Optimal. Leaf size=53 \[ \frac{\left (\frac{1}{5}+\frac{3 i}{5}\right ) 2^{1-i} c^2 (1-i a x)^{3+i} \, _2F_1\left (-2+i,3+i;4+i;\frac{1}{2} (1-i a x)\right )}{a} \]
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Rubi [A] time = 0.0389405, 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 = {5073, 69} \[ \frac{\left (\frac{1}{5}+\frac{3 i}{5}\right ) 2^{1-i} c^2 (1-i a x)^{3+i} \, _2F_1\left (-2+i,3+i;4+i;\frac{1}{2} (1-i a x)\right )}{a} \]
Antiderivative was successfully verified.
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Rule 5073
Rule 69
Rubi steps
\begin{align*} \int e^{2 \tan ^{-1}(a x)} \left (c+a^2 c x^2\right )^2 \, dx &=c^2 \int (1-i a x)^{2+i} (1+i a x)^{2-i} \, dx\\ &=\frac{\left (\frac{1}{5}+\frac{3 i}{5}\right ) 2^{1-i} c^2 (1-i a x)^{3+i} \, _2F_1\left (-2+i,3+i;4+i;\frac{1}{2} (1-i a x)\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0131317, size = 53, normalized size = 1. \[ \frac{\left (\frac{1}{5}+\frac{3 i}{5}\right ) 2^{1-i} c^2 (1-i a x)^{3+i} \, _2F_1\left (-2+i,3+i;4+i;\frac{1}{2} (1-i a x)\right )}{a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.191, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{2\,\arctan \left ( ax \right ) }} \left ({a}^{2}c{x}^{2}+c \right ) ^{2}\, dx \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{\left (a^{2} c x^{2} + c\right )}^{2} e^{\left (2 \, \arctan \left (a x\right )\right )}\,{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 ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} e^{\left (2 \, \arctan \left (a x\right )\right )}, 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*} c^{2} \left (\int 2 a^{2} x^{2} e^{2 \operatorname{atan}{\left (a x \right )}}\, dx + \int a^{4} x^{4} e^{2 \operatorname{atan}{\left (a x \right )}}\, dx + \int e^{2 \operatorname{atan}{\left (a x \right )}}\, dx\right ) \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{\left (a^{2} c x^{2} + c\right )}^{2} e^{\left (2 \, \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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