Optimal. Leaf size=68 \[ \frac {\sin ^{n+1}(c+d x)}{a^2 d (n+1)}-\frac {2 \sin ^{n+2}(c+d x)}{a^2 d (n+2)}+\frac {\sin ^{n+3}(c+d x)}{a^2 d (n+3)} \]
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Rubi [A] time = 0.13, antiderivative size = 68, 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} \[ \frac {\sin ^{n+1}(c+d x)}{a^2 d (n+1)}-\frac {2 \sin ^{n+2}(c+d x)}{a^2 d (n+2)}+\frac {\sin ^{n+3}(c+d x)}{a^2 d (n+3)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 \left (\frac {x}{a}\right )^n \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 \left (\frac {x}{a}\right )^n-2 a^2 \left (\frac {x}{a}\right )^{1+n}+a^2 \left (\frac {x}{a}\right )^{2+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\sin ^{1+n}(c+d x)}{a^2 d (1+n)}-\frac {2 \sin ^{2+n}(c+d x)}{a^2 d (2+n)}+\frac {\sin ^{3+n}(c+d x)}{a^2 d (3+n)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 50, normalized size = 0.74 \[ \frac {\sin ^{n+1}(c+d x) \left (\frac {\sin ^2(c+d x)}{n+3}-\frac {2 \sin (c+d x)}{n+2}+\frac {1}{n+1}\right )}{a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 105, normalized size = 1.54 \[ \frac {{\left (2 \, {\left (n^{2} + 4 \, n + 3\right )} \cos \left (d x + c\right )^{2} - 2 \, n^{2} - {\left ({\left (n^{2} + 3 \, n + 2\right )} \cos \left (d x + c\right )^{2} - 2 \, n^{2} - 8 \, n - 8\right )} \sin \left (d x + c\right ) - 8 \, n - 6\right )} \sin \left (d x + c\right )^{n}}{a^{2} d n^{3} + 6 \, a^{2} d n^{2} + 11 \, a^{2} d n + 6 \, a^{2} d} \]
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 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{5}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 13.92, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{\left (a +a \sin \left (d x +c \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 81, normalized size = 1.19 \[ \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} \sin \left (d x + c\right )^{3} - 2 \, {\left (n^{2} + 4 \, n + 3\right )} \sin \left (d x + c\right )^{2} + {\left (n^{2} + 5 \, n + 6\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.54, size = 146, normalized size = 2.15 \[ -\frac {\frac {{\sin \left (c+d\,x\right )}^n\,\left (24\,{\sin \left (c+d\,x\right )}^2-30\,\sin \left (c+d\,x\right )+2\,\sin \left (3\,c+3\,d\,x\right )\right )}{4}+\frac {n\,{\sin \left (c+d\,x\right )}^n\,\left (32\,{\sin \left (c+d\,x\right )}^2-29\,\sin \left (c+d\,x\right )+3\,\sin \left (3\,c+3\,d\,x\right )\right )}{4}+\frac {n^2\,{\sin \left (c+d\,x\right )}^n\,\left (8\,{\sin \left (c+d\,x\right )}^2-7\,\sin \left (c+d\,x\right )+\sin \left (3\,c+3\,d\,x\right )\right )}{4}}{a^2\,d\,\left (n^3+6\,n^2+11\,n+6\right )} \]
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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