Optimal. Leaf size=16 \[ \frac {a \log (1-\cos (c+d x))}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3879, 31} \[ \frac {a \log (1-\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3879
Rubi steps
\begin {align*} \int \cot (c+d x) (a+a \sec (c+d x)) \, dx &=-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{a-a x} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {a \log (1-\cos (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 29, normalized size = 1.81 \[ \frac {2 a \left (\log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 16, normalized size = 1.00 \[ \frac {a \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 58, normalized size = 3.62 \[ \frac {a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 29, normalized size = 1.81 \[ -\frac {a \ln \left (\sec \left (d x +c \right )\right )}{d}+\frac {a \ln \left (-1+\sec \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 14, normalized size = 0.88 \[ \frac {a \log \left (\cos \left (d x + c\right ) - 1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 34, normalized size = 2.12 \[ \frac {a\,\left (2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\right )}{d} \]
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 \[ a \left (\int \cot {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \cot {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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