Optimal. Leaf size=90 \[ \frac{i 2^{p+i} (1-i a x)^{p+(1-i)} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \, _2F_1\left (-p-i,p+(1-i);p+(2-i);\frac{1}{2} (1-i a x)\right )}{a (p+(1-i))} \]
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Rubi [A] time = 0.0675013, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5076, 5073, 69} \[ \frac{i 2^{p+i} (1-i a x)^{p+(1-i)} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \, _2F_1\left (-p-i,p+(1-i);p+(2-i);\frac{1}{2} (1-i a x)\right )}{a (p+(1-i))} \]
Antiderivative was successfully verified.
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Rule 5076
Rule 5073
Rule 69
Rubi steps
\begin{align*} \int e^{-2 \tan ^{-1}(a x)} \left (c+a^2 c x^2\right )^p \, dx &=\left (\left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p\right ) \int e^{-2 \tan ^{-1}(a x)} \left (1+a^2 x^2\right )^p \, dx\\ &=\left (\left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p\right ) \int (1-i a x)^{-i+p} (1+i a x)^{i+p} \, dx\\ &=\frac{i 2^{i+p} (1-i a x)^{(1-i)+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \, _2F_1\left (-i-p,(1-i)+p;(2-i)+p;\frac{1}{2} (1-i a x)\right )}{a ((1-i)+p)}\\ \end{align*}
Mathematica [A] time = 0.0222456, size = 90, normalized size = 1. \[ \frac{i 2^{p+i} (1-i a x)^{p+(1-i)} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \, _2F_1\left (-p-i,p+(1-i);p+(2-i);\frac{1}{2} (1-i a x)\right )}{a (p+(1-i))} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.326, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({a}^{2}c{x}^{2}+c \right ) ^{p}}{{{\rm e}^{2\,\arctan \left ( ax \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 )}^{p} 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 )}^{p} 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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{p} 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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