Optimal. Leaf size=467 \[ \frac {2 i b \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {4 i b \sqrt {1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.57, antiderivative size = 467, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {4705, 4713, 4709, 4183, 2531, 2282, 6589, 4657, 4181, 2279, 2391} \[ \frac {2 i b \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {4 i b \sqrt {1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 4181
Rule 4183
Rule 4657
Rule 4705
Rule 4709
Rule 4713
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {d-c^2 d x^2}} \, dx}{d}-\frac {\left (2 b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {1-c^2 x^2}} \, dx}{d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {4 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {4 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {4 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {4 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {4 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.89, size = 667, normalized size = 1.43 \[ \frac {a^2 \sqrt {d} \sqrt {d-c^2 d x^2} \log (c x)-a^2 \sqrt {d} \sqrt {d-c^2 d x^2} \log \left (\sqrt {d} \sqrt {d-c^2 d x^2}+d\right )+a^2 d+2 a b d \left (i \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-i \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )+\sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )-\sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )+\sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )-\sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+\sin ^{-1}(c x)\right )+b^2 d \left (2 i \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-2 i \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-2 i \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )+2 i \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )-2 \sqrt {1-c^2 x^2} \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )+2 \sqrt {1-c^2 x^2} \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )+\sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2 \log \left (1-e^{i \sin ^{-1}(c x)}\right )-\sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2 \log \left (1+e^{i \sin ^{-1}(c x)}\right )-2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+\sin ^{-1}(c x)^2\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )}}{c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.40, size = 1096, normalized size = 2.35 \[ \frac {a^{2}}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {4 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -a^{2} {\left (\frac {\log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right )}{d^{\frac {3}{2}}} - \frac {1}{\sqrt {-c^{2} d x^{2} + d} d}\right )} + \sqrt {d} \int \frac {{\left (b^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________