Optimal. Leaf size=53 \[ \frac {i (1-i a x)^{2 p+1} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p}{a (2 p+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5076, 5073, 32} \[ \frac {i (1-i a x)^{2 p+1} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p}{a (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 5073
Rule 5076
Rubi steps
\begin {align*} \int e^{-2 i p \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 i p \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)^{2 p} \, dx\\ &=\frac {i (1-i a x)^{1+2 p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p}{a (1+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 0.74 \[ \frac {(a x+i) \left (a^2 c x^2+c\right )^p e^{-2 i p \tan ^{-1}(a x)}}{2 a p+a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 42, normalized size = 0.79 \[ \frac {{\left (a x + i\right )} {\left (a^{2} c x^{2} + c\right )}^{p} \left (-\frac {a x + i}{a x - i}\right )^{p}}{2 \, a p + a} \]
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 [A] time = 0.10, size = 41, normalized size = 0.77 \[ \frac {\left (a x +i\right ) \left (a^{2} c \,x^{2}+c \right )^{p} {\mathrm e}^{-2 i p \arctan \left (a x \right )}}{a \left (1+2 p \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 76, normalized size = 1.43 \[ \frac {{\left (a c^{p} x + i \, c^{p}\right )} {\left (a^{2} x^{2} + 1\right )}^{p} \cos \left (2 \, p \arctan \left (a x\right )\right ) - {\left (i \, a c^{p} x - c^{p}\right )} {\left (a^{2} x^{2} + 1\right )}^{p} \sin \left (2 \, p \arctan \left (a x\right )\right )}{2 \, a p + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 54, normalized size = 1.02 \[ \left (\frac {x\,{\mathrm {e}}^{-p\,\mathrm {atan}\left (a\,x\right )\,2{}\mathrm {i}}}{2\,p+1}+\frac {{\mathrm {e}}^{-p\,\mathrm {atan}\left (a\,x\right )\,2{}\mathrm {i}}\,1{}\mathrm {i}}{a\,\left (2\,p+1\right )}\right )\,{\left (c\,a^2\,x^2+c\right )}^p \]
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 \[ \begin {cases} \frac {x}{\sqrt {c}} & \text {for}\: a = 0 \wedge p = - \frac {1}{2} \\c^{p} x & \text {for}\: a = 0 \\\int \frac {e^{i \operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {i a x \left (a^{2} c x^{2} + c\right )^{p}}{2 i a p e^{2 i p \operatorname {atan}{\left (a x \right )}} + i a e^{2 i p \operatorname {atan}{\left (a x \right )}}} - \frac {\left (a^{2} c x^{2} + c\right )^{p}}{2 i a p e^{2 i p \operatorname {atan}{\left (a x \right )}} + i a e^{2 i p \operatorname {atan}{\left (a x \right )}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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