Optimal. Leaf size=89 \[ \frac{2 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{m+2} F_1\left (m+\frac{3}{2};-\frac{1}{2},2;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )}{a^2 f (2 m+3)} \]
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Rubi [A] time = 0.10163, antiderivative size = 89, normalized size of antiderivative = 1., 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 = {2719, 137, 136} \[ \frac{2 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{m+2} F_1\left (m+\frac{3}{2};-\frac{1}{2},2;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )}{a^2 f (2 m+3)} \]
Antiderivative was successfully verified.
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Rule 2719
Rule 137
Rule 136
Rubi steps
\begin{align*} \int \cot ^2(e+f x) (a+a \sin (e+f x))^m \, dx &=\frac{\left (\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a-x} (a+x)^{\frac{1}{2}+m}}{x^2} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{\left (\sqrt{2} \sec (e+f x) (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{(a+x)^{\frac{1}{2}+m} \sqrt{\frac{1}{2}-\frac{x}{2 a}}}{x^2} \, dx,x,a \sin (e+f x)\right )}{a f \sqrt{\frac{a-a \sin (e+f x)}{a}}}\\ &=\frac{2 \sqrt{2} F_1\left (\frac{3}{2}+m;-\frac{1}{2},2;\frac{5}{2}+m;\frac{1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sec (e+f x) \sqrt{1-\sin (e+f x)} (a+a \sin (e+f x))^{2+m}}{a^2 f (3+2 m)}\\ \end{align*}
Mathematica [C] time = 99.9376, size = 47487, normalized size = 533.56 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.231, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( fx+e \right ) \right ) ^{2} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \left (\sin{\left (e + f x \right )} + 1\right )\right )^{m} \cot ^{2}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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