Optimal. Leaf size=92 \[ \frac {\sin ^{n+1}(c+d x)}{a^2 d (n+1)}-\frac {2 \sin ^{n+2}(c+d x)}{a^2 d (n+2)}+\frac {2 \sin ^{n+4}(c+d x)}{a^2 d (n+4)}-\frac {\sin ^{n+5}(c+d x)}{a^2 d (n+5)} \]
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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, 75} \[ \frac {\sin ^{n+1}(c+d x)}{a^2 d (n+1)}-\frac {2 \sin ^{n+2}(c+d x)}{a^2 d (n+2)}+\frac {2 \sin ^{n+4}(c+d x)}{a^2 d (n+4)}-\frac {\sin ^{n+5}(c+d x)}{a^2 d (n+5)} \]
Antiderivative was successfully verified.
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Rule 75
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 \left (\frac {x}{a}\right )^n (a+x) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^4 \left (\frac {x}{a}\right )^n-2 a^4 \left (\frac {x}{a}\right )^{1+n}+2 a^4 \left (\frac {x}{a}\right )^{3+n}-a^4 \left (\frac {x}{a}\right )^{4+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 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 {2 \sin ^{4+n}(c+d x)}{a^2 d (4+n)}-\frac {\sin ^{5+n}(c+d x)}{a^2 d (5+n)}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 117, normalized size = 1.27 \[ \frac {\sin ^{n+1}(c+d x) \left (-\left (\left (n^3+7 n^2+14 n+8\right ) \sin ^4(c+d x)\right )+2 \left (n^3+8 n^2+17 n+10\right ) \sin ^3(c+d x)-2 \left (n^3+10 n^2+29 n+20\right ) \sin (c+d x)+n^3+11 n^2+38 n+40\right )}{a^2 d (n+1) (n+2) (n+4) (n+5)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 169, normalized size = 1.84 \[ \frac {{\left (2 \, {\left (n^{3} + 8 \, n^{2} + 17 \, n + 10\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (n^{3} + 6 \, n^{2} + 5 \, n\right )} \cos \left (d x + c\right )^{2} - 4 \, n^{2} - {\left ({\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} \cos \left (d x + c\right )^{2} - 4 \, n^{2} - 24 \, n - 32\right )} \sin \left (d x + c\right ) - 24 \, n - 20\right )} \sin \left (d x + c\right )^{n}}{a^{2} d n^{4} + 12 \, a^{2} d n^{3} + 49 \, a^{2} d n^{2} + 78 \, a^{2} d n + 40 \, a^{2} d} \]
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 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 22.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{\left (a +a \sin \left (d x +c \right )\right )^{2}}\, dx \]
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 \[ -\frac {{\left ({\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} \sin \left (d x + c\right )^{5} - 2 \, {\left (n^{3} + 8 \, n^{2} + 17 \, n + 10\right )} \sin \left (d x + c\right )^{4} + 2 \, {\left (n^{3} + 10 \, n^{2} + 29 \, n + 20\right )} \sin \left (d x + c\right )^{2} - {\left (n^{3} + 11 \, n^{2} + 38 \, n + 40\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{{\left (n^{4} + 12 \, n^{3} + 49 \, n^{2} + 78 \, n + 40\right )} a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.26, size = 280, normalized size = 3.04 \[ \frac {{\sin \left (c+d\,x\right )}^n\,\left (560\,\sin \left (c+d\,x\right )-260\,n+160\,\cos \left (2\,c+2\,d\,x\right )+40\,\cos \left (4\,c+4\,d\,x\right )+40\,\sin \left (3\,c+3\,d\,x\right )-8\,\sin \left (5\,c+5\,d\,x\right )+468\,n\,\sin \left (c+d\,x\right )+192\,n\,\cos \left (2\,c+2\,d\,x\right )+68\,n\,\cos \left (4\,c+4\,d\,x\right )+70\,n\,\sin \left (3\,c+3\,d\,x\right )-14\,n\,\sin \left (5\,c+5\,d\,x\right )+106\,n^2\,\sin \left (c+d\,x\right )+6\,n^3\,\sin \left (c+d\,x\right )-64\,n^2-4\,n^3+32\,n^2\,\cos \left (2\,c+2\,d\,x\right )+32\,n^2\,\cos \left (4\,c+4\,d\,x\right )+4\,n^3\,\cos \left (4\,c+4\,d\,x\right )+35\,n^2\,\sin \left (3\,c+3\,d\,x\right )+5\,n^3\,\sin \left (3\,c+3\,d\,x\right )-7\,n^2\,\sin \left (5\,c+5\,d\,x\right )-n^3\,\sin \left (5\,c+5\,d\,x\right )-200\right )}{16\,a^2\,d\,\left (n^4+12\,n^3+49\,n^2+78\,n+40\right )} \]
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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