Optimal. Leaf size=50 \[ -\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {b \log (a+b \sin (c+d x))}{a^2 d}-\frac {\csc (c+d x)}{a d} \]
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Rubi [A] time = 0.07, antiderivative size = 50, 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} \[ -\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {b \log (a+b \sin (c+d x))}{a^2 d}-\frac {\csc (c+d x)}{a d} \]
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+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^2}{x^2 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {b \operatorname {Subst}\left (\int \frac {1}{x^2 (a+x)} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \left (\frac {1}{a x^2}-\frac {1}{a^2 x}+\frac {1}{a^2 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {\csc (c+d x)}{a d}-\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {b \log (a+b \sin (c+d x))}{a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 50, normalized size = 1.00 \[ -\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {b \log (a+b \sin (c+d x))}{a^2 d}-\frac {\csc (c+d x)}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 56, normalized size = 1.12 \[ \frac {b \log \left (b \sin \left (d x + c\right ) + a\right ) \sin \left (d x + c\right ) - b \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - a}{a^{2} d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 49, normalized size = 0.98 \[ \frac {\frac {b \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{2}} - \frac {b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} - \frac {1}{a \sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 35, normalized size = 0.70 \[ -\frac {\csc \left (d x +c \right )}{a d}+\frac {b \ln \left (a \csc \left (d x +c \right )+b \right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 47, normalized size = 0.94 \[ \frac {\frac {b \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{2}} - \frac {b \log \left (\sin \left (d x + c\right )\right )}{a^{2}} - \frac {1}{a \sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.81, size = 89, normalized size = 1.78 \[ \frac {b\,\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}{a^2\,d}-\frac {b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {1}{2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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