Optimal. Leaf size=69 \[ -\frac{\sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{a c \sqrt{a^2 c x^2+c}}-\frac{1}{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.204715, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4902, 4971, 4970, 3299} \[ -\frac{\sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{a c \sqrt{a^2 c x^2+c}}-\frac{1}{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4902
Rule 4971
Rule 4970
Rule 3299
Rubi steps
\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx &=-\frac{1}{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-a \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx\\ &=-\frac{1}{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-\frac{\left (a \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c \sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a c \sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-\frac{\sqrt{1+a^2 x^2} \text{Si}\left (\tan ^{-1}(a x)\right )}{a c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.100209, size = 53, normalized size = 0.77 \[ -\frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Si}\left (\tan ^{-1}(a x)\right )+1}{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.279, size = 212, normalized size = 3.1 \begin{align*}{\frac{{\frac{i}{2}}}{a{c}^{2}\arctan \left ( ax \right ) } \left ( \arctan \left ( ax \right ){\it Ei} \left ( 1,i\arctan \left ( ax \right ) \right ){x}^{2}{a}^{2}+{\it Ei} \left ( 1,i\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) +\sqrt{{a}^{2}{x}^{2}+1}xa+i\sqrt{{a}^{2}{x}^{2}+1} \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ({a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}-{\frac{{\frac{i}{2}}}{a{c}^{2}\arctan \left ( ax \right ) } \left ( \arctan \left ( ax \right ){\it Ei} \left ( 1,-i\arctan \left ( ax \right ) \right ){x}^{2}{a}^{2}+{\it Ei} \left ( 1,-i\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) +\sqrt{{a}^{2}{x}^{2}+1}xa-i\sqrt{{a}^{2}{x}^{2}+1} \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ({a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c}}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{2}}, x\right ) \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}} \operatorname{atan}^{2}{\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}} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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