Optimal. Leaf size=41 \[ \frac {a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {a \sin ^{n+2}(c+d x)}{d (n+2)} \]
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Rubi [A] time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2833, 43} \[ \frac {a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {a \sin ^{n+2}(c+d x)}{d (n+2)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \left (\frac {x}{a}\right )^n (a+x) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a \left (\frac {x}{a}\right )^n+a \left (\frac {x}{a}\right )^{1+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {a \sin ^{2+n}(c+d x)}{d (2+n)}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 38, normalized size = 0.93 \[ \frac {a \sin ^{n+1}(c+d x) ((n+1) \sin (c+d x)+n+2)}{d (n+1) (n+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 62, normalized size = 1.51 \[ -\frac {{\left ({\left (a n + a\right )} \cos \left (d x + c\right )^{2} - a n - {\left (a n + 2 \, a\right )} \sin \left (d x + c\right ) - a\right )} \sin \left (d x + c\right )^{n}}{d n^{2} + 3 \, d n + 2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 86, normalized size = 2.10 \[ \frac {a n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + a n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + a \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 2 \, a \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )}{{\left (n^{2} + 3 \, n + 2\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.79, size = 0, normalized size = 0.00 \[ \int \cos \left (d x +c \right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 39, normalized size = 0.95 \[ \frac {\frac {a \sin \left (d x + c\right )^{n + 2}}{n + 2} + \frac {a \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.18, size = 67, normalized size = 1.63 \[ \frac {a\,{\sin \left (c+d\,x\right )}^n\,\left (n+4\,\sin \left (c+d\,x\right )+2\,n\,\sin \left (c+d\,x\right )+n\,\left (2\,{\sin \left (c+d\,x\right )}^2-1\right )+2\,{\sin \left (c+d\,x\right )}^2\right )}{2\,d\,\left (n^2+3\,n+2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.08, size = 190, normalized size = 4.63 \[ \begin {cases} x \left (a \sin {\relax (c )} + a\right ) \sin ^{n}{\relax (c )} \cos {\relax (c )} & \text {for}\: d = 0 \\\frac {a \log {\left (\sin {\left (c + d x \right )} \right )}}{d} - \frac {a}{d \sin {\left (c + d x \right )}} & \text {for}\: n = -2 \\\frac {a \log {\left (\sin {\left (c + d x \right )} \right )}}{d} + \frac {a \sin {\left (c + d x \right )}}{d} & \text {for}\: n = -1 \\\frac {a n \sin ^{2}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{2} + 3 d n + 2 d} + \frac {a n \sin {\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{2} + 3 d n + 2 d} + \frac {a \sin ^{2}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{2} + 3 d n + 2 d} + \frac {2 a \sin {\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{2} + 3 d n + 2 d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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