Optimal. Leaf size=167 \[ \frac {\left (a^2-b^2\right )^2 \sin ^{n+1}(c+d x) \, _2F_1\left (1,n+1;n+2;-\frac {b \sin (c+d x)}{a}\right )}{a b^4 d (n+1)}-\frac {a \left (a^2-2 b^2\right ) \sin ^{n+1}(c+d x)}{b^4 d (n+1)}+\frac {\left (a^2-2 b^2\right ) \sin ^{n+2}(c+d x)}{b^3 d (n+2)}-\frac {a \sin ^{n+3}(c+d x)}{b^2 d (n+3)}+\frac {\sin ^{n+4}(c+d x)}{b d (n+4)} \]
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Rubi [A] time = 0.34, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2837, 952, 1620, 64} \[ \frac {\left (a^2-b^2\right )^2 \sin ^{n+1}(c+d x) \, _2F_1\left (1,n+1;n+2;-\frac {b \sin (c+d x)}{a}\right )}{a b^4 d (n+1)}-\frac {a \left (a^2-2 b^2\right ) \sin ^{n+1}(c+d x)}{b^4 d (n+1)}+\frac {\left (a^2-2 b^2\right ) \sin ^{n+2}(c+d x)}{b^3 d (n+2)}-\frac {a \sin ^{n+3}(c+d x)}{b^2 d (n+3)}+\frac {\sin ^{n+4}(c+d x)}{b d (n+4)} \]
Antiderivative was successfully verified.
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Rule 64
Rule 952
Rule 1620
Rule 2837
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) \sin ^n(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n \left (b^2-x^2\right )^2}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\sin ^{4+n}(c+d x)}{b d (4+n)}+\frac {\operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n \left (4+n-\frac {2 (4+n) x^2}{b^2}-\frac {a (4+n) x^3}{b^4}\right )}{a+x} \, dx,x,b \sin (c+d x)\right )}{b d (4+n)}\\ &=\frac {\sin ^{4+n}(c+d x)}{b d (4+n)}+\frac {\operatorname {Subst}\left (\int \left (-\frac {a \left (a^2-2 b^2\right ) (4+n) \left (\frac {x}{b}\right )^n}{b^4}-\frac {\left (-a^2+2 b^2\right ) (4+n) \left (\frac {x}{b}\right )^{1+n}}{b^3}-\frac {a (4+n) \left (\frac {x}{b}\right )^{2+n}}{b^2}+\frac {\left (-a^2+b^2\right )^2 (4+n) \left (\frac {x}{b}\right )^n}{b^4 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b d (4+n)}\\ &=-\frac {a \left (a^2-2 b^2\right ) \sin ^{1+n}(c+d x)}{b^4 d (1+n)}+\frac {\left (a^2-2 b^2\right ) \sin ^{2+n}(c+d x)}{b^3 d (2+n)}-\frac {a \sin ^{3+n}(c+d x)}{b^2 d (3+n)}+\frac {\sin ^{4+n}(c+d x)}{b d (4+n)}+\frac {\left (a^2-b^2\right )^2 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=-\frac {a \left (a^2-2 b^2\right ) \sin ^{1+n}(c+d x)}{b^4 d (1+n)}+\frac {\left (a^2-b^2\right )^2 \, _2F_1\left (1,1+n;2+n;-\frac {b \sin (c+d x)}{a}\right ) \sin ^{1+n}(c+d x)}{a b^4 d (1+n)}+\frac {\left (a^2-2 b^2\right ) \sin ^{2+n}(c+d x)}{b^3 d (2+n)}-\frac {a \sin ^{3+n}(c+d x)}{b^2 d (3+n)}+\frac {\sin ^{4+n}(c+d x)}{b d (4+n)}\\ \end {align*}
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Mathematica [A] time = 0.58, size = 133, normalized size = 0.80 \[ \frac {\sin ^{n+1}(c+d x) \left (-\frac {a^3-2 a b^2}{n+1}+\frac {\left (a^2-b^2\right )^2 \, _2F_1\left (1,n+1;n+2;-\frac {b \sin (c+d x)}{a}\right )}{a (n+1)}+\frac {b \left (a^2-2 b^2\right ) \sin (c+d x)}{n+2}-\frac {a b^2 \sin ^2(c+d x)}{n+3}+\frac {b^3 \sin ^3(c+d x)}{n+4}\right )}{b^4 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{5}}{b \sin \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{5}}{b \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.27, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}\, dx \]
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 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{5}}{b \sin \left (d x + c\right ) + a}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^5\,{\sin \left (c+d\,x\right )}^n}{a+b\,\sin \left (c+d\,x\right )} \,d x \]
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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