Optimal. Leaf size=176 \[ -\frac{5}{3} i b^2 c^3 d \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+\frac{5}{3} i b^2 c^3 d \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-\frac{b c d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}+\frac{10}{3} b c^3 d \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{b^2 c^2 d}{3 x} \]
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Rubi [A] time = 0.376732, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {4695, 4627, 4709, 4183, 2279, 2391, 4693, 30} \[ -\frac{5}{3} i b^2 c^3 d \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+\frac{5}{3} i b^2 c^3 d \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-\frac{b c d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}+\frac{10}{3} b c^3 d \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{b^2 c^2 d}{3 x} \]
Antiderivative was successfully verified.
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Rule 4695
Rule 4627
Rule 4709
Rule 4183
Rule 2279
Rule 2391
Rule 4693
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} (2 b c d) \int \frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx-\frac{1}{3} \left (2 c^2 d\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx\\ &=-\frac{b c d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (b^2 c^2 d\right ) \int \frac{1}{x^2} \, dx-\frac{1}{3} \left (b c^3 d\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx-\frac{1}{3} \left (4 b c^3 d\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{b^2 c^2 d}{3 x}-\frac{b c d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\frac{1}{3} \left (b c^3 d\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac{1}{3} \left (4 b c^3 d\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{b^2 c^2 d}{3 x}-\frac{b c d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{10}{3} b c^3 d \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+\frac{1}{3} \left (b^2 c^3 d\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )-\frac{1}{3} \left (b^2 c^3 d\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\frac{1}{3} \left (4 b^2 c^3 d\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )-\frac{1}{3} \left (4 b^2 c^3 d\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{b^2 c^2 d}{3 x}-\frac{b c d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{10}{3} b c^3 d \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\frac{1}{3} \left (i b^2 c^3 d\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )+\frac{1}{3} \left (i b^2 c^3 d\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )-\frac{1}{3} \left (4 i b^2 c^3 d\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )+\frac{1}{3} \left (4 i b^2 c^3 d\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )\\ &=-\frac{b^2 c^2 d}{3 x}-\frac{b c d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{10}{3} b c^3 d \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\frac{5}{3} i b^2 c^3 d \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )+\frac{5}{3} i b^2 c^3 d \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.743432, size = 266, normalized size = 1.51 \[ \frac{d \left (-5 i b^2 c^3 x^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+5 i b^2 c^3 x^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )+3 a^2 c^2 x^2-a^2-a b c x \sqrt{1-c^2 x^2}+6 a b c^2 x^2 \sin ^{-1}(c x)+5 a b c^3 x^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )-2 a b \sin ^{-1}(c x)-b^2 c^2 x^2+3 b^2 c^2 x^2 \sin ^{-1}(c x)^2-b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-5 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )+5 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )-b^2 \sin ^{-1}(c x)^2\right )}{3 x^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.361, size = 291, normalized size = 1.7 \begin{align*}{\frac{{c}^{2}d{a}^{2}}{x}}-{\frac{d{a}^{2}}{3\,{x}^{3}}}+{\frac{{c}^{2}d{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{x}}-{\frac{dc{b}^{2}\arcsin \left ( cx \right ) }{3\,{x}^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{d{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{3\,{x}^{3}}}-{\frac{{c}^{2}d{b}^{2}}{3\,x}}+{\frac{5\,d{c}^{3}{b}^{2}\arcsin \left ( cx \right ) }{3}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{5\,i}{3}}{b}^{2}{c}^{3}d{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -{\frac{5\,d{c}^{3}{b}^{2}\arcsin \left ( cx \right ) }{3}\ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{5\,i}{3}}{b}^{2}{c}^{3}d{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +2\,{\frac{{c}^{2}dab\arcsin \left ( cx \right ) }{x}}-{\frac{2\,dab\arcsin \left ( cx \right ) }{3\,{x}^{3}}}-{\frac{dcab}{3\,{x}^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{5\,d{c}^{3}ab}{3}{\it Artanh} \left ({\frac{1}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} 2 \,{\left (c \log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\arcsin \left (c x\right )}{x}\right )} a b c^{2} d - \frac{1}{3} \,{\left ({\left (c^{2} \log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\sqrt{-c^{2} x^{2} + 1}}{x^{2}}\right )} c + \frac{2 \, \arcsin \left (c x\right )}{x^{3}}\right )} a b d + \frac{a^{2} c^{2} d}{x} - \frac{a^{2} d}{3 \, x^{3}} + \frac{2 \, x^{3} \int \frac{{\left (3 \, b^{2} c^{3} d x^{2} - b^{2} c d\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{c^{2} x^{5} - x^{3}}\,{d x} +{\left (3 \, b^{2} c^{2} d x^{2} - b^{2} d\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2}}{3 \, x^{3}} \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{a^{2} c^{2} d x^{2} - a^{2} d +{\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )}{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*} - d \left (\int - \frac{a^{2}}{x^{4}}\, dx + \int \frac{a^{2} c^{2}}{x^{2}}\, dx + \int - \frac{b^{2} \operatorname{asin}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int - \frac{2 a b \operatorname{asin}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{b^{2} c^{2} \operatorname{asin}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{2 a b c^{2} \operatorname{asin}{\left (c x \right )}}{x^{2}}\, dx\right ) \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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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