Optimal. Leaf size=42 \[ \frac {\, _2F_1(2,1+m;2+m;1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \]
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Rubi [A]
time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2912, 12, 67}
\begin {gather*} \frac {(a \sin (c+d x)+a)^{m+1} \, _2F_1(2,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 67
Rule 2912
Rubi steps
\begin {align*} \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx &=\frac {\text {Subst}\left (\int \frac {a^2 (a+x)^m}{x^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a \text {Subst}\left (\int \frac {(a+x)^m}{x^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\, _2F_1(2,1+m;2+m;1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 42, normalized size = 1.00 \begin {gather*} \frac {\, _2F_1(2,1+m;2+m;1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \cos \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx\]
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 (c + d x \right )} + 1\right )\right )^{m} \cos {\left (c + d x \right )} \csc ^{2}{\left (c + d 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 \frac {\cos \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\sin \left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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