Optimal. Leaf size=22 \[ -\frac {1}{2 b d (a+b \sin (c+d x))^2} \]
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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} \[ -\frac {1}{2 b d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2668
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x)^3} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=-\frac {1}{2 b d (a+b \sin (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 22, normalized size = 1.00 \[ -\frac {1}{2 b d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 43, normalized size = 1.95 \[ \frac {1}{2 \, {\left (b^{3} d \cos \left (d x + c\right )^{2} - 2 \, a b^{2} d \sin \left (d x + c\right ) - {\left (a^{2} b + b^{3}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 20, normalized size = 0.91 \[ -\frac {1}{2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{2} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 21, normalized size = 0.95 \[ -\frac {1}{2 b d \left (a +b \sin \left (d x +c \right )\right )^{2}} \]
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 \[ -\frac {1}{2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{2} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 39, normalized size = 1.77 \[ -\frac {1}{d\,\left (2\,a^2\,b+4\,a\,b^2\,\sin \left (c+d\,x\right )+2\,b^3\,{\sin \left (c+d\,x\right )}^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.97, size = 73, normalized size = 3.32 \[ \begin {cases} \frac {x \cos {\relax (c )}}{a^{3}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sin {\left (c + d x \right )}}{a^{3} d} & \text {for}\: b = 0 \\\frac {x \cos {\relax (c )}}{\left (a + b \sin {\relax (c )}\right )^{3}} & \text {for}\: d = 0 \\- \frac {1}{2 a^{2} b d + 4 a b^{2} d \sin {\left (c + d x \right )} + 2 b^{3} d \sin ^{2}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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