Optimal. Leaf size=52 \[ \frac {\sin (c+d x)}{a^2 d}-\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )}-\frac {2 \log (\sin (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.07, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ \frac {\sin (c+d x)}{a^2 d}-\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )}-\frac {2 \log (\sin (c+d x)+1)}{a^2 d} \]
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 ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{a^2 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {a^2}{(a+x)^2}-\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}-\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 55, normalized size = 1.06 \[ \frac {4 \sin (c+d x)-8 \log (\sin (c+d x)+1)-\frac {4}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}}{4 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 57, normalized size = 1.10 \[ -\frac {\cos \left (d x + c\right )^{2} + 2 \, {\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - \sin \left (d x + c\right )}{a^{2} d \sin \left (d x + c\right ) + a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 70, normalized size = 1.35 \[ \frac {\frac {2 \, \log \left (\frac {{\left | a \sin \left (d x + c\right ) + a \right |}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2} {\left | a \right |}}\right )}{a^{2}} + \frac {a \sin \left (d x + c\right ) + a}{a^{3}} - \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )} a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 50, normalized size = 0.96 \[ \frac {\sin \left (d x +c \right )}{a^{2} d}-\frac {2 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{2} d}-\frac {1}{d \,a^{2} \left (1+\sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 47, normalized size = 0.90 \[ -\frac {\frac {1}{a^{2} \sin \left (d x + c\right ) + a^{2}} + \frac {2 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} - \frac {\sin \left (d x + c\right )}{a^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 45, normalized size = 0.87 \[ \frac {{\sin \left (c+d\,x\right )}^2-2}{a^2\,d\,\left (\sin \left (c+d\,x\right )+1\right )}-\frac {2\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.45, size = 126, normalized size = 2.42 \[ \begin {cases} - \frac {2 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} - \frac {2 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} + \frac {\sin ^{2}{\left (c + d x \right )}}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} - \frac {2}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{2}{\relax (c )} \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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