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 = 0.10, antiderivative size = 91, 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} \[ \frac {a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac {3 a^3 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac {a^3 \sin ^{n+4}(c+d x)}{d (n+4)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2833
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*}
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Mathematica [A] time = 0.17, 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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fricas [B] time = 0.52, size = 210, normalized size = 2.31 \[ \frac {{\left (4 \, a^{3} n^{3} + 30 \, a^{3} n^{2} + {\left (a^{3} n^{3} + 6 \, a^{3} n^{2} + 11 \, a^{3} n + 6 \, a^{3}\right )} \cos \left (d x + c\right )^{4} + 68 \, a^{3} n + 42 \, a^{3} - {\left (5 \, a^{3} n^{3} + 36 \, a^{3} n^{2} + 79 \, a^{3} n + 48 \, a^{3}\right )} \cos \left (d x + c\right )^{2} + {\left (4 \, a^{3} n^{3} + 30 \, a^{3} n^{2} + 68 \, a^{3} n + 48 \, a^{3} - 3 \, {\left (a^{3} n^{3} + 7 \, a^{3} n^{2} + 14 \, a^{3} n + 8 \, a^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{4} + 10 \, d n^{3} + 35 \, d n^{2} + 50 \, d n + 24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 379, normalized size = 4.16 \[ \frac {a^{3} n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 3 \, a^{3} n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 6 \, a^{3} n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 3 \, a^{3} n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 21 \, a^{3} n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 11 \, a^{3} n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + a^{3} n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 24 \, a^{3} n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 42 \, a^{3} n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 9 \, a^{3} n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 57 \, a^{3} n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 24 \, a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 26 \, a^{3} n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 36 \, a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 24 \, a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.39, size = 0, normalized size = 0.00 \[ \int \cos \left (d x +c \right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 83, normalized size = 0.91 \[ \frac {\frac {a^{3} \sin \left (d x + c\right )^{n + 4}}{n + 4} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 3}}{n + 3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 2}}{n + 2} + \frac {a^{3} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.42, size = 242, normalized size = 2.66 \[ \frac {a^3\,{\sin \left (c+d\,x\right )}^n\,\left (261\,n+336\,\sin \left (c+d\,x\right )-168\,\cos \left (2\,c+2\,d\,x\right )+6\,\cos \left (4\,c+4\,d\,x\right )-48\,\sin \left (3\,c+3\,d\,x\right )+460\,n\,\sin \left (c+d\,x\right )-272\,n\,\cos \left (2\,c+2\,d\,x\right )+11\,n\,\cos \left (4\,c+4\,d\,x\right )-84\,n\,\sin \left (3\,c+3\,d\,x\right )+198\,n^2\,\sin \left (c+d\,x\right )+26\,n^3\,\sin \left (c+d\,x\right )+114\,n^2+15\,n^3-120\,n^2\,\cos \left (2\,c+2\,d\,x\right )-16\,n^3\,\cos \left (2\,c+2\,d\,x\right )+6\,n^2\,\cos \left (4\,c+4\,d\,x\right )+n^3\,\cos \left (4\,c+4\,d\,x\right )-42\,n^2\,\sin \left (3\,c+3\,d\,x\right )-6\,n^3\,\sin \left (3\,c+3\,d\,x\right )+162\right )}{8\,d\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 27.52, size = 1061, normalized size = 11.66 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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