Optimal. Leaf size=69 \[ \frac {(a \cot (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac {1}{2} (-m+n+1);\frac {1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{b f (-m+n+1)} \]
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Rubi [A] time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2604, 3476, 364} \[ \frac {(a \cot (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac {1}{2} (-m+n+1);\frac {1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{b f (-m+n+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2604
Rule 3476
Rubi steps
\begin {align*} \int (a \cot (e+f x))^m (b \tan (e+f x))^n \, dx &=\left ((a \cot (e+f x))^m (b \tan (e+f x))^m\right ) \int (b \tan (e+f x))^{-m+n} \, dx\\ &=\frac {\left (b (a \cot (e+f x))^m (b \tan (e+f x))^m\right ) \operatorname {Subst}\left (\int \frac {x^{-m+n}}{b^2+x^2} \, dx,x,b \tan (e+f x)\right )}{f}\\ &=\frac {(a \cot (e+f x))^m \, _2F_1\left (1,\frac {1}{2} (1-m+n);\frac {1}{2} (3-m+n);-\tan ^2(e+f x)\right ) (b \tan (e+f x))^{1+n}}{b f (1-m+n)}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 67, normalized size = 0.97 \[ \frac {a (a \cot (e+f x))^{m-1} (b \tan (e+f x))^n \, _2F_1\left (1,\frac {1}{2} (-m+n+1);\frac {1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{f (-m+n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (a \cot \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int \left (a \cot \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.74, size = 0, normalized size = 0.00 \[ \int \left (a \cot \left (f x +e \right )\right )^{m} \left (b \tan \left (f x +e \right )\right )^{n}\, dx \]
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 \[ \int \left (a \cot \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,\mathrm {cot}\left (e+f\,x\right )\right )}^m\,{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \]
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 \[ \int \left (a \cot {\left (e + f x \right )}\right )^{m} \left (b \tan {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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