Optimal. Leaf size=86 \[ -\frac {i 2^{-1-\frac {m}{2}} (e \cos (c+d x))^m \, _2F_1\left (-\frac {m}{2},\frac {4+m}{2};1-\frac {m}{2};\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{m/2}}{a d m} \]
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Rubi [A]
time = 0.16, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3596, 3586,
3604, 72, 71} \begin {gather*} -\frac {i 2^{-\frac {m}{2}-1} (1+i \tan (c+d x))^{m/2} (e \cos (c+d x))^m \, _2F_1\left (-\frac {m}{2},\frac {m+4}{2};1-\frac {m}{2};\frac {1}{2} (1-i \tan (c+d x))\right )}{a d m} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 3586
Rule 3596
Rule 3604
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^m}{a+i a \tan (c+d x)} \, dx &=\left ((e \cos (c+d x))^m (e \sec (c+d x))^m\right ) \int \frac {(e \sec (c+d x))^{-m}}{a+i a \tan (c+d x)} \, dx\\ &=\left ((e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{m/2}\right ) \int (a-i a \tan (c+d x))^{-m/2} (a+i a \tan (c+d x))^{-1-\frac {m}{2}} \, dx\\ &=\frac {\left (a^2 (e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{m/2}\right ) \text {Subst}\left (\int (a-i a x)^{-1-\frac {m}{2}} (a+i a x)^{-2-\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\left (2^{-2-\frac {m}{2}} (e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} \left (\frac {a+i a \tan (c+d x)}{a}\right )^{m/2}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {i x}{2}\right )^{-2-\frac {m}{2}} (a-i a x)^{-1-\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {i 2^{-1-\frac {m}{2}} (e \cos (c+d x))^m \, _2F_1\left (-\frac {m}{2},\frac {4+m}{2};1-\frac {m}{2};\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{m/2}}{a d 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. \(201\) vs. \(2(86)=172\).
time = 59.20, size = 201, normalized size = 2.34 \begin {gather*} \frac {2^{-1-m} e^{-i (c+2 d x)} \left (1+e^{2 i (c+d x)}\right )^{-m} \left (e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^m \cos ^{-1-m}(c+d x) (e \cos (c+d x))^m \left (m \, _2F_1\left (-1-\frac {m}{2},-m;-\frac {m}{2};-e^{2 i (c+d x)}\right )+e^{2 i (c+d x)} (2+m) \, _2F_1\left (-m,-\frac {m}{2};1-\frac {m}{2};-e^{2 i (c+d x)}\right )\right ) (\cos (d x)+i \sin (d x))}{a d m (2+m) (-i+\tan (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.97, size = 0, normalized size = 0.00 \[\int \frac {\left (e \cos \left (d x +c \right )\right )^{m}}{a +i a \tan \left (d x +c \right )}\, dx\]
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 (e \cos {\left (c + d x \right )}\right )^{m}}{\tan {\left (c + d 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 (e\,\cos \left (c+d\,x\right )\right )}^m}{a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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