Optimal. Leaf size=22 \[ \frac {(a+b \sin (c+d x))^4}{4 b d} \]
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Rubi [A] time = 0.03, 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 = {2668, 32} \[ \frac {(a+b \sin (c+d x))^4}{4 b d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2668
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^3 \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {(a+b \sin (c+d x))^4}{4 b d}\\ \end {align*}
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Mathematica [B] time = 0.06, size = 57, normalized size = 2.59 \[ \frac {\sin (c+d x) \left (4 a^3+6 a^2 b \sin (c+d x)+4 a b^2 \sin ^2(c+d x)+b^3 \sin ^3(c+d x)\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 71, normalized size = 3.23 \[ \frac {b^{3} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left (a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - a b^{2}\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 20, normalized size = 0.91 \[ \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{4}}{4 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 21, normalized size = 0.95 \[ \frac {\left (a +b \sin \left (d x +c \right )\right )^{4}}{4 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 20, normalized size = 0.91 \[ \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{4}}{4 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 55, normalized size = 2.50 \[ \frac {a^3\,\sin \left (c+d\,x\right )+\frac {3\,a^2\,b\,{\sin \left (c+d\,x\right )}^2}{2}+a\,b^2\,{\sin \left (c+d\,x\right )}^3+\frac {b^3\,{\sin \left (c+d\,x\right )}^4}{4}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.27, size = 73, normalized size = 3.32 \[ \begin {cases} \frac {a^{3} \sin {\left (c + d x \right )}}{d} + \frac {3 a^{2} b \sin ^{2}{\left (c + d x \right )}}{2 d} + \frac {a b^{2} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {b^{3} \sin ^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{3} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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