Optimal. Leaf size=24 \[ \frac {a \log (\sin (c+d x))}{d}+\frac {a \sin (c+d x)}{d} \]
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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2786, 45}
\begin {gather*} \frac {a \sin (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2786
Rubi steps
\begin {align*} \int \cot (c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\text {Subst}\left (\int \frac {a+x}{x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (1+\frac {a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \log (\sin (c+d x))}{d}+\frac {a \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 26, normalized size = 1.08 \begin {gather*} \frac {a (\log (\cos (c+d x))+\log (\tan (c+d x))+\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 20, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {a \left (\sin \left (d x +c \right )+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(20\) |
default | \(\frac {a \left (\sin \left (d x +c \right )+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(20\) |
risch | \(-i a x -\frac {2 i a c}{d}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {a \sin \left (d x +c \right )}{d}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 22, normalized size = 0.92 \begin {gather*} \frac {a \log \left (\sin \left (d x + c\right )\right ) + a \sin \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 24, normalized size = 1.00 \begin {gather*} \frac {a \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + a \sin \left (d x + c\right )}{d} \end {gather*}
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 \begin {gather*} a \left (\int \sin {\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int \cot {\left (c + d x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.04, size = 23, normalized size = 0.96 \begin {gather*} \frac {a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + a \sin \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.58, size = 38, normalized size = 1.58 \begin {gather*} \frac {a\,\left (\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )+\sin \left (c+d\,x\right )-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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