Optimal. Leaf size=102 \[ \frac {i 2^{p+\left (1-\frac {i}{2}\right )} (1-i a x)^{p+\left (1+\frac {i}{2}\right )} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \, _2F_1\left (\frac {i}{2}-p,p+\left (1+\frac {i}{2}\right );p+\left (2+\frac {i}{2}\right );\frac {1}{2} (1-i a x)\right )}{a (2 p+(2+i))} \]
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Rubi [A] time = 0.08, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5076, 5073, 69} \[ \frac {i 2^{p+\left (1-\frac {i}{2}\right )} (1-i a x)^{p+\left (1+\frac {i}{2}\right )} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \, _2F_1\left (\frac {i}{2}-p,p+\left (1+\frac {i}{2}\right );p+\left (2+\frac {i}{2}\right );\frac {1}{2} (1-i a x)\right )}{a (2 p+(2+i))} \]
Antiderivative was successfully verified.
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Rule 69
Rule 5073
Rule 5076
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 {i 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 ((2+i)+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 102, normalized size = 1.00 \[ \frac {i 2^{p-\frac {i}{2}} (1-i a x)^{p+\left (1+\frac {i}{2}\right )} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \, _2F_1\left (\frac {i}{2}-p,p+\left (1+\frac {i}{2}\right );p+\left (2+\frac {i}{2}\right );\frac {1}{2} (1-i a x)\right )}{a \left (p+\left (1+\frac {i}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (\arctan \left (a x\right )\right )}, x\right ) \]
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 = 0.31, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{p}\, dx \]
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 \[ \int {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (\arctan \left (a x\right )\right )}\,{d x} \]
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 \[ \int {\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^p \,d x \]
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 \[ \int \left (c \left (a^{2} x^{2} + 1\right )\right )^{p} e^{\operatorname {atan}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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