Optimal. Leaf size=160 \[ -\frac {(a-a \sin (c+d x))^2 \sin ^{n+1}(c+d x)}{d (n+2) \left (a^7 \sin (c+d x)+a^7\right )}+\frac {\sin ^{n+1}(c+d x) \left (a (2 n+7) \sin (c+d x)+a \left (8 n^2+30 n+27\right )\right )}{d \left (n^2+3 n+2\right ) \left (a^6 \sin (c+d x)+a^6\right )}-\frac {4 (2 n+3) \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^5 d (n+1)} \]
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Rubi [A] time = 0.20, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2836, 100, 146, 64} \[ -\frac {4 (2 n+3) \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^5 d (n+1)}+\frac {\sin ^{n+1}(c+d x) \left (a (2 n+7) \sin (c+d x)+a \left (8 n^2+30 n+27\right )\right )}{d \left (n^2+3 n+2\right ) \left (a^6 \sin (c+d x)+a^6\right )}-\frac {(a-a \sin (c+d x))^2 \sin ^{n+1}(c+d x)}{d (n+2) \left (a^7 \sin (c+d x)+a^7\right )} \]
Antiderivative was successfully verified.
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Rule 64
Rule 100
Rule 146
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 \left (\frac {x}{a}\right )^n}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\frac {\sin ^{1+n}(c+d x) (a-a \sin (c+d x))^2}{d (2+n) \left (a^7+a^7 \sin (c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {(a-x) \left (\frac {x}{a}\right )^n (a (3+2 n)+(-7-2 n) x)}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^6 d (2+n)}\\ &=-\frac {\sin ^{1+n}(c+d x) (a-a \sin (c+d x))^2}{d (2+n) \left (a^7+a^7 \sin (c+d x)\right )}+\frac {\sin ^{1+n}(c+d x) \left (a \left (27+30 n+8 n^2\right )+a (7+2 n) \sin (c+d x)\right )}{d (1+n) (2+n) \left (a^6+a^6 \sin (c+d x)\right )}-\frac {(4 (3+2 n)) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{a}\right )^n}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac {4 (3+2 n) \, _2F_1(1,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^5 d (1+n)}-\frac {\sin ^{1+n}(c+d x) (a-a \sin (c+d x))^2}{d (2+n) \left (a^7+a^7 \sin (c+d x)\right )}+\frac {\sin ^{1+n}(c+d x) \left (a \left (27+30 n+8 n^2\right )+a (7+2 n) \sin (c+d x)\right )}{d (1+n) (2+n) \left (a^6+a^6 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 108, normalized size = 0.68 \[ \frac {\sin ^{n+1}(c+d x) \left (-4 \left (2 n^2+7 n+6\right ) (\sin (c+d x)+1) \, _2F_1(1,n+1;n+2;-\sin (c+d x))-(n+1) \sin ^2(c+d x)+(4 n+9) \sin (c+d x)+8 n^2+29 n+26\right )}{a^5 d (n+1) (n+2) (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{5 \, a^{5} \cos \left (d x + c\right )^{4} - 20 \, a^{5} \cos \left (d x + c\right )^{2} + 16 \, a^{5} + {\left (a^{5} \cos \left (d x + c\right )^{4} - 12 \, a^{5} \cos \left (d x + c\right )^{2} + 16 \, a^{5}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.80, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{\left (a +a \sin \left (d x +c \right )\right )^{5}}\, dx \]
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 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{5}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^7\,{\sin \left (c+d\,x\right )}^n}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^5} \,d x \]
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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