Optimal. Leaf size=93 \[ \frac{2 \sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 x}{a c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}} \]
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Rubi [A] time = 0.178184, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4942, 4905, 4904, 3304, 3352} \[ \frac{2 \sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 x}{a c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 4942
Rule 4905
Rule 4904
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 x}{a c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{2 \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a}\\ &=-\frac{2 x}{a c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \int \frac{1}{\left (1+a^2 x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x}{a c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x}{a c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a^2 c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x}{a c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{2 \sqrt{2 \pi } \sqrt{1+a^2 x^2} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^2 c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [F] time = 0.341656, size = 0, normalized size = 0. \[ \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.83, size = 0, normalized size = 0. \begin{align*} \int{x \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}} \left ( \arctan \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \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(-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{x}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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