Optimal. Leaf size=52 \[ -\frac {i \, _2F_1\left (2,m;1+m;\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{4 c f m} \]
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Rubi [A]
time = 0.09, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3603, 3568, 70}
\begin {gather*} -\frac {i (a+i a \tan (e+f x))^m \, _2F_1\left (2,m;m+1;\frac {1}{2} (i \tan (e+f x)+1)\right )}{4 c f m} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^m}{c-i c \tan (e+f x)} \, dx &=\frac {\int \cos ^2(e+f x) (a+i a \tan (e+f x))^{1+m} \, dx}{a c}\\ &=-\frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+m}}{(a-x)^2} \, dx,x,i a \tan (e+f x)\right )}{c f}\\ &=-\frac {i \, _2F_1\left (2,m;1+m;\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{4 c f m}\\ \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. \(133\) vs. \(2(52)=104\).
time = 16.05, size = 133, normalized size = 2.56 \begin {gather*} -\frac {i 2^{-2+m} \left (e^{i f x}\right )^m \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \left (1+e^{2 i (e+f x)}\right )^2 \, _2F_1\left (1,2;1+m;-e^{2 i (e+f x)}\right ) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{c f m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.99, size = 0, normalized size = 0.00 \[\int \frac {\left (a +i a \tan \left (f x +e \right )\right )^{m}}{c -i c \tan \left (f x +e \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {i \int \frac {\left (i a \tan {\left (e + f x \right )} + a\right )^{m}}{\tan {\left (e + f x \right )} + i}\, dx}{c} \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 = -0.02 \begin {gather*} \int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m}{c-c\,\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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