Optimal. Leaf size=49 \[ (g \cot (e+f x))^p (g \tan (e+f x))^p \text {Int}\left ((a+b \cos (e+f x))^m (g \cot (e+f x))^{-p},x\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx &=\left ((g \cot (e+f x))^p (g \tan (e+f x))^p\right ) \int (a+b \cos (e+f x))^m (g \cot (e+f x))^{-p} \, dx\\ \end {align*}
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Mathematica [A]
time = 2.56, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.18, size = 0, normalized size = 0.00 \[\int \left (a +b \cos \left (f x +e \right )\right )^{m} \left (g \tan \left (f x +e \right )\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 = 0.00, 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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (g \tan {\left (e + f x \right )}\right )^{p} \left (a + b \cos {\left (e + f x \right )}\right )^{m}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (g\,\mathrm {tan}\left (e+f\,x\right )\right )}^p\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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