Optimal. Leaf size=47 \[ \frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a c^2}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2} \]
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Rubi [A] time = 0.0695104, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4904, 3312, 3304, 3352} \[ \frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a c^2}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2} \]
Antiderivative was successfully verified.
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Rule 4904
Rule 3312
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos ^2(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c^2}\\ &=\frac{\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 )}{a c^2}\\ &=\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2}+\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{2 a c^2}\\ &=\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2}+\frac{\operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a c^2}\\ &=\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2}+\frac{\sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a c^2}\\ \end{align*}
Mathematica [A] time = 0.232991, size = 43, normalized size = 0.91 \[ \frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )+2 \sqrt{\tan ^{-1}(a x)}}{2 a c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.138, size = 38, normalized size = 0.8 \begin{align*}{\frac{\sqrt{\pi }}{2\,a{c}^{2}}{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) }+{\frac{1}{a{c}^{2}}\sqrt{\arctan \left ( ax \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{1}{a^{4} x^{4} \sqrt{\operatorname{atan}{\left (a x \right )}} + 2 a^{2} x^{2} \sqrt{\operatorname{atan}{\left (a x \right )}} + \sqrt{\operatorname{atan}{\left (a x \right )}}}\, dx}{c^{2}} \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{1}{{\left (a^{2} c x^{2} + c\right )}^{2} \sqrt{\arctan \left (a x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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