Optimal. Leaf size=109 \[ \frac {8 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^4 d (n+1)}-\frac {7 \sin ^{n+1}(c+d x)}{a^4 d (n+1)}+\frac {4 \sin ^{n+2}(c+d x)}{a^4 d (n+2)}-\frac {\sin ^{n+3}(c+d x)}{a^4 d (n+3)} \]
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Rubi [A] time = 0.17, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2836, 88, 43, 64} \[ \frac {8 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^4 d (n+1)}-\frac {7 \sin ^{n+1}(c+d x)}{a^4 d (n+1)}+\frac {4 \sin ^{n+2}(c+d x)}{a^4 d (n+2)}-\frac {\sin ^{n+3}(c+d x)}{a^4 d (n+3)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 64
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(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 (-4 a^2 \left (\frac {x}{a}\right )^n-2 a (a-x) \left (\frac {x}{a}\right )^n-(a-x)^2 \left (\frac {x}{a}\right )^n+\frac {8 a^3 \left (\frac {x}{a}\right )^n}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\frac {4 \sin ^{1+n}(c+d x)}{a^4 d (1+n)}-\frac {\operatorname {Subst}\left (\int (a-x)^2 \left (\frac {x}{a}\right )^n \, dx,x,a \sin (c+d x)\right )}{a^7 d}-\frac {2 \operatorname {Subst}\left (\int (a-x) \left (\frac {x}{a}\right )^n \, dx,x,a \sin (c+d x)\right )}{a^6 d}+\frac {8 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{a}\right )^n}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=-\frac {4 \sin ^{1+n}(c+d x)}{a^4 d (1+n)}+\frac {8 \, _2F_1(1,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^4 d (1+n)}-\frac {\operatorname {Subst}\left (\int \left (a^2 \left (\frac {x}{a}\right )^n-2 a^2 \left (\frac {x}{a}\right )^{1+n}+a^2 \left (\frac {x}{a}\right )^{2+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}-\frac {2 \operatorname {Subst}\left (\int \left (a \left (\frac {x}{a}\right )^n-a \left (\frac {x}{a}\right )^{1+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=-\frac {7 \sin ^{1+n}(c+d x)}{a^4 d (1+n)}+\frac {8 \, _2F_1(1,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^4 d (1+n)}+\frac {4 \sin ^{2+n}(c+d x)}{a^4 d (2+n)}-\frac {\sin ^{3+n}(c+d x)}{a^4 d (3+n)}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 104, normalized size = 0.95 \[ \frac {\frac {8 a^3 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{n+1}-\frac {7 a^3 \sin ^{n+1}(c+d x)}{n+1}+\frac {4 a^3 \sin ^{n+2}(c+d x)}{n+2}-\frac {a^3 \sin ^{n+3}(c+d x)}{n+3}}{a^7 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 9.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{\left (a +a \sin \left (d x +c \right )\right )^{4}}\, dx \]
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 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^7\,{\sin \left (c+d\,x\right )}^n}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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