Integrand size = 27, antiderivative size = 296 \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(a+b \arcsin (c x))^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}+\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )}{2 d^3} \]
[Out]
Time = 0.34 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {4793, 4769, 4504, 4268, 2611, 2320, 6724, 4745, 266, 4747, 267} \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=-\frac {2 \text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{d^3}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {(a+b \arcsin (c x))^2}{2 d^3 \left (1-c^2 x^2\right )}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^3}-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^3}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )}{2 d^3}+\frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3} \]
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Rule 266
Rule 267
Rule 2320
Rule 2611
Rule 4268
Rule 4504
Rule 4745
Rule 4747
Rule 4769
Rule 4793
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(b c) \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{2 d^3}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^2} \, dx}{d} \\ & = -\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {(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 \left (1-c^2 x^2\right )}-\frac {(b c) \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^3}-\frac {(b c) \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^3}+\frac {\left (b^2 c^2\right ) \int \frac {x}{\left (1-c^2 x^2\right )^2} \, dx}{6 d^3}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )} \, dx}{d^2} \\ & = \frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(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 \left (1-c^2 x^2\right )}+\frac {\text {Subst}\left (\int (a+b x)^2 \csc (x) \sec (x) \, dx,x,\arcsin (c x)\right )}{d^3}+\frac {\left (b^2 c^2\right ) \int \frac {x}{1-c^2 x^2} \, dx}{3 d^3}+\frac {\left (b^2 c^2\right ) \int \frac {x}{1-c^2 x^2} \, dx}{d^3} \\ & = \frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(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 \left (1-c^2 x^2\right )}-\frac {2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}+\frac {2 \text {Subst}\left (\int (a+b x)^2 \csc (2 x) \, dx,x,\arcsin (c x)\right )}{d^3} \\ & = \frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(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 \left (1-c^2 x^2\right )}-\frac {2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}-\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^3}+\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1+e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^3} \\ & = \frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(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 \left (1-c^2 x^2\right )}-\frac {2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}+\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^3}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^3} \\ & = \frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(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 \left (1-c^2 x^2\right )}-\frac {2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}+\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{2 d^3}+\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{2 d^3} \\ & = \frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(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 \left (1-c^2 x^2\right )}-\frac {2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}+\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )}{2 d^3} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(756\) vs. \(2(296)=592\).
Time = 4.56 (sec) , antiderivative size = 756, normalized size of antiderivative = 2.55 \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\frac {\frac {6 a^2}{\left (-1+c^2 x^2\right )^2}-\frac {12 a^2}{-1+c^2 x^2}+\frac {15 a b \left (\sqrt {1-c^2 x^2}-\arcsin (c x)\right )}{-1+c x}+\frac {15 a b \left (\sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{1+c x}+\frac {a b \left ((-2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)\right )}{(-1+c x)^2}+\frac {a b \left ((2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)\right )}{(1+c x)^2}+48 a b \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+24 a^2 \log (c x)-12 a^2 \log \left (1-c^2 x^2\right )+12 a b \left (i \arcsin (c x)^2+\arcsin (c x) \left (-3 i \pi -4 \log \left (1+i e^{i \arcsin (c x)}\right )\right )+2 \pi \left (-2 \log \left (1+e^{-i \arcsin (c x)}\right )+\log \left (1+i e^{i \arcsin (c x)}\right )+2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )-\log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )+4 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )\right )+12 a b \left (i \arcsin (c x)^2+\arcsin (c x) \left (-i \pi -4 \log \left (1-i e^{i \arcsin (c x)}\right )\right )+2 \pi \left (-2 \log \left (1+e^{-i \arcsin (c x)}\right )-\log \left (1-i e^{i \arcsin (c x)}\right )+2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+\log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )+4 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-24 i a b \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-b^2 \left (i \pi ^3+\frac {2}{-1+c^2 x^2}+\frac {4 c x \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}+\frac {32 c x \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {6 \arcsin (c x)^2}{\left (-1+c^2 x^2\right )^2}+\frac {12 \arcsin (c x)^2}{-1+c^2 x^2}-16 i \arcsin (c x)^3-24 \arcsin (c x)^2 \log \left (1-e^{-2 i \arcsin (c x)}\right )+24 \arcsin (c x)^2 \log \left (1+e^{2 i \arcsin (c x)}\right )+16 \log \left (1-c^2 x^2\right )-24 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c x)}\right )-24 i \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-12 \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c x)}\right )+12 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )\right )}{24 d^3} \]
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[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 758 vs. \(2 (324 ) = 648\).
Time = 0.44 (sec) , antiderivative size = 759, normalized size of antiderivative = 2.56
method | result | size |
parts | \(-\frac {a^{2} \left (-\ln \left (x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b^{2} \left (\frac {16 i \arcsin \left (c x \right ) x^{4} c^{4}-16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-32 i \arcsin \left (c x \right ) x^{2} c^{2}+18 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -9 \arcsin \left (c x \right )^{2}+16 i \arcsin \left (c x \right )+c^{2} x^{2}-1}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}+\frac {4 \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}-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \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^{4} x^{4}-8 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+6 c^{2} x^{2} \arcsin \left (c x \right )-16 i c^{2} x^{2}+9 c x \sqrt {-c^{2} x^{2}+1}-9 \arcsin \left (c x \right )+8 i}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}\) | \(759\) |
derivativedivides | \(-\frac {a^{2} \left (-\ln \left (c x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b^{2} \left (\frac {16 i \arcsin \left (c x \right ) x^{4} c^{4}-16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-32 i \arcsin \left (c x \right ) x^{2} c^{2}+18 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -9 \arcsin \left (c x \right )^{2}+16 i \arcsin \left (c x \right )+c^{2} x^{2}-1}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}+\frac {4 \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}-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \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^{4} x^{4}-8 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+6 c^{2} x^{2} \arcsin \left (c x \right )-16 i c^{2} x^{2}+9 c x \sqrt {-c^{2} x^{2}+1}-9 \arcsin \left (c x \right )+8 i}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}\) | \(761\) |
default | \(-\frac {a^{2} \left (-\ln \left (c x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b^{2} \left (\frac {16 i \arcsin \left (c x \right ) x^{4} c^{4}-16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-32 i \arcsin \left (c x \right ) x^{2} c^{2}+18 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -9 \arcsin \left (c x \right )^{2}+16 i \arcsin \left (c x \right )+c^{2} x^{2}-1}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}+\frac {4 \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}-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \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^{4} x^{4}-8 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+6 c^{2} x^{2} \arcsin \left (c x \right )-16 i c^{2} x^{2}+9 c x \sqrt {-c^{2} x^{2}+1}-9 \arcsin \left (c x \right )+8 i}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}\) | \(761\) |
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\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx}{d^{3}} \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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