Optimal. Leaf size=22 \[ \frac {(a \sin (c+d x)+a)^3}{3 a d} \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2667, 32} \[ \frac {(a \sin (c+d x)+a)^3}{3 a d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {(a+a \sin (c+d x))^3}{3 a d}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 47, normalized size = 2.14 \[ \frac {a^2 \sin ^3(c+d x)}{3 d}+\frac {a^2 \sin ^2(c+d x)}{d}+\frac {a^2 \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 44, normalized size = 2.00 \[ -\frac {3 \, a^{2} \cos \left (d x + c\right )^{2} + {\left (a^{2} \cos \left (d x + c\right )^{2} - 4 \, a^{2}\right )} \sin \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 20, normalized size = 0.91 \[ \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{3 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 21, normalized size = 0.95 \[ \frac {\left (a +a \sin \left (d x +c \right )\right )^{3}}{3 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 20, normalized size = 0.91 \[ \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{3 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.55, size = 32, normalized size = 1.45 \[ \frac {a^2\,\sin \left (c+d\,x\right )\,\left ({\sin \left (c+d\,x\right )}^2+3\,\sin \left (c+d\,x\right )+3\right )}{3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.82, size = 53, normalized size = 2.41 \[ \begin {cases} \frac {a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{2} \sin ^{2}{\left (c + d x \right )}}{d} + \frac {a^{2} \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{2} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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