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 = 0.0849619, antiderivative size = 91, normalized size of antiderivative = 1., 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 2833
Rule 12
Rule 43
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*}
Mathematica [A] time = 0.812519, size = 100, normalized size = 1.1 \[ \frac{a^4 (52290 \sin (c+d x)-30660 \sin (3 (c+d x))+9828 \sin (5 (c+d x))-1395 \sin (7 (c+d x))+35 \sin (9 (c+d x))-42840 \cos (2 (c+d x))+18900 \cos (4 (c+d x))-4200 \cos (6 (c+d x))+315 \cos (8 (c+d x))+4095)}{80640 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 71, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{9}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{2}}+{\frac{6\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{2\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{3}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10056, size = 96, normalized size = 1.05 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98013, size = 324, normalized size = 3.56 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 24.9305, size = 97, normalized size = 1.07 \begin{align*} \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{\left (c \right )} + a\right )^{4} \sin ^{4}{\left (c \right )} \cos{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28994, size = 96, normalized size = 1.05 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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