Optimal. Leaf size=91 \[ \frac {a^4 \sin ^9(c+d x)}{9 d}+\frac {a^4 \sin ^8(c+d x)}{2 d}+\frac {6 a^4 \sin ^7(c+d x)}{7 d}+\frac {2 a^4 \sin ^6(c+d x)}{3 d}+\frac {a^4 \sin ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.08, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ \frac {a^4 \sin ^9(c+d x)}{9 d}+\frac {a^4 \sin ^8(c+d x)}{2 d}+\frac {6 a^4 \sin ^7(c+d x)}{7 d}+\frac {2 a^4 \sin ^6(c+d x)}{3 d}+\frac {a^4 \sin ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4 (a+x)^4}{a^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int x^4 (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^4 x^4+4 a^3 x^5+6 a^2 x^6+4 a x^7+x^8\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {a^4 \sin ^5(c+d x)}{5 d}+\frac {2 a^4 \sin ^6(c+d x)}{3 d}+\frac {6 a^4 \sin ^7(c+d x)}{7 d}+\frac {a^4 \sin ^8(c+d x)}{2 d}+\frac {a^4 \sin ^9(c+d x)}{9 d}\\ \end {align*}
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Mathematica [A] time = 0.84, size = 100, normalized size = 1.10 \[ \frac {a^4 (52290 \sin (c+d x)-30660 \sin (3 (c+d x))+9828 \sin (5 (c+d x))-1395 \sin (7 (c+d x))+35 \sin (9 (c+d x))-42840 \cos (2 (c+d x))+18900 \cos (4 (c+d x))-4200 \cos (6 (c+d x))+315 \cos (8 (c+d x))+4095)}{80640 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 124, normalized size = 1.36 \[ \frac {315 \, a^{4} \cos \left (d x + c\right )^{8} - 1680 \, a^{4} \cos \left (d x + c\right )^{6} + 3150 \, a^{4} \cos \left (d x + c\right )^{4} - 2520 \, a^{4} \cos \left (d x + c\right )^{2} + 2 \, {\left (35 \, a^{4} \cos \left (d x + c\right )^{8} - 410 \, a^{4} \cos \left (d x + c\right )^{6} + 1083 \, a^{4} \cos \left (d x + c\right )^{4} - 1076 \, a^{4} \cos \left (d x + c\right )^{2} + 368 \, a^{4}\right )} \sin \left (d x + c\right )}{630 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 71, normalized size = 0.78 \[ \frac {70 \, a^{4} \sin \left (d x + c\right )^{9} + 315 \, a^{4} \sin \left (d x + c\right )^{8} + 540 \, a^{4} \sin \left (d x + c\right )^{7} + 420 \, a^{4} \sin \left (d x + c\right )^{6} + 126 \, a^{4} \sin \left (d x + c\right )^{5}}{630 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 71, normalized size = 0.78 \[ \frac {\frac {a^{4} \left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {a^{4} \left (\sin ^{8}\left (d x +c \right )\right )}{2}+\frac {6 a^{4} \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {2 a^{4} \left (\sin ^{6}\left (d x +c \right )\right )}{3}+\frac {a^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{5}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 71, normalized size = 0.78 \[ \frac {70 \, a^{4} \sin \left (d x + c\right )^{9} + 315 \, a^{4} \sin \left (d x + c\right )^{8} + 540 \, a^{4} \sin \left (d x + c\right )^{7} + 420 \, a^{4} \sin \left (d x + c\right )^{6} + 126 \, a^{4} \sin \left (d x + c\right )^{5}}{630 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.46, size = 70, normalized size = 0.77 \[ \frac {\frac {a^4\,{\sin \left (c+d\,x\right )}^9}{9}+\frac {a^4\,{\sin \left (c+d\,x\right )}^8}{2}+\frac {6\,a^4\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {2\,a^4\,{\sin \left (c+d\,x\right )}^6}{3}+\frac {a^4\,{\sin \left (c+d\,x\right )}^5}{5}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.92, size = 97, normalized size = 1.07 \[ \begin {cases} \frac {a^{4} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac {a^{4} \sin ^{8}{\left (c + d x \right )}}{2 d} + \frac {6 a^{4} \sin ^{7}{\left (c + d x \right )}}{7 d} + \frac {2 a^{4} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac {a^{4} \sin ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{4} \sin ^{4}{\relax (c )} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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