Optimal. Leaf size=85 \[ \frac {4 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^3 d (n+1)}-\frac {3 \sin ^{n+1}(c+d x)}{a^3 d (n+1)}+\frac {\sin ^{n+2}(c+d x)}{a^3 d (n+2)} \]
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Rubi [A] time = 0.14, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 88, 64} \[ \frac {4 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^3 d (n+1)}-\frac {3 \sin ^{n+1}(c+d x)}{a^3 d (n+1)}+\frac {\sin ^{n+2}(c+d x)}{a^3 d (n+2)} \]
Antiderivative was successfully verified.
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Rule 64
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(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 (-3 a \left (\frac {x}{a}\right )^n+a \left (\frac {x}{a}\right )^{1+n}+\frac {4 a^2 \left (\frac {x}{a}\right )^n}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac {3 \sin ^{1+n}(c+d x)}{a^3 d (1+n)}+\frac {\sin ^{2+n}(c+d x)}{a^3 d (2+n)}+\frac {4 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{a}\right )^n}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac {3 \sin ^{1+n}(c+d x)}{a^3 d (1+n)}+\frac {4 \, _2F_1(1,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^3 d (1+n)}+\frac {\sin ^{2+n}(c+d x)}{a^3 d (2+n)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 64, normalized size = 0.75 \[ \frac {\sin ^{n+1}(c+d x) (4 (n+2) \, _2F_1(1,n+1;n+2;-\sin (c+d x))+(n+1) \sin (c+d x)-3 (n+2))}{a^3 d (n+1) (n+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{5}}{3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{5}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{\left (a +a \sin \left (d x +c \right )\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{5}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^5\,{\sin \left (c+d\,x\right )}^n}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3} \,d x \]
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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