Optimal. Leaf size=53 \[ \frac {i (1-i a x)^{1+2 p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p}{a (1+2 p)} \]
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Rubi [A]
time = 0.05, 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 = {5184, 5181, 32}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 5181
Rule 5184
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 = 0.02, size = 39, normalized size = 0.74 \begin {gather*} \frac {e^{-2 i p \text {ArcTan}(a x)} (i+a x) \left (c+a^2 c x^2\right )^p}{a+2 a p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 41, normalized size = 0.77
method | result | size |
gosper | \(\frac {\left (a x +i\right ) \left (a^{2} c \,x^{2}+c \right )^{p} {\mathrm e}^{-2 i p \arctan \left (a x \right )}}{a \left (1+2 p \right )}\) | \(41\) |
risch | \(\frac {\left (a x +i\right ) {\mathrm e}^{\frac {p \left (i \mathrm {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right ) \mathrm {csgn}\left (i c \left (a x +i\right ) \left (a x -i\right )\right )^{2} \pi +i \mathrm {csgn}\left (a x -i\right )^{2} \pi +i \mathrm {csgn}\left (a x +i\right )^{2} \pi -i \mathrm {csgn}\left (a x -i\right )^{2} \mathrm {csgn}\left (i \left (a x -i\right )\right ) \pi -i \mathrm {csgn}\left (a x +i\right ) \mathrm {csgn}\left (i \left (a x +i\right )\right ) \pi +i \mathrm {csgn}\left (a x -i\right ) \mathrm {csgn}\left (i \left (a x -i\right )\right ) \pi -i \mathrm {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right ) \mathrm {csgn}\left (i c \left (a x +i\right ) \left (a x -i\right )\right ) \mathrm {csgn}\left (i c \right ) \pi -i \mathrm {csgn}\left (a x +i\right )^{3} \pi -i \mathrm {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right ) \mathrm {csgn}\left (i \left (a x +i\right )\right ) \mathrm {csgn}\left (i \left (a x -i\right )\right ) \pi -i \mathrm {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right )^{3} \pi +i \mathrm {csgn}\left (a x +i\right )^{2} \mathrm {csgn}\left (i \left (a x +i\right )\right ) \pi -i \mathrm {csgn}\left (i c \left (a x +i\right ) \left (a x -i\right )\right )^{3} \pi +i \mathrm {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right )^{2} \mathrm {csgn}\left (i \left (a x +i\right )\right ) \pi +i \mathrm {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right )^{2} \mathrm {csgn}\left (i \left (a x -i\right )\right ) \pi -i \mathrm {csgn}\left (a x -i\right )^{3} \pi -2 i \pi +i \mathrm {csgn}\left (i c \left (a x +i\right ) \left (a x -i\right )\right )^{2} \mathrm {csgn}\left (i c \right ) \pi +2 \ln \left (c \right )+4 \ln \left (a x +i\right )\right )}{2}}}{\left (1+2 p \right ) a}\) | \(438\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 76, normalized size = 1.43 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.34, size = 42, normalized size = 0.79 \begin {gather*} \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} \end {gather*}
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 {gather*} \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 {a x \left (a^{2} c x^{2} + c\right )^{p}}{2 a p e^{2 i p \operatorname {atan}{\left (a x \right )}} + a e^{2 i p \operatorname {atan}{\left (a x \right )}}} + \frac {i \left (a^{2} c x^{2} + c\right )^{p}}{2 a p e^{2 i p \operatorname {atan}{\left (a x \right )}} + a e^{2 i p \operatorname {atan}{\left (a x \right )}}} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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