Optimal. Leaf size=57 \[ (d \cos (e+f x))^m \left (\frac {\sec (e+f x)}{d}\right )^m \text {Int}\left (\left (\frac {\sec (e+f x)}{d}\right )^{-m} \left (a+b (c \tan (e+f x))^n\right )^p,x\right ) \]
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Rubi [A]
time = 0.09, 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 (d \cos (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int (d \cos (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx &=\left ((d \cos (e+f x))^m \left (\frac {\sec (e+f x)}{d}\right )^m\right ) \int \left (\frac {\sec (e+f x)}{d}\right )^{-m} \left (a+b (c \tan (e+f x))^n\right )^p \, dx\\ \end {align*}
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Mathematica [A]
time = 1.75, size = 0, normalized size = 0.00 \begin {gather*} \int (d \cos (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.29, size = 0, normalized size = 0.00 \[\int \left (d \cos \left (f x +e \right )\right )^{m} \left (a +b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx\]
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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 (d\,\cos \left (e+f\,x\right )\right )}^m\,{\left (a+b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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