Optimal. Leaf size=30 \[ -\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d} \]
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Rubi [A] time = 0.08, antiderivative size = 30, 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 = {2836, 12, 43} \[ -\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^2 (a-x)}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a-x}{x^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a}{x^2}-\frac {1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 22, normalized size = 0.73 \[ -\frac {\csc (c+d x)+\log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 34, normalized size = 1.13 \[ -\frac {\log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 1}{a d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 30, normalized size = 1.00 \[ -\frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {1}{a \sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 30, normalized size = 1.00 \[ -\frac {\csc \left (d x +c \right )}{a d}+\frac {\ln \left (\csc \left (d x +c \right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 29, normalized size = 0.97 \[ -\frac {\frac {\log \left (\sin \left (d x + c\right )\right )}{a} + \frac {1}{a \sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.70, size = 59, normalized size = 1.97 \[ -\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+\frac {1}{2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}}{a\,d} \]
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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