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 = 0.09, antiderivative size = 115, normalized size of antiderivative = 1.00, 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 69
Rule 5073
Rule 5076
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*}
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Mathematica [A] time = 0.04, size = 115, normalized size = 1.00 \[ \frac {2^{-\frac {i n}{2}+p+1} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p (1-i a x)^{\frac {i n}{2}+p+1} \, _2F_1\left (\frac {i n}{2}-p,\frac {i n}{2}+p+1;\frac {i n}{2}+p+2;\frac {1}{2} (1-i a x)\right )}{a (n-2 i (p+1))} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (n \arctan \left (a x\right )\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (n \arctan \left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \left (a^{2} x^{2} + 1\right )\right )^{p} e^{n \operatorname {atan}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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