Optimal. Leaf size=439 \[ \frac{5 i b c^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2}-\frac{5 i b c^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2}+\frac{13 i b^2 c^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{3 d^2}-\frac{13 i b^2 c^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{3 d^2}-\frac{5 b^2 c^3 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )}{d^2}+\frac{5 b^2 c^3 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )}{d^2}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac{2 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2}}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x \left (1-c^2 x^2\right )}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 i c^3 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d^2}-\frac{26 b c^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2}-\frac{b^2 c^2}{3 d^2 x}+\frac{b^2 c^3 \tanh ^{-1}(c x)}{d^2} \]
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Rubi [A] time = 0.949499, antiderivative size = 439, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 15, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {4701, 4655, 4657, 4181, 2531, 2282, 6589, 4677, 206, 4705, 4709, 4183, 2279, 2391, 325} \[ \frac{5 i b c^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2}-\frac{5 i b c^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2}+\frac{13 i b^2 c^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{3 d^2}-\frac{13 i b^2 c^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{3 d^2}-\frac{5 b^2 c^3 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )}{d^2}+\frac{5 b^2 c^3 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )}{d^2}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac{2 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2}}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x \left (1-c^2 x^2\right )}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 i c^3 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d^2}-\frac{26 b c^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2}-\frac{b^2 c^2}{3 d^2 x}+\frac{b^2 c^3 \tanh ^{-1}(c x)}{d^2} \]
Antiderivative was successfully verified.
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Rule 4701
Rule 4655
Rule 4657
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rule 4677
Rule 206
Rule 4705
Rule 4709
Rule 4183
Rule 2279
Rule 2391
Rule 325
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac{1}{3} \left (5 c^2\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx+\frac{(2 b c) \int \frac{a+b \sin ^{-1}(c x)}{x^3 \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2}\\ &=-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x \left (1-c^2 x^2\right )}+\left (5 c^4\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^2} \, dx+\frac{\left (b^2 c^2\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx}{3 d^2}+\frac{\left (b c^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac{\left (10 b c^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2}\\ &=-\frac{b^2 c^2}{3 d^2 x}+\frac{13 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}+\frac{\left (b c^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx}{d^2}+\frac{\left (10 b c^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx}{3 d^2}+\frac{\left (b^2 c^4\right ) \int \frac{1}{1-c^2 x^2} \, dx}{3 d^2}-\frac{\left (b^2 c^4\right ) \int \frac{1}{1-c^2 x^2} \, dx}{d^2}-\frac{\left (10 b^2 c^4\right ) \int \frac{1}{1-c^2 x^2} \, dx}{3 d^2}-\frac{\left (5 b c^5\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac{\left (5 c^4\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx}{2 d}\\ &=-\frac{b^2 c^2}{3 d^2 x}-\frac{2 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac{4 b^2 c^3 \tanh ^{-1}(c x)}{d^2}+\frac{\left (5 c^3\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}+\frac{\left (10 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2}+\frac{\left (5 b^2 c^4\right ) \int \frac{1}{1-c^2 x^2} \, dx}{d^2}\\ &=-\frac{b^2 c^2}{3 d^2 x}-\frac{2 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac{5 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2}-\frac{26 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2}+\frac{b^2 c^3 \tanh ^{-1}(c x)}{d^2}-\frac{\left (5 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}+\frac{\left (5 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}-\frac{\left (b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}+\frac{\left (b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}-\frac{\left (10 b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2}+\frac{\left (10 b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2}\\ &=-\frac{b^2 c^2}{3 d^2 x}-\frac{2 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac{5 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2}-\frac{26 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2}+\frac{b^2 c^3 \tanh ^{-1}(c x)}{d^2}+\frac{5 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d^2}-\frac{5 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d^2}+\frac{\left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2}-\frac{\left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2}+\frac{\left (10 i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 d^2}-\frac{\left (10 i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 d^2}-\frac{\left (5 i b^2 c^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}+\frac{\left (5 i b^2 c^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}\\ &=-\frac{b^2 c^2}{3 d^2 x}-\frac{2 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac{5 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2}-\frac{26 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2}+\frac{b^2 c^3 \tanh ^{-1}(c x)}{d^2}+\frac{13 i b^2 c^3 \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{3 d^2}+\frac{5 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d^2}-\frac{5 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d^2}-\frac{13 i b^2 c^3 \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2}-\frac{\left (5 b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2}+\frac{\left (5 b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2}\\ &=-\frac{b^2 c^2}{3 d^2 x}-\frac{2 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac{5 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2}-\frac{26 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2}+\frac{b^2 c^3 \tanh ^{-1}(c x)}{d^2}+\frac{13 i b^2 c^3 \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{3 d^2}+\frac{5 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d^2}-\frac{5 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d^2}-\frac{13 i b^2 c^3 \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2}-\frac{5 b^2 c^3 \text{Li}_3\left (-i e^{i \sin ^{-1}(c x)}\right )}{d^2}+\frac{5 b^2 c^3 \text{Li}_3\left (i e^{i \sin ^{-1}(c x)}\right )}{d^2}\\ \end{align*}
Mathematica [B] time = 12.9268, size = 1514, normalized size = 3.45 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.382, size = 1019, normalized size = 2.3 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{12} \,{\left (\frac{15 \, c^{3} \log \left (c x + 1\right )}{d^{2}} - \frac{15 \, c^{3} \log \left (c x - 1\right )}{d^{2}} - \frac{2 \,{\left (15 \, c^{4} x^{4} - 10 \, c^{2} x^{2} - 2\right )}}{c^{2} d^{2} x^{5} - d^{2} x^{3}}\right )} a^{2} + \frac{15 \,{\left (b^{2} c^{5} x^{5} - b^{2} c^{3} x^{3}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} \log \left (c x + 1\right ) - 15 \,{\left (b^{2} c^{5} x^{5} - b^{2} c^{3} x^{3}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} \log \left (-c x + 1\right ) - 2 \,{\left (15 \, b^{2} c^{4} x^{4} - 10 \, b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + 2 \,{\left (c^{2} d^{2} x^{5} - d^{2} x^{3}\right )} \int \frac{12 \, a b \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) +{\left (15 \,{\left (b^{2} c^{6} x^{6} - b^{2} c^{4} x^{4}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (c x + 1\right ) - 15 \,{\left (b^{2} c^{6} x^{6} - b^{2} c^{4} x^{4}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (-c x + 1\right ) - 2 \,{\left (15 \, b^{2} c^{5} x^{5} - 10 \, b^{2} c^{3} x^{3} - 2 \, b^{2} c x\right )} \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^{8} - 2 \, c^{2} d^{2} x^{6} + d^{2} x^{4}}\,{d x}}{12 \,{\left (c^{2} d^{2} x^{5} - d^{2} x^{3}\right )}} \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{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}{c^{4} d^{2} x^{8} - 2 \, c^{2} d^{2} x^{6} + d^{2} x^{4}}, 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*} \frac{\int \frac{a^{2}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac{b^{2} \operatorname{asin}^{2}{\left (c x \right )}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac{2 a b \operatorname{asin}{\left (c x \right )}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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