Optimal. Leaf size=78 \[ \frac {\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \cot (e+f x))^m \, _2F_1\left (1,\frac {1}{2} (-m+2 p+1);\frac {1}{2} (-m+2 p+3);-\tan ^2(e+f x)\right )}{f (-m+2 p+1)} \]
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Rubi [A] time = 0.12, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3658, 2604, 3476, 364} \[ \frac {\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \cot (e+f x))^m \, _2F_1\left (1,\frac {1}{2} (-m+2 p+1);\frac {1}{2} (-m+2 p+3);-\tan ^2(e+f x)\right )}{f (-m+2 p+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2604
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx &=\left (\tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \int (d \cot (e+f x))^m \tan ^{2 p}(e+f x) \, dx\\ &=\left ((d \cot (e+f x))^m \tan ^{m-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \int \tan ^{-m+2 p}(e+f x) \, dx\\ &=\frac {\left ((d \cot (e+f x))^m \tan ^{m-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \operatorname {Subst}\left (\int \frac {x^{-m+2 p}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(d \cot (e+f x))^m \, _2F_1\left (1,\frac {1}{2} (1-m+2 p);\frac {1}{2} (3-m+2 p);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1-m+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 70, normalized size = 0.90 \[ -\frac {d \left (b \tan ^2(e+f x)\right )^p (d \cot (e+f x))^{m-1} \, _2F_1\left (1,-\frac {m}{2}+p+\frac {1}{2};-\frac {m}{2}+p+\frac {3}{2};-\tan ^2(e+f x)\right )}{f (m-2 p-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cot \left (f x + e\right )\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cot \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.00, size = 0, normalized size = 0.00 \[ \int \left (d \cot \left (f x +e \right )\right )^{m} \left (b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]
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 \[ \int \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cot \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^m\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}^p \,d x \]
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 \[ \int \left (b \tan ^{2}{\left (e + f x \right )}\right )^{p} \left (d \cot {\left (e + f x \right )}\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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