Optimal. Leaf size=37 \[ \frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {\log (\sin (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 43} \[ \frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {\log (\sin (c+d x)+1)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{a (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{(a+x)^2}+\frac {1}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\log (1+\sin (c+d x))}{a^2 d}+\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 27, normalized size = 0.73 \[ \frac {\frac {1}{\sin (c+d x)+1}+\log (\sin (c+d x)+1)}{a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 40, normalized size = 1.08 \[ \frac {{\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 1}{a^{2} d \sin \left (d x + c\right ) + a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 56, normalized size = 1.51 \[ -\frac {\frac {\log \left (\frac {{\left | a \sin \left (d x + c\right ) + a \right |}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2} {\left | a \right |}}\right )}{a} - \frac {1}{a \sin \left (d x + c\right ) + a}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 35, normalized size = 0.95 \[ \frac {\ln \left (1+\sin \left (d x +c \right )\right )}{a^{2} d}+\frac {1}{d \,a^{2} \left (1+\sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 34, normalized size = 0.92 \[ \frac {\frac {1}{a^{2} \sin \left (d x + c\right ) + a^{2}} + \frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 34, normalized size = 0.92 \[ \frac {1}{a^2\,d\,\left (\sin \left (c+d\,x\right )+1\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.04, size = 95, normalized size = 2.57 \[ \begin {cases} \frac {\log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} + \frac {\log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} + \frac {1}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin {\relax (c )} \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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