Optimal. Leaf size=13 \[ -\frac{B \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0066428, antiderivative size = 13, 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} \[ -\frac{B \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 3475
Rubi steps
\begin{align*} \int \frac{\tan (c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx &=B \int \tan (c+d x) \, dx\\ &=-\frac{B \log (\cos (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0068276, size = 13, normalized size = 1. \[ -\frac{B \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 18, normalized size = 1.4 \begin{align*}{\frac{B\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67684, size = 23, normalized size = 1.77 \begin{align*} \frac{B \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93919, size = 51, normalized size = 3.92 \begin{align*} -\frac{B \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.688866, size = 37, normalized size = 2.85 \begin{align*} \begin{cases} \frac{B \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text{for}\: d \neq 0 \\\frac{x \left (B a + B b \tan{\left (c \right )}\right ) \tan{\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 = 1.22798, size = 134, normalized size = 10.31 \begin{align*} \frac{B \log \left ({\left | -\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} - \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 2 \right |}\right ) - B \log \left ({\left | -\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} - \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 2 \right |}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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