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 = 0.01, 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 = {3740, 3556}
\begin {gather*} -\frac {\cot (e+f x) \log (\cos (e+f x)) \left (b \tan ^n(e+f x)\right )^{\frac {1}{n}}}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3740
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 \begin {gather*} -\frac {\cot (e+f x) \log (\cos (e+f x)) \left (b \tan ^n(e+f x)\right )^{\frac {1}{n}}}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 4.99, size = 12884, normalized size = 402.62
method | result | size |
risch | \(\text {Expression too large to display}\) | \(12884\) |
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 \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 = 1.81, size = 24, normalized size = 0.75 \begin {gather*} -\frac {b^{\left (\frac {1}{n}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \, f} \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*} \int \left (b \tan ^{n}{\left (e + f x \right )}\right )^{\frac {1}{n}}\, dx \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 = -0.03 \begin {gather*} \int {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^n\right )}^{1/n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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