Optimal. Leaf size=20 \[ -\frac {1}{b d (a+b \sin (c+d x))} \]
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Rubi [A] time = 0.03, antiderivative size = 20, 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} \[ -\frac {1}{b d (a+b \sin (c+d x))} \]
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))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=-\frac {1}{b d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 1.00 \[ -\frac {1}{b d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 20, normalized size = 1.00 \[ -\frac {1}{b^{2} d \sin \left (d x + c\right ) + a b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 20, normalized size = 1.00 \[ -\frac {1}{{\left (b \sin \left (d x + c\right ) + a\right )} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 21, normalized size = 1.05 \[ -\frac {1}{b d \left (a +b \sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 20, normalized size = 1.00 \[ -\frac {1}{{\left (b \sin \left (d x + c\right ) + a\right )} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.07, size = 20, normalized size = 1.00 \[ -\frac {1}{b\,d\,\left (a+b\,\sin \left (c+d\,x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.24, size = 51, normalized size = 2.55 \[ \begin {cases} \frac {x \cos {\relax (c )}}{a^{2}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sin {\left (c + d x \right )}}{a^{2} d} & \text {for}\: b = 0 \\\frac {x \cos {\relax (c )}}{\left (a + b \sin {\relax (c )}\right )^{2}} & \text {for}\: d = 0 \\- \frac {1}{a b d + b^{2} d \sin {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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