Optimal. Leaf size=12 \[ \frac{B \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0066853, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {21, 3475} \[ \frac{B \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 3475
Rubi steps
\begin{align*} \int \frac{\cot (c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx &=B \int \cot (c+d x) \, dx\\ &=\frac{B \log (\sin (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0106897, size = 20, normalized size = 1.67 \[ \frac{B (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 13, normalized size = 1.1 \begin{align*}{\frac{B\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.69498, size = 39, normalized size = 3.25 \begin{align*} -\frac{B \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, B \log \left (\tan \left (d x + c\right )\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05819, size = 57, normalized size = 4.75 \begin{align*} \frac{B \log \left (-\frac{1}{2} \, \cos \left (2 \, d x + 2 \, c\right ) + \frac{1}{2}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.18723, size = 49, normalized size = 4.08 \begin{align*} \begin{cases} - \frac{B \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{B \log{\left (\tan{\left (c + d x \right )} \right )}}{d} & \text{for}\: d \neq 0 \\\frac{x \left (B a + B b \tan{\left (c \right )}\right ) \cot{\left (c \right )}}{a + b \tan{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27845, size = 80, normalized size = 6.67 \begin{align*} \frac{B \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 2 \, B \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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