Optimal. Leaf size=22 \[ -\frac {1}{2 a d (a \sin (c+d x)+a)^2} \]
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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} \[ -\frac {1}{2 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac {1}{2 a d (a+a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 33, normalized size = 1.50 \[ -\frac {1}{2 a^3 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 36, normalized size = 1.64 \[ \frac {1}{2 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.91, size = 20, normalized size = 0.91 \[ -\frac {1}{2 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a d} \]
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 \[ -\frac {1}{2 a d \left (a +a \sin \left (d x +c \right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 20, normalized size = 0.91 \[ -\frac {1}{2 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.46, size = 18, normalized size = 0.82 \[ -\frac {1}{2\,a^3\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.89, size = 51, normalized size = 2.32 \[ \begin {cases} - \frac {1}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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