Optimal. Leaf size=53 \[ \frac{(a x+1) e^{2 \tan ^{-1}(a x)}}{4 a c^2 \left (a^2 x^2+1\right )}+\frac{e^{2 \tan ^{-1}(a x)}}{8 a c^2} \]
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Rubi [A] time = 0.057281, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5070, 5071} \[ \frac{(a x+1) e^{2 \tan ^{-1}(a x)}}{4 a c^2 \left (a^2 x^2+1\right )}+\frac{e^{2 \tan ^{-1}(a x)}}{8 a c^2} \]
Antiderivative was successfully verified.
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Rule 5070
Rule 5071
Rubi steps
\begin{align*} \int \frac{e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac{e^{2 \tan ^{-1}(a x)} (1+a x)}{4 a c^2 \left (1+a^2 x^2\right )}+\frac{\int \frac{e^{2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{4 c}\\ &=\frac{e^{2 \tan ^{-1}(a x)}}{8 a c^2}+\frac{e^{2 \tan ^{-1}(a x)} (1+a x)}{4 a c^2 \left (1+a^2 x^2\right )}\\ \end{align*}
Mathematica [C] time = 0.0184911, size = 55, normalized size = 1.04 \[ \frac{(1-i a x)^i (1+i a x)^{-i} \left (a^2 x^2+2 a x+3\right )}{8 c^2 \left (a^3 x^2+a\right )} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.037, size = 40, normalized size = 0.8 \begin{align*}{\frac{{{\rm e}^{2\,\arctan \left ( ax \right ) }} \left ({a}^{2}{x}^{2}+2\,ax+3 \right ) }{ \left ( 8\,{a}^{2}{x}^{2}+8 \right ) a{c}^{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{e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92883, size = 90, normalized size = 1.7 \begin{align*} \frac{{\left (a^{2} x^{2} + 2 \, a x + 3\right )} e^{\left (2 \, \arctan \left (a x\right )\right )}}{8 \,{\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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{e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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