Optimal. Leaf size=13 \[ -\frac {B \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 13, 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} \[ -\frac {B \log (\cos (c+d x))}{d} \]
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*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ -\frac {B \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 19, normalized size = 1.46 \[ -\frac {B \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 99, normalized size = 7.62 \[ \frac {B \log \left ({\left | -\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} - \frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 2 \right |}\right ) - B \log \left ({\left | -\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} - \frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 2 \right |}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 18, normalized size = 1.38 \[ \frac {B \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 17, normalized size = 1.31 \[ \frac {B \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.24, size = 17, normalized size = 1.31 \[ \frac {B\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 37, normalized size = 2.85 \[ \begin {cases} \frac {B \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text {for}\: d \neq 0 \\\frac {x \left (B a + B b \tan {\relax (c )}\right ) \tan {\relax (c )}}{a + b \tan {\relax (c )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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