Optimal. Leaf size=32 \[ -\frac {\cot (e+f x) \log (\cos (e+f x)) \left (b \tan ^n(e+f x)\right )^{\frac {1}{n}}}{f} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3659, 3475} \[ -\frac {\cot (e+f x) \log (\cos (e+f x)) \left (b \tan ^n(e+f x)\right )^{\frac {1}{n}}}{f} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3659
Rubi steps
\begin {align*} \int \left (b \tan ^n(e+f x)\right )^{\frac {1}{n}} \, dx &=\left (\cot (e+f x) \left (b \tan ^n(e+f x)\right )^{\frac {1}{n}}\right ) \int \tan (e+f x) \, dx\\ &=-\frac {\cot (e+f x) \log (\cos (e+f x)) \left (b \tan ^n(e+f x)\right )^{\frac {1}{n}}}{f}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 1.00 \[ -\frac {\cot (e+f x) \log (\cos (e+f x)) \left (b \tan ^n(e+f x)\right )^{\frac {1}{n}}}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 23, normalized size = 0.72 \[ -\frac {b^{\left (\frac {1}{n}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \, f} \]
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 \[ \int \left (b \tan \left (f x + e\right )^{n}\right )^{\left (\frac {1}{n}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (b \left (\tan ^{n}\left (f x +e \right )\right )\right )^{\frac {1}{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \tan \left (f x + e\right )^{n}\right )^{\left (\frac {1}{n}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^n\right )}^{1/n} \,d x \]
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 \[ \int \left (b \tan ^{n}{\left (e + f x \right )}\right )^{\frac {1}{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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