Optimal. Leaf size=107 \[ \frac {\tan (e+f x) (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {1-m}{2};1,-p;\frac {3-m}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )}{f (1-m)} \]
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Rubi [A] time = 0.20, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3674, 3670, 511, 510} \[ \frac {\tan (e+f x) (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {1-m}{2};1,-p;\frac {3-m}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )}{f (1-m)} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rule 3670
Rule 3674
Rubi steps
\begin {align*} \int (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\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 \tan ^2(e+f x)\right )^p \, dx\\ &=\frac {\left ((d \cot (e+f x))^m \left (\frac {\tan (e+f x)}{d}\right )^m\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{d}\right )^{-m} \left (a+b x^2\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\left ((d \cot (e+f x))^m \left (\frac {\tan (e+f x)}{d}\right )^m \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{d}\right )^{-m} \left (1+\frac {b x^2}{a}\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {F_1\left (\frac {1-m}{2};1,-p;\frac {3-m}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) (d \cot (e+f x))^m \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{f (1-m)}\\ \end {align*}
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Mathematica [B] time = 2.58, size = 265, normalized size = 2.48 \[ -\frac {a (m-3) \cos ^2(e+f x) \cot (e+f x) (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p F_1\left (\frac {1-m}{2};-p,1;\frac {3-m}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )}{f (m-1) \left (-2 b p F_1\left (\frac {3-m}{2};1-p,1;\frac {5-m}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 a F_1\left (\frac {3-m}{2};-p,2;\frac {5-m}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+a (m-3) \cot ^2(e+f x) F_1\left (\frac {1-m}{2};-p,1;\frac {3-m}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.92, size = 0, normalized size = 0.00 \[ \int \left (d \cot \left (f x +e \right )\right )^{m} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]
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 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^m\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \]
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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