Optimal. Leaf size=91 \[ \frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{3 a^3 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{a^3 \sin ^{n+4}(c+d x)}{d (n+4)} \]
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Rubi [A] time = 0.0953491, antiderivative size = 91, normalized size of antiderivative = 1., 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} \[ \frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{3 a^3 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{a^3 \sin ^{n+4}(c+d x)}{d (n+4)} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 43
Rubi steps
\begin{align*} \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{x}{a}\right )^n (a+x)^3 \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^3 \left (\frac{x}{a}\right )^n+3 a^3 \left (\frac{x}{a}\right )^{1+n}+3 a^3 \left (\frac{x}{a}\right )^{2+n}+a^3 \left (\frac{x}{a}\right )^{3+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^3 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac{3 a^3 \sin ^{2+n}(c+d x)}{d (2+n)}+\frac{3 a^3 \sin ^{3+n}(c+d x)}{d (3+n)}+\frac{a^3 \sin ^{4+n}(c+d x)}{d (4+n)}\\ \end{align*}
Mathematica [A] time = 0.148885, size = 65, normalized size = 0.71 \[ \frac{a^3 \sin ^{n+1}(c+d x) \left (\frac{\sin ^3(c+d x)}{n+4}+\frac{3 \sin ^2(c+d x)}{n+3}+\frac{3 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.313, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{n} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85824, size = 468, normalized size = 5.14 \begin{align*} \frac{{\left (4 \, a^{3} n^{3} + 30 \, a^{3} n^{2} +{\left (a^{3} n^{3} + 6 \, a^{3} n^{2} + 11 \, a^{3} n + 6 \, a^{3}\right )} \cos \left (d x + c\right )^{4} + 68 \, a^{3} n + 42 \, a^{3} -{\left (5 \, a^{3} n^{3} + 36 \, a^{3} n^{2} + 79 \, a^{3} n + 48 \, a^{3}\right )} \cos \left (d x + c\right )^{2} +{\left (4 \, a^{3} n^{3} + 30 \, a^{3} n^{2} + 68 \, a^{3} n + 48 \, a^{3} - 3 \,{\left (a^{3} n^{3} + 7 \, a^{3} n^{2} + 14 \, a^{3} n + 8 \, a^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{4} + 10 \, d n^{3} + 35 \, d n^{2} + 50 \, d n + 24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 77.423, size = 1061, normalized size = 11.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18669, size = 136, normalized size = 1.49 \begin{align*} \frac{\frac{a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4}}{n + 4} + \frac{3 \, a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3}}{n + 3} + \frac{3 \, a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}}{n + 2} + \frac{a^{3} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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