Optimal. Leaf size=109 \[ -\frac {\cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n)}-\frac {2^{\frac {1}{2}+n} n \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-\frac {1}{2}-n} (a+a \sin (c+d x))^n}{d (1+n)} \]
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Rubi [A]
time = 0.05, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2830, 2731,
2730} \begin {gather*} -\frac {2^{n+\frac {1}{2}} n \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac {1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d (n+1)}-\frac {\cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2730
Rule 2731
Rule 2830
Rubi steps
\begin {align*} \int \sin (c+d x) (a+a \sin (c+d x))^n \, dx &=-\frac {\cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n)}+\frac {n \int (a+a \sin (c+d x))^n \, dx}{1+n}\\ &=-\frac {\cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n)}+\frac {\left (n (1+\sin (c+d x))^{-n} (a+a \sin (c+d x))^n\right ) \int (1+\sin (c+d x))^n \, dx}{1+n}\\ &=-\frac {\cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n)}-\frac {2^{\frac {1}{2}+n} n \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-\frac {1}{2}-n} (a+a \sin (c+d x))^n}{d (1+n)}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.42, size = 178, normalized size = 1.63 \begin {gather*} -\frac {\sqrt [4]{-1} 2^{-1-2 n} e^{-\frac {3}{2} i (c+d x)} \left (-(-1)^{3/4} e^{-\frac {1}{2} i (c+d x)} \left (i+e^{i (c+d x)}\right )\right )^{1+2 n} \left (e^{2 i (c+d x)} (-1+n) \, _2F_1\left (1,n;-n;-i e^{-i (c+d x)}\right )-(1+n) \, _2F_1\left (1,2+n;2-n;-i e^{-i (c+d x)}\right )\right ) (a (1+\sin (c+d x)))^n \sin ^{-2 n}\left (\frac {1}{4} (2 c+\pi +2 d x)\right )}{d (-1+n) (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \sin \left (d x +c \right ) \left (a +a \sin \left (d x +c \right )\right )^{n}\, dx\]
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 )^{n} \sin {\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 = -0.01 \begin {gather*} \int \sin \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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