Optimal. Leaf size=248 \[ \frac{3 i b c^2 \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}-\frac{3 i b c^2 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}-\frac{6 c^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^3}-\frac{2 b c^3 x}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.346334, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4701, 4705, 4679, 4419, 4183, 2279, 2391, 191, 192, 271} \[ \frac{3 i b c^2 \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}-\frac{3 i b c^2 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}-\frac{6 c^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^3}-\frac{2 b c^3 x}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4701
Rule 4705
Rule 4679
Rule 4419
Rule 4183
Rule 2279
Rule 2391
Rule 191
Rule 192
Rule 271
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx &=-\frac{a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\left (3 c^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx+\frac{(b c) \int \frac{1}{x^2 \left (1-c^2 x^2\right )^{5/2}} \, dx}{2 d^3}\\ &=-\frac{b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}-\frac{\left (3 b c^3\right ) \int \frac{1}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{4 d^3}+\frac{\left (2 b c^3\right ) \int \frac{1}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^3}+\frac{\left (3 c^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx}{d}\\ &=-\frac{b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac{5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac{\left (b c^3\right ) \int \frac{1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^3}+\frac{\left (4 b c^3\right ) \int \frac{1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^3}-\frac{\left (3 b c^3\right ) \int \frac{1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^3}+\frac{\left (3 c^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d^2}\\ &=-\frac{b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac{5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{2 b c^3 x}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=-\frac{b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac{5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{2 b c^3 x}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac{\left (6 c^2\right ) \operatorname{Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=-\frac{b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac{5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{2 b c^3 x}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac{6 c^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac{\left (3 b c^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}+\frac{\left (3 b c^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=-\frac{b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac{5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{2 b c^3 x}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac{6 c^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}+\frac{\left (3 i b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}-\frac{\left (3 i b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}\\ &=-\frac{b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac{5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{2 b c^3 x}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac{6 c^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}+\frac{3 i b c^2 \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}-\frac{3 i b c^2 \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}\\ \end{align*}
Mathematica [A] time = 1.47888, size = 256, normalized size = 1.03 \[ -\frac{b c^2 \left (-18 i \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )+18 i \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+\frac{14 c x}{\sqrt{1-c^2 x^2}}+\frac{c x}{\left (1-c^2 x^2\right )^{3/2}}+\frac{6 \sqrt{1-c^2 x^2}}{c x}+\frac{12 \sin ^{-1}(c x)}{c^2 x^2-1}-\frac{3 \sin ^{-1}(c x)}{\left (c^2 x^2-1\right )^2}+\frac{6 \sin ^{-1}(c x)}{c^2 x^2}-36 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+36 \sin ^{-1}(c x) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )\right )+\frac{12 a c^2}{c^2 x^2-1}-\frac{3 a c^2}{\left (c^2 x^2-1\right )^2}+18 a c^2 \log \left (1-c^2 x^2\right )-36 a c^2 \log (x)+\frac{6 a}{x^2}}{12 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.264, size = 635, normalized size = 2.6 \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}{4} \, a{\left (\frac{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}} + \frac{6 \, c^{2} \log \left (c x + 1\right )}{d^{3}} + \frac{6 \, c^{2} \log \left (c x - 1\right )}{d^{3}} - \frac{12 \, c^{2} \log \left (x\right )}{d^{3}}\right )} - b \int \frac{\arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{c^{6} d^{3} x^{9} - 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} - d^{3} x^{3}}\,{d x} \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 \arcsin \left (c x\right ) + a}{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\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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