Optimal. Leaf size=98 \[ \frac {(c+d x)^{1+m}}{2 a d (1+m)}+\frac {i 2^{-2-m} e^{-2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (1+m,\frac {2 i f (c+d x)}{d}\right )}{a f} \]
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Rubi [A]
time = 0.08, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3808, 2212}
\begin {gather*} \frac {(c+d x)^{m+1}}{2 a d (m+1)}+\frac {i 2^{-m-2} e^{-2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {2 i f (c+d x)}{d}\right )}{a f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3808
Rubi steps
\begin {align*} \int \frac {(c+d x)^m}{a+i a \tan (e+f x)} \, dx &=\frac {(c+d x)^{1+m}}{2 a d (1+m)}+\frac {\int e^{-2 i (e+f x)} (c+d x)^m \, dx}{2 a}\\ &=\frac {(c+d x)^{1+m}}{2 a d (1+m)}+\frac {i 2^{-2-m} e^{-2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 i f (c+d x)}{d}\right )}{a f}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(205\) vs. \(2(98)=196\).
time = 1.40, size = 205, normalized size = 2.09 \begin {gather*} \frac {2^{-2-m} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^m \left (\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} \sec (e+f x) \left (2^{1+m} f (c+d x) \left (\frac {i f (c+d x)}{d}\right )^m \left (\cos \left (e-\frac {c f}{d}\right )+i \sin \left (e-\frac {c f}{d}\right )\right )+d (1+m) \text {Gamma}\left (1+m,\frac {2 i f (c+d x)}{d}\right ) \left (i \cos \left (e-\frac {c f}{d}\right )+\sin \left (e-\frac {c f}{d}\right )\right )\right ) \left (-i \cos \left (f \left (\frac {c}{d}+x\right )\right )+\sin \left (f \left (\frac {c}{d}+x\right )\right )\right )}{a d f (1+m) (-i+\tan (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.30, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{m}}{a +i a \tan \left (f x +e \right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.12, size = 86, normalized size = 0.88 \begin {gather*} \frac {{\left (i \, d m + i \, d\right )} e^{\left (-\frac {d m \log \left (\frac {2 i \, f}{d}\right ) - 2 i \, c f + 2 i \, d e}{d}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) + 2 \, {\left (d f x + c f\right )} {\left (d x + c\right )}^{m}}{4 \, {\left (a d f m + a d f\right )}} \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*} - \frac {i \int \frac {\left (c + d x\right )^{m}}{\tan {\left (e + f x \right )} - i}\, dx}{a} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^m}{a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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