Optimal. Leaf size=60 \[ \frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0931624, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {4905, 4904, 3304, 3352} \[ \frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4905
Rule 4904
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{1}{\left (1+a^2 x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{c \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c \sqrt{c+a^2 c x^2}}\\ &=\frac{\left (2 \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 c \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{2 \pi } \sqrt{1+a^2 x^2} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.253614, size = 60, normalized size = 1. \[ \frac{\sqrt{2 \pi } \sqrt{a^2 c x^2+c} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a c^2 \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.719, size = 0, normalized size = 0. \begin{align*} \int{ \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{\arctan \left ( ax \right ) }}}}\, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \sqrt{\operatorname{atan}{\left (a x \right )}}}\, dx \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 )}^{\frac{3}{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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