Optimal. Leaf size=91 \[ \frac {\sin ^{n+1}(c+d x)}{a d (n+1)}-\frac {\sin ^{n+2}(c+d x)}{a d (n+2)}-\frac {\sin ^{n+3}(c+d x)}{a d (n+3)}+\frac {\sin ^{n+4}(c+d x)}{a d (n+4)} \]
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Rubi [A] time = 0.14, antiderivative size = 91, 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} \[ \frac {\sin ^{n+1}(c+d x)}{a d (n+1)}-\frac {\sin ^{n+2}(c+d x)}{a d (n+2)}-\frac {\sin ^{n+3}(c+d x)}{a d (n+3)}+\frac {\sin ^{n+4}(c+d x)}{a d (n+4)} \]
Antiderivative was successfully verified.
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Rule 75
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 \left (\frac {x}{a}\right )^n (a+x) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^3 \left (\frac {x}{a}\right )^n-a^3 \left (\frac {x}{a}\right )^{1+n}-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^5 d}\\ &=\frac {\sin ^{1+n}(c+d x)}{a d (1+n)}-\frac {\sin ^{2+n}(c+d x)}{a d (2+n)}-\frac {\sin ^{3+n}(c+d x)}{a d (3+n)}+\frac {\sin ^{4+n}(c+d x)}{a d (4+n)}\\ \end {align*}
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Mathematica [A] time = 0.70, size = 74, normalized size = 0.81 \[ \frac {\sin ^{n+1}(c+d x) \left (-\frac {(n+4) \sin ^2(c+d x)}{n+3}-\frac {(n+4) \sin (c+d x)}{n+2}+\sin ^3(c+d x)+\frac {n+4}{n+1}\right )}{a d (n+4)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 134, normalized size = 1.47 \[ \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} \cos \left (d x + c\right )^{4} - {\left (n^{3} + 4 \, n^{2} + 3 \, n\right )} \cos \left (d x + c\right )^{2} - 2 \, n^{2} + {\left ({\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} \cos \left (d x + c\right )^{2} + 2 \, n^{2} + 12 \, n + 16\right )} \sin \left (d x + c\right ) - 8 \, n - 6\right )} \sin \left (d x + c\right )^{n}}{a d n^{4} + 10 \, a d n^{3} + 35 \, a d n^{2} + 50 \, a d n + 24 \, a d} \]
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 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{5}}{a \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.92, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{a +a \sin \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.80, size = 124, normalized size = 1.36 \[ \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} \sin \left (d x + c\right )^{4} - {\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} \sin \left (d x + c\right )^{3} - {\left (n^{3} + 8 \, n^{2} + 19 \, n + 12\right )} \sin \left (d x + c\right )^{2} + {\left (n^{3} + 9 \, n^{2} + 26 \, n + 24\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.21, size = 228, normalized size = 2.51 \[ \frac {{\sin \left (c+d\,x\right )}^n\,\left (144\,\sin \left (c+d\,x\right )-43\,n+24\,\cos \left (2\,c+2\,d\,x\right )+6\,\cos \left (4\,c+4\,d\,x\right )+16\,\sin \left (3\,c+3\,d\,x\right )+124\,n\,\sin \left (c+d\,x\right )+32\,n\,\cos \left (2\,c+2\,d\,x\right )+11\,n\,\cos \left (4\,c+4\,d\,x\right )+28\,n\,\sin \left (3\,c+3\,d\,x\right )+30\,n^2\,\sin \left (c+d\,x\right )+2\,n^3\,\sin \left (c+d\,x\right )-14\,n^2-n^3+8\,n^2\,\cos \left (2\,c+2\,d\,x\right )+6\,n^2\,\cos \left (4\,c+4\,d\,x\right )+n^3\,\cos \left (4\,c+4\,d\,x\right )+14\,n^2\,\sin \left (3\,c+3\,d\,x\right )+2\,n^3\,\sin \left (3\,c+3\,d\,x\right )-30\right )}{8\,a\,d\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )} \]
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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