Optimal. Leaf size=55 \[ \frac {a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \sin ^4(c+d x)}{2 d}+\frac {a^2 \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 55, 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} \[ \frac {a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \sin ^4(c+d x)}{2 d}+\frac {a^2 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 (a+x)^2}{a^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int x^2 (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 x^2+2 a x^3+x^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {a^2 \sin ^3(c+d x)}{3 d}+\frac {a^2 \sin ^4(c+d x)}{2 d}+\frac {a^2 \sin ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 53, normalized size = 0.96 \[ \frac {a^2 \left (104 \sin ^3(c+d x)+15 \cos (4 (c+d x))-12 \left (2 \sin ^3(c+d x)+5\right ) \cos (2 (c+d x))\right )}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 72, normalized size = 1.31 \[ \frac {15 \, a^{2} \cos \left (d x + c\right )^{4} - 30 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{2} \cos \left (d x + c\right )^{4} - 11 \, a^{2} \cos \left (d x + c\right )^{2} + 8 \, a^{2}\right )} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 45, normalized size = 0.82 \[ \frac {6 \, a^{2} \sin \left (d x + c\right )^{5} + 15 \, a^{2} \sin \left (d x + c\right )^{4} + 10 \, a^{2} \sin \left (d x + c\right )^{3}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 45, normalized size = 0.82 \[ \frac {\frac {\left (\sin ^{5}\left (d x +c \right )\right ) a^{2}}{5}+\frac {a^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{2}+\frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 45, normalized size = 0.82 \[ \frac {6 \, a^{2} \sin \left (d x + c\right )^{5} + 15 \, a^{2} \sin \left (d x + c\right )^{4} + 10 \, a^{2} \sin \left (d x + c\right )^{3}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.49, size = 36, normalized size = 0.65 \[ \frac {a^2\,{\sin \left (c+d\,x\right )}^3\,\left (6\,{\sin \left (c+d\,x\right )}^2+15\,\sin \left (c+d\,x\right )+10\right )}{30\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.60, size = 63, normalized size = 1.15 \[ \begin {cases} \frac {a^{2} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {a^{2} \sin ^{4}{\left (c + d x \right )}}{2 d} + \frac {a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{2} \sin ^{2}{\relax (c )} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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