Optimal. Leaf size=43 \[ -\frac {\, _2F_1(1,1+m;2+m;1+\sin (e+f x)) (a+a \sin (e+f x))^{1+m}}{a f (1+m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2786, 67}
\begin {gather*} -\frac {(a \sin (e+f x)+a)^{m+1} \, _2F_1(1,m+1;m+2;\sin (e+f x)+1)}{a f (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 2786
Rubi steps
\begin {align*} \int \cot (e+f x) (a+a \sin (e+f x))^m \, dx &=\frac {\text {Subst}\left (\int \frac {(a+x)^m}{x} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=-\frac {\, _2F_1(1,1+m;2+m;1+\sin (e+f x)) (a+a \sin (e+f x))^{1+m}}{a f (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 43, normalized size = 1.00 \begin {gather*} -\frac {\, _2F_1(1,1+m;2+m;1+\sin (e+f x)) (a+a \sin (e+f x))^{1+m}}{a f (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \cot \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \cot {\left (e + f x \right )}\, 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 = -0.02 \begin {gather*} \int \mathrm {cot}\left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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