Optimal. Leaf size=634 \[ \frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{3 b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{4 i b c^2 \sqrt{1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{3 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.927775, antiderivative size = 634, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 15, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.517, Rules used = {4701, 4705, 4713, 4709, 4183, 2531, 2282, 6589, 4657, 4181, 2279, 2391, 266, 63, 208} \[ \frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{3 b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{4 i b c^2 \sqrt{1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{3 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 4701
Rule 4705
Rule 4713
Rule 4709
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rule 4657
Rule 4181
Rule 2279
Rule 2391
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{1}{2} \left (3 c^2\right ) \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 (b c \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 c^2\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt{d-c^2 d x^2}} \, dx}{2 d}+\frac{\left (b^2 c^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x \sqrt{1-c^2 x^2}} \, dx}{d \sqrt{d-c^2 d x^2}}+\frac{\left (b c^3 \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 (3 b c^3 \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{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 c^2 \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt{1-c^2 x^2}} \, dx}{2 d \sqrt{d-c^2 d x^2}}+\frac{\left (b c^2 \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 (3 b c^2 \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 (b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{2 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{4 i b c^2 \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{\left (b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (3 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 c^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 (b^2 c^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 (3 b^2 c^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 (3 b^2 c^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{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{4 i b c^2 \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{3 c^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{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{\left (3 b c^2 \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 (3 b c^2 \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 (i b^2 c^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 (i b^2 c^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 (3 i b^2 c^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 (3 i b^2 c^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{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{4 i b c^2 \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{3 c^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{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 i b c^2 \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 c^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 c^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{3 i b c^2 \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 (3 i b^2 c^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 (3 i b^2 c^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{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{4 i b c^2 \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{3 c^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{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 i b c^2 \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 c^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 c^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{3 i b c^2 \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 (3 b^2 c^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 (3 b^2 c^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{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{4 i b c^2 \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{3 c^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{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 i b c^2 \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 c^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 c^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{3 i b c^2 \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{3 b^2 c^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{3 b^2 c^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 = 8.16105, size = 844, normalized size = 1.33 \[ \frac{3 a^2 \log (x) c^2}{2 d^{3/2}}-\frac{3 a^2 \log \left (d+\sqrt{-d \left (c^2 x^2-1\right )} \sqrt{d}\right ) c^2}{2 d^{3/2}}+\frac{b^2 \sqrt{1-c^2 x^2} \left (-\csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)^2+\sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)^2+\frac{8 \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)^2}{\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )}-\frac{8 \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)^2}{\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )}+8 \sin ^{-1}(c x)^2-4 \cot \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)-4 \tan \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)+8 \log \left (\tan \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-16 \left (\sin ^{-1}(c x) \left (\log \left (1-i e^{i \sin ^{-1}(c x)}\right )-\log \left (1+i e^{i \sin ^{-1}(c x)}\right )\right )+i \left (\text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )-\text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )\right )\right )+12 \left (\left (\log \left (1-e^{i \sin ^{-1}(c x)}\right )-\log \left (1+e^{i \sin ^{-1}(c x)}\right )\right ) \sin ^{-1}(c x)^2+2 i \left (\text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )\right ) \sin ^{-1}(c x)+2 \left (\text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )-\text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )\right )\right )\right ) c^2}{8 d \sqrt{d \left (1-c^2 x^2\right )}}+\frac{a b \left (6 i \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \sin \left (2 \sin ^{-1}(c x)\right )-6 i \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \sin \left (2 \sin ^{-1}(c x)\right )-\frac{6 \cos \left (2 \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)+3 \cos \left (3 \sin ^{-1}(c x)\right ) \log \left (1-e^{i \sin ^{-1}(c x)}\right ) \sin ^{-1}(c x)-3 \cos \left (3 \sin ^{-1}(c x)\right ) \log \left (1+e^{i \sin ^{-1}(c x)}\right ) \sin ^{-1}(c x)-2 \sin ^{-1}(c x)+2 \cos \left (3 \sin ^{-1}(c x)\right ) \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-2 \cos \left (3 \sin ^{-1}(c x)\right ) \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} \left (-3 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )+3 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )-2 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+2 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )+2 \sin \left (2 \sin ^{-1}(c x)\right )}{c x}\right ) c}{4 d x \sqrt{d \left (1-c^2 x^2\right )}}+\sqrt{-d \left (c^2 x^2-1\right )} \left (-\frac{c^2 a^2}{d^2 \left (c^2 x^2-1\right )}-\frac{a^2}{2 d^2 x^2}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.383, size = 1490, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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^{7} - 2 \, c^{2} d^{2} x^{5} + d^{2} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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