Optimal. Leaf size=83 \[ -\frac {\csc ^2(e+f x) (a+a \sin (e+f x))^{2+m}}{2 a^2 f}-\frac {(2-m) \, _2F_1(2,2+m;3+m;1+\sin (e+f x)) (a+a \sin (e+f x))^{2+m}}{2 a^2 f (2+m)} \]
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Rubi [A]
time = 0.05, antiderivative size = 83, 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 = {2786, 79, 67}
\begin {gather*} -\frac {(2-m) (a \sin (e+f x)+a)^{m+2} \, _2F_1(2,m+2;m+3;\sin (e+f x)+1)}{2 a^2 f (m+2)}-\frac {\csc ^2(e+f x) (a \sin (e+f x)+a)^{m+2}}{2 a^2 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 79
Rule 2786
Rubi steps
\begin {align*} \int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x) (a+x)^{1+m}}{x^3} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=-\frac {\csc ^2(e+f x) (a+a \sin (e+f x))^{2+m}}{2 a^2 f}-\frac {(2-m) \text {Subst}\left (\int \frac {(a+x)^{1+m}}{x^2} \, dx,x,a \sin (e+f x)\right )}{2 f}\\ &=-\frac {\csc ^2(e+f x) (a+a \sin (e+f x))^{2+m}}{2 a^2 f}-\frac {(2-m) \, _2F_1(2,2+m;3+m;1+\sin (e+f x)) (a+a \sin (e+f x))^{2+m}}{2 a^2 f (2+m)}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 68, normalized size = 0.82 \begin {gather*} -\frac {\left ((2+m) \csc ^2(e+f x)-(-2+m) \, _2F_1(2,2+m;3+m;1+\sin (e+f x))\right ) (1+\sin (e+f x))^2 (a (1+\sin (e+f x)))^m}{2 f (2+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \left (\cot ^{3}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx\]
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 ^{3}{\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 = -0.01 \begin {gather*} \int {\mathrm {cot}\left (e+f\,x\right )}^3\,{\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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