Optimal. Leaf size=101 \[ \frac{2^{p+\left (1+\frac{i}{2}\right )} (1-i a x)^{p+\left (1-\frac{i}{2}\right )} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \, _2F_1\left (-p-\frac{i}{2},p+\left (1-\frac{i}{2}\right );p+\left (2-\frac{i}{2}\right );\frac{1}{2} (1-i a x)\right )}{a (-2 i p-(1+2 i))} \]
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Rubi [A] time = 0.0732523, antiderivative size = 101, 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} \[ \frac{2^{p+\left (1+\frac{i}{2}\right )} (1-i a x)^{p+\left (1-\frac{i}{2}\right )} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \, _2F_1\left (-p-\frac{i}{2},p+\left (1-\frac{i}{2}\right );p+\left (2-\frac{i}{2}\right );\frac{1}{2} (1-i a x)\right )}{a (-2 i p-(1+2 i))} \]
Antiderivative was successfully verified.
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Rule 5076
Rule 5073
Rule 69
Rubi steps
\begin{align*} \int e^{-\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^{-\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)^{-\frac{i}{2}+p} (1+i a x)^{\frac{i}{2}+p} \, dx\\ &=\frac{2^{\left (1+\frac{i}{2}\right )+p} (1-i a x)^{\left (1-\frac{i}{2}\right )+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \, _2F_1\left (-\frac{i}{2}-p,\left (1-\frac{i}{2}\right )+p;\left (2-\frac{i}{2}\right )+p;\frac{1}{2} (1-i a x)\right )}{a ((-1-2 i)-2 i p)}\\ \end{align*}
Mathematica [A] time = 0.0256839, size = 102, normalized size = 1.01 \[ \frac{i 2^{p+\frac{i}{2}} (1-i a x)^{p+\left (1-\frac{i}{2}\right )} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \, _2F_1\left (-p-\frac{i}{2},p+\left (1-\frac{i}{2}\right );p+\left (2-\frac{i}{2}\right );\frac{1}{2} (1-i a x)\right )}{a \left (p+\left (1-\frac{i}{2}\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.323, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({a}^{2}c{x}^{2}+c \right ) ^{p}}{{{\rm e}^{\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 (-\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 (-\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 (-\arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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