Optimal. Leaf size=38 \[ \frac {\, _2F_1(3,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^3 d (1+n)} \]
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Rubi [A]
time = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2912, 66}
\begin {gather*} \frac {\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;-\sin (c+d x))}{a^3 d (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 2912
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (\frac {x}{a}\right )^n}{(a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\, _2F_1(3,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^3 d (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 38, normalized size = 1.00 \begin {gather*} \frac {\, _2F_1(3,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^3 d (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.44, size = 0, normalized size = 0.00 \[\int \frac {\cos \left (d x +c \right ) \left (\sin ^{n}\left (d x +c \right )\right )}{\left (a +a \sin \left (d x +c \right )\right )^{3}}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.03 \begin {gather*} \int \frac {\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^n}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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