Optimal. Leaf size=67 \[ \frac{(a \sin (c+d x)+a)^7}{7 a^3 d}-\frac{(a \sin (c+d x)+a)^6}{3 a^2 d}+\frac{(a \sin (c+d x)+a)^5}{5 a d} \]
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Rubi [A] time = 0.0753, antiderivative size = 67, 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} \[ \frac{(a \sin (c+d x)+a)^7}{7 a^3 d}-\frac{(a \sin (c+d x)+a)^6}{3 a^2 d}+\frac{(a \sin (c+d x)+a)^5}{5 a d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 (a+x)^4}{a^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int x^2 (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 (a+x)^4-2 a (a+x)^5+(a+x)^6\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{(a+a \sin (c+d x))^5}{5 a d}-\frac{(a+a \sin (c+d x))^6}{3 a^2 d}+\frac{(a+a \sin (c+d x))^7}{7 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.332869, size = 80, normalized size = 1.19 \[ -\frac{a^4 (-7245 \sin (c+d x)+3395 \sin (3 (c+d x))-609 \sin (5 (c+d x))+15 \sin (7 (c+d x))+5460 \cos (2 (c+d x))-1680 \cos (4 (c+d x))+140 \cos (6 (c+d x))-630)}{6720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 70, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{{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{6\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12522, size = 96, normalized size = 1.43 \begin{align*} \frac{15 \, a^{4} \sin \left (d x + c\right )^{7} + 70 \, a^{4} \sin \left (d x + c\right )^{6} + 126 \, a^{4} \sin \left (d x + c\right )^{5} + 105 \, a^{4} \sin \left (d x + c\right )^{4} + 35 \, a^{4} \sin \left (d x + c\right )^{3}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95417, size = 247, normalized size = 3.69 \begin{align*} -\frac{70 \, a^{4} \cos \left (d x + c\right )^{6} - 315 \, a^{4} \cos \left (d x + c\right )^{4} + 420 \, a^{4} \cos \left (d x + c\right )^{2} +{\left (15 \, a^{4} \cos \left (d x + c\right )^{6} - 171 \, a^{4} \cos \left (d x + c\right )^{4} + 332 \, a^{4} \cos \left (d x + c\right )^{2} - 176 \, a^{4}\right )} \sin \left (d x + c\right )}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.2798, size = 119, normalized size = 1.78 \begin{align*} \begin{cases} \frac{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{6 a^{4} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac{a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{2 a^{4} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac{a^{4} \cos ^{4}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{4} \sin ^{2}{\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 = 1.18746, size = 96, normalized size = 1.43 \begin{align*} \frac{15 \, a^{4} \sin \left (d x + c\right )^{7} + 70 \, a^{4} \sin \left (d x + c\right )^{6} + 126 \, a^{4} \sin \left (d x + c\right )^{5} + 105 \, a^{4} \sin \left (d x + c\right )^{4} + 35 \, a^{4} \sin \left (d x + c\right )^{3}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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