Optimal. Leaf size=215 \[ -\frac{2^{n+\frac{1}{2}} n \left (n^2+3 n+5\right ) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d (n+1) (n+2) (n+3)}-\frac{n \cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d \left (n^2+5 n+6\right )}-\frac{\sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+3)}-\frac{(n+4) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1) (n+2) (n+3)} \]
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Rubi [A] time = 0.294057, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2783, 2968, 3023, 2751, 2652, 2651} \[ -\frac{2^{n+\frac{1}{2}} n \left (n^2+3 n+5\right ) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d (n+1) (n+2) (n+3)}-\frac{n \cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d \left (n^2+5 n+6\right )}-\frac{\sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+3)}-\frac{(n+4) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1) (n+2) (n+3)} \]
Antiderivative was successfully verified.
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Rule 2783
Rule 2968
Rule 3023
Rule 2751
Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int \sin ^3(c+d x) (a+a \sin (c+d x))^n \, dx &=-\frac{\cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n)}+\frac{\int \sin (c+d x) (a+a \sin (c+d x))^n (2 a+a n \sin (c+d x)) \, dx}{a (3+n)}\\ &=-\frac{\cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n)}+\frac{\int (a+a \sin (c+d x))^n \left (2 a \sin (c+d x)+a n \sin ^2(c+d x)\right ) \, dx}{a (3+n)}\\ &=-\frac{\cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n)}-\frac{n \cos (c+d x) (a+a \sin (c+d x))^{1+n}}{a d \left (6+5 n+n^2\right )}+\frac{\int (a+a \sin (c+d x))^n \left (a^2 n (1+n)+a^2 (4+n) \sin (c+d x)\right ) \, dx}{a^2 (2+n) (3+n)}\\ &=-\frac{(4+n) \cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n) (2+n) (3+n)}-\frac{\cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n)}-\frac{n \cos (c+d x) (a+a \sin (c+d x))^{1+n}}{a d \left (6+5 n+n^2\right )}+\frac{\left (n \left (5+3 n+n^2\right )\right ) \int (a+a \sin (c+d x))^n \, dx}{(1+n) (2+n) (3+n)}\\ &=-\frac{(4+n) \cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n) (2+n) (3+n)}-\frac{\cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n)}-\frac{n \cos (c+d x) (a+a \sin (c+d x))^{1+n}}{a d \left (6+5 n+n^2\right )}+\frac{\left (n \left (5+3 n+n^2\right ) (1+\sin (c+d x))^{-n} (a+a \sin (c+d x))^n\right ) \int (1+\sin (c+d x))^n \, dx}{(1+n) (2+n) (3+n)}\\ &=-\frac{(4+n) \cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n) (2+n) (3+n)}-\frac{\cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n)}-\frac{2^{\frac{1}{2}+n} n \left (5+3 n+n^2\right ) \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) (2+n) (3+n)}-\frac{n \cos (c+d x) (a+a \sin (c+d x))^{1+n}}{a d \left (6+5 n+n^2\right )}\\ \end{align*}
Mathematica [C] time = 127.957, size = 60244, normalized size = 280.2 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.111, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (\cos \left (d x + c\right )^{2} - 1\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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