Optimal. Leaf size=316 \[ \frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \sqrt{d-c^2 d x^2}}-\frac{3 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2}}{2 d x \sqrt{d-c^2 d x^2}}-\frac{b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.441389, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4701, 4705, 4713, 4709, 4183, 2279, 2391, 206, 325} \[ \frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \sqrt{d-c^2 d x^2}}-\frac{3 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2}}{2 d x \sqrt{d-c^2 d x^2}}-\frac{b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 4701
Rule 4705
Rule 4713
Rule 4709
Rule 4183
Rule 2279
Rule 2391
Rule 206
Rule 325
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{1}{2} \left (3 c^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{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{1}{x^2 \left (1-c^2 x^2\right )} \, dx}{2 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 c^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{d-c^2 d x^2}} \, dx}{2 d}+\frac{\left (b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{1-c^2 x^2} \, dx}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (3 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{1-c^2 x^2} \, dx}{2 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \sqrt{d-c^2 d x^2}}-\frac{b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}}+\frac{\left (3 c^2 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx}{2 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \sqrt{d-c^2 d x^2}}-\frac{b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{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) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \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 ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{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 \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 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 \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \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 ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}}+\frac{\left (3 i b c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (3 i b c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \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 ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}}+\frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 2.11962, size = 404, normalized size = 1.28 \[ \frac{\frac{b \sqrt{d} \left (6 i c x \sin \left (2 \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-6 i c x \sin \left (2 \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )+\sqrt{1-c^2 x^2} \left (2 \left (\log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-\log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )+3 \sin ^{-1}(c x) \left (\log \left (1-e^{i \sin ^{-1}(c x)}\right )-\log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )\right )+2 \sin ^{-1}(c x)-2 \sin \left (2 \sin ^{-1}(c x)\right )-6 \sin ^{-1}(c x) \cos \left (2 \sin ^{-1}(c x)\right )-3 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right ) \cos \left (3 \sin ^{-1}(c x)\right )+3 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right ) \cos \left (3 \sin ^{-1}(c x)\right )-2 \cos \left (3 \sin ^{-1}(c x)\right ) \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+2 \cos \left (3 \sin ^{-1}(c x)\right ) \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )}{x^2 \sqrt{d-c^2 d x^2}}+\frac{4 a \sqrt{d} \left (3 c^2 x^2-1\right )}{x^2 \sqrt{d-c^2 d x^2}}-12 a c^2 \log \left (\sqrt{d} \sqrt{d-c^2 d x^2}+d\right )+12 a c^2 \log (x)}{8 d^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.23, size = 474, normalized size = 1.5 \begin{align*} -{\frac{a}{2\,d{x}^{2}}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}+{\frac{3\,a{c}^{2}}{2\,d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}-{\frac{3\,a{c}^{2}}{2}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{-{c}^{2}d{x}^{2}+d} \right ) } \right ){d}^{-{\frac{3}{2}}}}-{\frac{3\,b\arcsin \left ( cx \right ){c}^{2}}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{bc}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) x}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{b\arcsin \left ( cx \right ) }{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ){x}^{2}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{3\,b\arcsin \left ( cx \right ){c}^{2}}{ \left ( 2\,{c}^{2}{x}^{2}-2 \right ){d}^{2}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{2\,ib{c}^{2}}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}\arctan \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{{\frac{3\,i}{2}}b{c}^{2}}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}{\it dilog} \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{{\frac{3\,i}{2}}b{c}^{2}}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}{\it dilog} \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \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 \arcsin \left (c x\right ) + a\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(-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 )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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