Optimal. Leaf size=101 \[ -\frac{2 \text{Unintegrable}\left (\frac{1}{x^2 \left (a^2 c x^2+c\right )^2 \sqrt{\tan ^{-1}(a x)}},x\right )}{a}-\frac{2}{a c^2 x \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{3 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{c^2}-\frac{6 \sqrt{\tan ^{-1}(a x)}}{c^2} \]
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Rubi [A] time = 0.199116, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac{2}{a c^2 x \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{2 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{a}-(6 a) \int \frac{1}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=-\frac{2}{a c^2 x \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{2 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{a}-\frac{6 \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac{2}{a c^2 x \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{2 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{a}-\frac{6 \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac{2}{a c^2 x \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{6 \sqrt{\tan ^{-1}(a x)}}{c^2}-\frac{2 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{a}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac{2}{a c^2 x \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{6 \sqrt{\tan ^{-1}(a x)}}{c^2}-\frac{2 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{a}-\frac{6 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{c^2}\\ &=-\frac{2}{a c^2 x \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{6 \sqrt{\tan ^{-1}(a x)}}{c^2}-\frac{3 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{c^2}-\frac{2 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{a}\\ \end{align*}
Mathematica [A] time = 4.5515, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.525, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({a}^{2}c{x}^{2}+c \right ) ^{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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{4} x^{5} \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )} + 2 a^{2} x^{3} \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )} + x \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )}}\, dx}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{2} x \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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