Optimal. Leaf size=12 \[ \frac {B \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.67, size = 20, normalized size = 1.67 \[ \frac {B \log \left (-\frac {1}{2} \, \cos \left (2 \, d x + 2 \, c\right ) + \frac {1}{2}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 59, normalized size = 4.92 \[ \frac {B \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 2 \, B \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 13, normalized size = 1.08 \[ \frac {B \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.95, size = 29, normalized size = 2.42 \[ -\frac {B \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, B \log \left (\tan \left (d x + c\right )\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.27, size = 27, normalized size = 2.25 \[ -\frac {B\,\left (\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )-2\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.80, size = 49, normalized size = 4.08 \[ \begin {cases} - \frac {B \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B \log {\left (\tan {\left (c + d x \right )} \right )}}{d} & \text {for}\: d \neq 0 \\\frac {x \left (B a + B b \tan {\relax (c )}\right ) \cot {\relax (c )}}{a + b \tan {\relax (c )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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