Optimal. Leaf size=25 \[ \frac {a \log (\sin (c+d x))}{d}-\frac {a \csc (c+d x)}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2833, 12, 43} \[ \frac {a \log (\sin (c+d x))}{d}-\frac {a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cot (c+d x) \csc (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 d}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {a+x}{x^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (\frac {a}{x^2}+\frac {1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 33, normalized size = 1.32 \[ \frac {a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}-\frac {a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 33, normalized size = 1.32 \[ \frac {a \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - a}{d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 26, normalized size = 1.04 \[ \frac {a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {a}{\sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 28, normalized size = 1.12 \[ -\frac {a}{d \sin \left (d x +c \right )}+\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 25, normalized size = 1.00 \[ \frac {a \log \left (\sin \left (d x + c\right )\right ) - \frac {a}{\sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.58, size = 55, normalized size = 2.20 \[ -\frac {a\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )+2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+\frac {1}{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2\,d} \]
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 \[ a \left (\int \cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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