Optimal. Leaf size=32 \[ \frac {2 \log (\sin (c+d x)+1)}{a^2 d}-\frac {\sin (c+d x)}{a^2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac {2 \log (\sin (c+d x)+1)}{a^2 d}-\frac {\sin (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a-x}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-1+\frac {2 a}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {2 \log (1+\sin (c+d x))}{a^2 d}-\frac {\sin (c+d x)}{a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 26, normalized size = 0.81 \[ -\frac {\sin (c+d x)-2 \log (\sin (c+d x)+1)}{a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 27, normalized size = 0.84 \[ \frac {2 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - \sin \left (d x + c\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 54, normalized size = 1.69 \[ -\frac {\frac {2 \, \log \left (\frac {{\left | a \sin \left (d x + c\right ) + a \right |}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2} {\left | a \right |}}\right )}{a^{2}} + \frac {a \sin \left (d x + c\right ) + a}{a^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 33, normalized size = 1.03 \[ \frac {2 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{2} d}-\frac {\sin \left (d x +c \right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 30, normalized size = 0.94 \[ \frac {\frac {2 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} - \frac {\sin \left (d x + c\right )}{a^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 27, normalized size = 0.84 \[ \frac {2\,\ln \left (\sin \left (c+d\,x\right )+1\right )-\sin \left (c+d\,x\right )}{a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.79, size = 150, normalized size = 4.69 \[ \begin {cases} \frac {2 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} + \frac {2 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} - \frac {2 \sin ^{2}{\left (c + d x \right )}}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} - \frac {\cos ^{2}{\left (c + d x \right )}}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} + \frac {2}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{3}{\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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