Optimal. Leaf size=294 \[ -\frac {2^{n+\frac {1}{2}} \left (n^4+6 n^3+17 n^2+12 n+9\right ) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac {1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d (n+1) (n+2) (n+3) (n+4)}+\frac {\left (-n^2-n+9\right ) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1) (n+2) (n+3) (n+4)}-\frac {\left (n^2+3 n+9\right ) \cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d (n+2) (n+3) (n+4)}-\frac {\sin ^3(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+4)}-\frac {n \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+3) (n+4)} \]
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Rubi [A] time = 0.51, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2783, 2983, 2968, 3023, 2751, 2652, 2651} \[ -\frac {2^{n+\frac {1}{2}} \left (n^4+6 n^3+17 n^2+12 n+9\right ) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac {1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d (n+1) (n+2) (n+3) (n+4)}+\frac {\left (-n^2-n+9\right ) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1) (n+2) (n+3) (n+4)}-\frac {\left (n^2+3 n+9\right ) \cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d (n+2) (n+3) (n+4)}-\frac {\sin ^3(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+4)}-\frac {n \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+3) (n+4)} \]
Antiderivative was successfully verified.
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Rule 2651
Rule 2652
Rule 2751
Rule 2783
Rule 2968
Rule 2983
Rule 3023
Rubi steps
\begin {align*} \int \sin ^4(c+d x) (a+a \sin (c+d x))^n \, dx &=-\frac {\cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^n}{d (4+n)}+\frac {\int \sin ^2(c+d x) (a+a \sin (c+d x))^n (3 a+a n \sin (c+d x)) \, dx}{a (4+n)}\\ &=-\frac {n \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n) (4+n)}-\frac {\cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^n}{d (4+n)}+\frac {\int \sin (c+d x) (a+a \sin (c+d x))^n \left (2 a^2 n+a^2 \left (9+3 n+n^2\right ) \sin (c+d x)\right ) \, dx}{a^2 (3+n) (4+n)}\\ &=-\frac {n \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n) (4+n)}-\frac {\cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^n}{d (4+n)}+\frac {\int (a+a \sin (c+d x))^n \left (2 a^2 n \sin (c+d x)+a^2 \left (9+3 n+n^2\right ) \sin ^2(c+d x)\right ) \, dx}{a^2 (3+n) (4+n)}\\ &=-\frac {n \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n) (4+n)}-\frac {\cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^n}{d (4+n)}-\frac {\left (9+3 n+n^2\right ) \cos (c+d x) (a+a \sin (c+d x))^{1+n}}{a d (2+n) (3+n) (4+n)}+\frac {\int (a+a \sin (c+d x))^n \left (a^3 (1+n) \left (9+3 n+n^2\right )-a^3 \left (9-n-n^2\right ) \sin (c+d x)\right ) \, dx}{a^3 (2+n) (3+n) (4+n)}\\ &=\frac {\left (9-n-n^2\right ) \cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n) (2+n) (3+n) (4+n)}-\frac {n \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n) (4+n)}-\frac {\cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^n}{d (4+n)}-\frac {\left (9+3 n+n^2\right ) \cos (c+d x) (a+a \sin (c+d x))^{1+n}}{a d (2+n) (3+n) (4+n)}+\frac {\left (9+12 n+17 n^2+6 n^3+n^4\right ) \int (a+a \sin (c+d x))^n \, dx}{(1+n) (2+n) (3+n) (4+n)}\\ &=\frac {\left (9-n-n^2\right ) \cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n) (2+n) (3+n) (4+n)}-\frac {n \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n) (4+n)}-\frac {\cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^n}{d (4+n)}-\frac {\left (9+3 n+n^2\right ) \cos (c+d x) (a+a \sin (c+d x))^{1+n}}{a d (2+n) (3+n) (4+n)}+\frac {\left (\left (9+12 n+17 n^2+6 n^3+n^4\right ) (1+\sin (c+d x))^{-n} (a+a \sin (c+d x))^n\right ) \int (1+\sin (c+d x))^n \, dx}{(1+n) (2+n) (3+n) (4+n)}\\ &=\frac {\left (9-n-n^2\right ) \cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n) (2+n) (3+n) (4+n)}-\frac {n \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n) (4+n)}-\frac {\cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^n}{d (4+n)}-\frac {2^{\frac {1}{2}+n} \left (9+12 n+17 n^2+6 n^3+n^4\right ) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-\frac {1}{2}-n} (a+a \sin (c+d x))^n}{d (1+n) (2+n) (3+n) (4+n)}-\frac {\left (9+3 n+n^2\right ) \cos (c+d x) (a+a \sin (c+d x))^{1+n}}{a d (2+n) (3+n) (4+n)}\\ \end {align*}
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Mathematica [F] time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{n}, x\right ) \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.04, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{4}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\sin \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{n} \sin ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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