Optimal. Leaf size=92 \[ \frac {\sin ^{n+1}(c+d x)}{a^3 d (n+1)}-\frac {3 \sin ^{n+2}(c+d x)}{a^3 d (n+2)}+\frac {3 \sin ^{n+3}(c+d x)}{a^3 d (n+3)}-\frac {\sin ^{n+4}(c+d x)}{a^3 d (n+4)} \]
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Rubi [A] time = 0.14, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2836, 43} \[ \frac {\sin ^{n+1}(c+d x)}{a^3 d (n+1)}-\frac {3 \sin ^{n+2}(c+d x)}{a^3 d (n+2)}+\frac {3 \sin ^{n+3}(c+d x)}{a^3 d (n+3)}-\frac {\sin ^{n+4}(c+d x)}{a^3 d (n+4)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 \left (\frac {x}{a}\right )^n \, dx,x,a \sin (c+d x)\right )}{a^7 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^7 d}\\ &=\frac {\sin ^{1+n}(c+d x)}{a^3 d (1+n)}-\frac {3 \sin ^{2+n}(c+d x)}{a^3 d (2+n)}+\frac {3 \sin ^{3+n}(c+d x)}{a^3 d (3+n)}-\frac {\sin ^{4+n}(c+d x)}{a^3 d (4+n)}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 66, normalized size = 0.72 \[ \frac {\sin ^{n+1}(c+d x) \left (-\frac {\sin ^3(c+d x)}{n+4}+\frac {3 \sin ^2(c+d x)}{n+3}-\frac {3 \sin (c+d x)}{n+2}+\frac {1}{n+1}\right )}{a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 160, normalized size = 1.74 \[ -\frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} \cos \left (d x + c\right )^{4} + 4 \, n^{3} - {\left (5 \, n^{3} + 36 \, n^{2} + 79 \, n + 48\right )} \cos \left (d x + c\right )^{2} + 30 \, n^{2} - {\left (4 \, n^{3} - 3 \, {\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} \cos \left (d x + c\right )^{2} + 30 \, n^{2} + 68 \, n + 48\right )} \sin \left (d x + c\right ) + 68 \, n + 42\right )} \sin \left (d x + c\right )^{n}}{a^{3} d n^{4} + 10 \, a^{3} d n^{3} + 35 \, a^{3} d n^{2} + 50 \, a^{3} d n + 24 \, a^{3} d} \]
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 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.84, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{\left (a +a \sin \left (d x +c \right )\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 126, normalized size = 1.37 \[ -\frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} \sin \left (d x + c\right )^{4} - 3 \, {\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} \sin \left (d x + c\right )^{3} + 3 \, {\left (n^{3} + 8 \, n^{2} + 19 \, n + 12\right )} \sin \left (d x + c\right )^{2} - {\left (n^{3} + 9 \, n^{2} + 26 \, n + 24\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.61, size = 242, normalized size = 2.63 \[ -\frac {{\sin \left (c+d\,x\right )}^n\,\left (261\,n-336\,\sin \left (c+d\,x\right )-168\,\cos \left (2\,c+2\,d\,x\right )+6\,\cos \left (4\,c+4\,d\,x\right )+48\,\sin \left (3\,c+3\,d\,x\right )-460\,n\,\sin \left (c+d\,x\right )-272\,n\,\cos \left (2\,c+2\,d\,x\right )+11\,n\,\cos \left (4\,c+4\,d\,x\right )+84\,n\,\sin \left (3\,c+3\,d\,x\right )-198\,n^2\,\sin \left (c+d\,x\right )-26\,n^3\,\sin \left (c+d\,x\right )+114\,n^2+15\,n^3-120\,n^2\,\cos \left (2\,c+2\,d\,x\right )-16\,n^3\,\cos \left (2\,c+2\,d\,x\right )+6\,n^2\,\cos \left (4\,c+4\,d\,x\right )+n^3\,\cos \left (4\,c+4\,d\,x\right )+42\,n^2\,\sin \left (3\,c+3\,d\,x\right )+6\,n^3\,\sin \left (3\,c+3\,d\,x\right )+162\right )}{8\,a^3\,d\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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