Optimal. Leaf size=43 \[ \frac {\sin (c+d x)}{a^2 d}-\frac {\csc (c+d x)}{a^2 d}-\frac {2 \log (\sin (c+d x))}{a^2 d} \]
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Rubi [A] time = 0.10, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 43} \[ \frac {\sin (c+d x)}{a^2 d}-\frac {\csc (c+d x)}{a^2 d}-\frac {2 \log (\sin (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^2 (a-x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {a^2}{x^2}-\frac {2 a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac {\csc (c+d x)}{a^2 d}-\frac {2 \log (\sin (c+d x))}{a^2 d}+\frac {\sin (c+d x)}{a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 32, normalized size = 0.74 \[ -\frac {-\sin (c+d x)+\csc (c+d x)+2 \log (\sin (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 42, normalized size = 0.98 \[ -\frac {\cos \left (d x + c\right )^{2} + 2 \, \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right )}{a^{2} d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 53, normalized size = 1.23 \[ -\frac {\frac {2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} - \frac {\sin \left (d x + c\right )}{a^{2}} - \frac {2 \, \sin \left (d x + c\right ) - 1}{a^{2} \sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 46, normalized size = 1.07 \[ \frac {\sin \left (d x +c \right )}{a^{2} d}-\frac {1}{a^{2} d \sin \left (d x +c \right )}-\frac {2 \ln \left (\sin \left (d x +c \right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 41, normalized size = 0.95 \[ -\frac {\frac {2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{2}} - \frac {\sin \left (d x + c\right )}{a^{2}} + \frac {1}{a^{2} \sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.92, size = 110, normalized size = 2.56 \[ \frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1}{d\,\left (2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^2\,d}+\frac {2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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