x])^2/(2*( 1 - c^2*x^2)) - b*c*((x*(a + b*ArcSin[c*x]))/Sqrt[1 - c^2*x^2] + (b*Log[1 - c^2*x^2])/(2*c)) - (b*c*(-1/6*b/(c*(1 - c^2*x^2)) + (x*(a + b*ArcSin[c*x ]))/(3*(1 - c^2*x^2)^(3/2)) + (2*((x*(a + b*ArcSin[c*x]))/Sqrt[1 - c^2*x^2 ] + (b*Log[1 - c^2*x^2])/(2*c)))/3))/2 + 2*(-((a + b*ArcSin[c*x])^2*ArcTan h[E^((2*I)*ArcSin[c*x])]) + b*((I/2)*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2 *I)*ArcSin[c*x])] - (b*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/4) - b*((I/2)*( a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] - (b*PolyLog[3, E^((2 *I)*ArcSin[c*x])])/4))))/d^3
3.3.8.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
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_.) + (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[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n , x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[x*((a + b*ArcSin[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp[b *c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcSin[c*x ])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 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_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[1/d Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcSi n[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_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) , x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin [c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[Sim plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) ) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Simp[b* c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*( 1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp[b*c *(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)* (1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b , c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && !G tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
Int[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (427 ) = 854\).
Time = 0.51 (sec) , antiderivative size = 921, normalized size of antiderivative = 2.29
| method | result | size |
| derivativedivides | \(c^{2} \left (-\frac {a^{2} \left (\frac {1}{2 c^{2} x^{2}}-3 \ln \left (c x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {9}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {9}{16 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b^{2} \left (\frac {16 i \arcsin \left (c x \right ) c^{6} x^{6}-16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{5} x^{5}+18 \arcsin \left (c x \right )^{2} x^{4} c^{4}-32 i \arcsin \left (c x \right ) x^{4} c^{4}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-27 \arcsin \left (c x \right )^{2} x^{2} c^{2}+16 i \arcsin \left (c x \right ) x^{2} c^{2}+c^{4} x^{4}+12 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +6 \arcsin \left (c x \right )^{2}-c^{2} x^{2}}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\frac {7 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3}-\frac {8 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}-\frac {2 a b \left (\frac {8 i c^{6} x^{6}-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+18 c^{4} x^{4} \arcsin \left (c x \right )-16 i c^{4} x^{4}+3 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-27 c^{2} x^{2} \arcsin \left (c x \right )+8 i c^{2} x^{2}+6 c x \sqrt {-c^{2} x^{2}+1}+6 \arcsin \left (c x \right )}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+3 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}\right )\) | \(921\) |
| default | \(c^{2} \left (-\frac {a^{2} \left (\frac {1}{2 c^{2} x^{2}}-3 \ln \left (c x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {9}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {9}{16 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b^{2} \left (\frac {16 i \arcsin \left (c x \right ) c^{6} x^{6}-16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{5} x^{5}+18 \arcsin \left (c x \right )^{2} x^{4} c^{4}-32 i \arcsin \left (c x \right ) x^{4} c^{4}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-27 \arcsin \left (c x \right )^{2} x^{2} c^{2}+16 i \arcsin \left (c x \right ) x^{2} c^{2}+c^{4} x^{4}+12 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +6 \arcsin \left (c x \right )^{2}-c^{2} x^{2}}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\frac {7 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3}-\frac {8 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}-\frac {2 a b \left (\frac {8 i c^{6} x^{6}-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+18 c^{4} x^{4} \arcsin \left (c x \right )-16 i c^{4} x^{4}+3 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-27 c^{2} x^{2} \arcsin \left (c x \right )+8 i c^{2} x^{2}+6 c x \sqrt {-c^{2} x^{2}+1}+6 \arcsin \left (c x \right )}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+3 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}\right )\) | \(921\) |
| parts | \(-\frac {a^{2} \left (\frac {1}{2 x^{2}}-3 c^{2} \ln \left (x \right )-\frac {c^{2}}{16 \left (c x -1\right )^{2}}+\frac {9 c^{2}}{16 \left (c x -1\right )}+\frac {3 c^{2} \ln \left (c x -1\right )}{2}-\frac {c^{2}}{16 \left (c x +1\right )^{2}}-\frac {9 c^{2}}{16 \left (c x +1\right )}+\frac {3 c^{2} \ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b^{2} c^{2} \left (\frac {16 i \arcsin \left (c x \right ) c^{6} x^{6}-16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{5} x^{5}+18 \arcsin \left (c x \right )^{2} x^{4} c^{4}-32 i \arcsin \left (c x \right ) x^{4} c^{4}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-27 \arcsin \left (c x \right )^{2} x^{2} c^{2}+16 i \arcsin \left (c x \right ) x^{2} c^{2}+c^{4} x^{4}+12 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +6 \arcsin \left (c x \right )^{2}-c^{2} x^{2}}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\frac {7 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3}-\frac {8 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}-\frac {2 a b \,c^{2} \left (\frac {8 i c^{6} x^{6}-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+18 c^{4} x^{4} \arcsin \left (c x \right )-16 i c^{4} x^{4}+3 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-27 c^{2} x^{2} \arcsin \left (c x \right )+8 i c^{2} x^{2}+6 c x \sqrt {-c^{2} x^{2}+1}+6 \arcsin \left (c x \right )}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+3 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}\) | \(939\) |
c^2*(-a^2/d^3*(1/2/c^2/x^2-3*ln(c*x)-1/16/(c*x-1)^2+9/16/(c*x-1)+3/2*ln(c* x-1)-1/16/(c*x+1)^2-9/16/(c*x+1)+3/2*ln(c*x+1))-b^2/d^3*(1/12/(c^4*x^4-2*c ^2*x^2+1)/c^2/x^2*(16*I*arcsin(c*x)*c^6*x^6-16*(-c^2*x^2+1)^(1/2)*arcsin(c *x)*c^5*x^5+18*arcsin(c*x)^2*x^4*c^4-32*I*arcsin(c*x)*c^4*x^4+6*(-c^2*x^2+ 1)^(1/2)*arcsin(c*x)*c^3*x^3-27*arcsin(c*x)^2*x^2*c^2+16*I*arcsin(c*x)*c^2 *x^2+c^4*x^4+12*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x*c+6*arcsin(c*x)^2-c^2*x^2 )-ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)+7/3*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-8/ 3*ln(I*c*x+(-c^2*x^2+1)^(1/2))-ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-3*arcsin(c*x )^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+6*I*arcsin(c*x)*polylog(2,-I*c*x-(-c^2* x^2+1)^(1/2))-6*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+3*arcsin(c*x)^2*ln(1+ (I*c*x+(-c^2*x^2+1)^(1/2))^2)-3*I*arcsin(c*x)*polylog(2,-(I*c*x+(-c^2*x^2+ 1)^(1/2))^2)+3/2*polylog(3,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-3*arcsin(c*x)^2* ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+6*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1 )^(1/2))-6*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2)))-2*a*b/d^3*(1/12/(c^4*x^4-2 *c^2*x^2+1)/c^2/x^2*(8*I*c^6*x^6-8*c^5*x^5*(-c^2*x^2+1)^(1/2)+18*c^4*x^4*a rcsin(c*x)-16*I*c^4*x^4+3*c^3*x^3*(-c^2*x^2+1)^(1/2)-27*c^2*x^2*arcsin(c*x )+8*I*c^2*x^2+6*c*x*(-c^2*x^2+1)^(1/2)+6*arcsin(c*x))-3*arcsin(c*x)*ln(1+I *c*x+(-c^2*x^2+1)^(1/2))+3*I*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+3*arcsin (c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-3/2*I*polylog(2,-(I*c*x+(-c^2*x^2 +1)^(1/2))^2)-3*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+3*I*polylog(...
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{3}} \,d x } \]
integral(-(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/(c^6*d^3*x^9 - 3*c ^4*d^3*x^7 + 3*c^2*d^3*x^5 - d^3*x^3), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2}}{c^{6} x^{9} - 3 c^{4} x^{7} + 3 c^{2} x^{5} - x^{3}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{9} - 3 c^{4} x^{7} + 3 c^{2} x^{5} - x^{3}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{9} - 3 c^{4} x^{7} + 3 c^{2} x^{5} - x^{3}}\, dx}{d^{3}} \]
-(Integral(a**2/(c**6*x**9 - 3*c**4*x**7 + 3*c**2*x**5 - x**3), x) + Integ ral(b**2*asin(c*x)**2/(c**6*x**9 - 3*c**4*x**7 + 3*c**2*x**5 - x**3), x) + Integral(2*a*b*asin(c*x)/(c**6*x**9 - 3*c**4*x**7 + 3*c**2*x**5 - x**3), x))/d**3
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{3}} \,d x } \]
-1/4*a^2*((6*c^4*x^4 - 9*c^2*x^2 + 2)/(c^4*d^3*x^6 - 2*c^2*d^3*x^4 + d^3*x ^2) + 6*c^2*log(c*x + 1)/d^3 + 6*c^2*log(c*x - 1)/d^3 - 12*c^2*log(x)/d^3) - integrate((b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*arc tan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3* c^2*d^3*x^5 - d^3*x^3), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^3\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]