Optimal. Leaf size=46 \[ -\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\log (\sin (c+d x)+1)}{a d} \]
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Rubi [A] time = 0.06, antiderivative size = 46, 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, 44} \[ -\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2833
Rubi steps
\begin {align*} \int \frac {\cot (c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^2}{x^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {1}{x^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (\frac {1}{a x^2}-\frac {1}{a^2 x}+\frac {1}{a^2 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\log (1+\sin (c+d x))}{a d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 1.00 \[ -\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 51, normalized size = 1.11 \[ -\frac {\log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 1}{a d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 45, normalized size = 0.98 \[ \frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {1}{a \sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 49, normalized size = 1.07 \[ -\frac {1}{d a \sin \left (d x +c \right )}-\frac {\ln \left (\sin \left (d x +c \right )\right )}{a d}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 43, normalized size = 0.93 \[ \frac {\frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {\log \left (\sin \left (d x + c\right )\right )}{a} - \frac {1}{a \sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.60, size = 55, normalized size = 1.20 \[ -\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {1}{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}}{2\,a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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