Optimal. Leaf size=51 \[ \frac{\left (\frac{1}{5}+\frac{2 i}{5}\right ) 2^{1-i} c (1-i a x)^{2+i} \, _2F_1\left (-1+i,2+i;3+i;\frac{1}{2} (1-i a x)\right )}{a} \]
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Rubi [A] time = 0.0256805, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {5073, 69} \[ \frac{\left (\frac{1}{5}+\frac{2 i}{5}\right ) 2^{1-i} c (1-i a x)^{2+i} \, _2F_1\left (-1+i,2+i;3+i;\frac{1}{2} (1-i a x)\right )}{a} \]
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 ) \, dx &=c \int (1-i a x)^{1+i} (1+i a x)^{1-i} \, dx\\ &=\frac{\left (\frac{1}{5}+\frac{2 i}{5}\right ) 2^{1-i} c (1-i a x)^{2+i} \, _2F_1\left (-1+i,2+i;3+i;\frac{1}{2} (1-i a x)\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0122826, size = 51, normalized size = 1. \[ \frac{\left (\frac{1}{5}+\frac{2 i}{5}\right ) 2^{1-i} c (1-i a x)^{2+i} \, _2F_1\left (-1+i,2+i;3+i;\frac{1}{2} (1-i a x)\right )}{a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{2\,\arctan \left ( ax \right ) }} \left ({a}^{2}c{x}^{2}+c \right ) \, 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 )} 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^{2} c x^{2} + c\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 \left (\int a^{2} x^{2} 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 )} 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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