Optimal. Leaf size=57 \[ \left (\frac {\tan (e+f x)}{d}\right )^m (d \cot (e+f x))^m \text {Int}\left (\left (\frac {\tan (e+f x)}{d}\right )^{-m} \left (a+b (c \tan (e+f x))^n\right )^p,x\right ) \]
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Rubi [A] time = 0.14, 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 = {} \[ \int (d \cot (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (d \cot (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx &=\left ((d \cot (e+f x))^m \left (\frac {\tan (e+f x)}{d}\right )^m\right ) \int \left (\frac {\tan (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 = 9.86, size = 0, normalized size = 0.00 \[ \int (d \cot (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m}, x\right ) \]
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 \[ \int {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.00, size = 0, normalized size = 0.00 \[ \int \left (d \cot \left (f x +e \right )\right )^{m} \left (a +b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]
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 \[ \int {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^m\,{\left (a+b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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