Optimal. Leaf size=115 \[ \frac{2^{-\frac{i n}{2}+p+1} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p (1-i a x)^{\frac{i n}{2}+p+1} \, _2F_1\left (\frac{i n}{2}-p,\frac{i n}{2}+p+1;\frac{i n}{2}+p+2;\frac{1}{2} (1-i a x)\right )}{a (n-2 i (p+1))} \]
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Rubi [A] time = 0.0867331, antiderivative size = 115, 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} \[ \frac{2^{-\frac{i n}{2}+p+1} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p (1-i a x)^{\frac{i n}{2}+p+1} \, _2F_1\left (\frac{i n}{2}-p,\frac{i n}{2}+p+1;\frac{i n}{2}+p+2;\frac{1}{2} (1-i a x)\right )}{a (n-2 i (p+1))} \]
Antiderivative was successfully verified.
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Rule 5076
Rule 5073
Rule 69
Rubi steps
\begin{align*} \int e^{n \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^{n \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 n}{2}+p} (1+i a x)^{-\frac{i n}{2}+p} \, dx\\ &=\frac{2^{1-\frac{i n}{2}+p} (1-i a x)^{1+\frac{i n}{2}+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \, _2F_1\left (\frac{i n}{2}-p,1+\frac{i n}{2}+p;2+\frac{i n}{2}+p;\frac{1}{2} (1-i a x)\right )}{a (n-2 i (1+p))}\\ \end{align*}
Mathematica [A] time = 0.0320095, size = 115, normalized size = 1. \[ \frac{2^{-\frac{i n}{2}+p+1} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p (1-i a x)^{\frac{i n}{2}+p+1} \, _2F_1\left (\frac{i n}{2}-p,\frac{i n}{2}+p+1;\frac{i n}{2}+p+2;\frac{1}{2} (1-i a x)\right )}{a (n-2 i (p+1))} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.31, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n\arctan \left ( ax \right ) }} \left ({a}^{2}c{x}^{2}+c \right ) ^{p}\, 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 (n \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 (n \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*} \int \left (c \left (a^{2} x^{2} + 1\right )\right )^{p} e^{n \operatorname{atan}{\left (a x \right )}}\, dx \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 (n \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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