Optimal. Leaf size=71 \[ \frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{4 a^4 c^3}-\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{8 a^4 c^3} \]
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Rubi [A] time = 0.12819, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4970, 4406, 3305, 3351} \[ \frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{4 a^4 c^3}-\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{8 a^4 c^3} \]
Antiderivative was successfully verified.
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Rule 4970
Rule 4406
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{x^3}{\left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos (x) \sin ^3(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 \sqrt{x}}-\frac{\sin (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^4 c^3}+\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^4 c^3}\\ &=-\frac{\operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{4 a^4 c^3}+\frac{\operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{2 a^4 c^3}\\ &=-\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{8 a^4 c^3}+\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{4 a^4 c^3}\\ \end{align*}
Mathematica [C] time = 0.143132, size = 131, normalized size = 1.85 \[ \frac{-2 \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \tan ^{-1}(a x)\right )-2 \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \tan ^{-1}(a x)\right )+\sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )+\sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )}{32 a^4 c^3 \sqrt{\tan ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.11, size = 54, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}\sqrt{\pi }}{16\,{c}^{3}{a}^{4}}{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) }+{\frac{\sqrt{\pi }}{4\,{c}^{3}{a}^{4}}{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{x^{3}}{a^{6} x^{6} \sqrt{\operatorname{atan}{\left (a x \right )}} + 3 a^{4} x^{4} \sqrt{\operatorname{atan}{\left (a x \right )}} + 3 a^{2} x^{2} \sqrt{\operatorname{atan}{\left (a x \right )}} + \sqrt{\operatorname{atan}{\left (a x \right )}}}\, dx}{c^{3}} \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{x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} \sqrt{\arctan \left (a x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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