Optimal. Leaf size=68 \[ \frac {a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac {a^2 \sin ^{n+3}(c+d x)}{d (n+3)} \]
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Rubi [A] time = 0.09, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2833, 43} \[ \frac {a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac {a^2 \sin ^{n+3}(c+d x)}{d (n+3)} \]
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))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (\frac {x}{a}\right )^n (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 \left (\frac {x}{a}\right )^n+2 a^2 \left (\frac {x}{a}\right )^{1+n}+a^2 \left (\frac {x}{a}\right )^{2+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^2 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {2 a^2 \sin ^{2+n}(c+d x)}{d (2+n)}+\frac {a^2 \sin ^{3+n}(c+d x)}{d (3+n)}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 50, normalized size = 0.74 \[ \frac {a^2 \sin ^{n+1}(c+d x) \left (\frac {\sin ^2(c+d x)}{n+3}+\frac {2 \sin (c+d x)}{n+2}+\frac {1}{n+1}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 135, normalized size = 1.99 \[ \frac {{\left (2 \, a^{2} n^{2} + 8 \, a^{2} n - 2 \, {\left (a^{2} n^{2} + 4 \, a^{2} n + 3 \, a^{2}\right )} \cos \left (d x + c\right )^{2} + 6 \, a^{2} + {\left (2 \, a^{2} n^{2} + 8 \, a^{2} n - {\left (a^{2} n^{2} + 3 \, a^{2} n + 2 \, a^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{3} + 6 \, d n^{2} + 11 \, d n + 6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 213, normalized size = 3.13 \[ \frac {a^{2} n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 2 \, a^{2} n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 3 \, a^{2} n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + a^{2} n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 8 \, a^{2} n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 5 \, a^{2} n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 6 \, a^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 6 \, a^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.52, size = 0, normalized size = 0.00 \[ \int \cos \left (d x +c \right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 63, normalized size = 0.93 \[ \frac {\frac {a^{2} \sin \left (d x + c\right )^{n + 3}}{n + 3} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 2}}{n + 2} + \frac {a^{2} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.64, size = 147, normalized size = 2.16 \[ \frac {a^2\,{\sin \left (c+d\,x\right )}^n\,\left (16\,n+30\,\sin \left (c+d\,x\right )-2\,\sin \left (3\,c+3\,d\,x\right )+29\,n\,\sin \left (c+d\,x\right )+16\,n\,\left (2\,{\sin \left (c+d\,x\right )}^2-1\right )-3\,n\,\sin \left (3\,c+3\,d\,x\right )+7\,n^2\,\sin \left (c+d\,x\right )+4\,n^2\,\left (2\,{\sin \left (c+d\,x\right )}^2-1\right )+4\,n^2+24\,{\sin \left (c+d\,x\right )}^2-n^2\,\sin \left (3\,c+3\,d\,x\right )\right )}{4\,d\,\left (n^3+6\,n^2+11\,n+6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.06, size = 530, normalized size = 7.79 \[ \begin {cases} x \left (a \sin {\relax (c )} + a\right )^{2} \sin ^{n}{\relax (c )} \cos {\relax (c )} & \text {for}\: d = 0 \\\frac {a^{2} \log {\left (\sin {\left (c + d x \right )} \right )}}{d} - \frac {2 a^{2}}{d \sin {\left (c + d x \right )}} - \frac {a^{2}}{2 d \sin ^{2}{\left (c + d x \right )}} & \text {for}\: n = -3 \\\frac {2 a^{2} \log {\left (\sin {\left (c + d x \right )} \right )}}{d} + \frac {a^{2} \sin {\left (c + d x \right )}}{d} - \frac {a^{2}}{d \sin {\left (c + d x \right )}} & \text {for}\: n = -2 \\\frac {a^{2} \log {\left (\sin {\left (c + d x \right )} \right )}}{d} + \frac {a^{2} \sin ^{2}{\left (c + d x \right )}}{2 d} + \frac {2 a^{2} \sin {\left (c + d x \right )}}{d} & \text {for}\: n = -1 \\\frac {a^{2} n^{2} \sin ^{3}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac {2 a^{2} n^{2} \sin ^{2}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac {a^{2} n^{2} \sin {\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac {3 a^{2} n \sin ^{3}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac {8 a^{2} n \sin ^{2}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac {5 a^{2} n \sin {\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac {2 a^{2} \sin ^{3}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac {6 a^{2} \sin ^{2}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac {6 a^{2} \sin {\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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