Optimal. Leaf size=17 \[ -\frac {B \cot (c+d x)}{d}-B x \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {21, 3473, 8} \[ -\frac {B \cot (c+d x)}{d}-B x \]
Antiderivative was successfully verified.
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Rule 8
Rule 21
Rule 3473
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx &=B \int \cot ^2(c+d x) \, dx\\ &=-\frac {B \cot (c+d x)}{d}-B \int 1 \, dx\\ &=-B x-\frac {B \cot (c+d x)}{d}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 30, normalized size = 1.76 \[ -\frac {B \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 42, normalized size = 2.47 \[ -\frac {B d x \sin \left (2 \, d x + 2 \, c\right ) + B \cos \left (2 \, d x + 2 \, c\right ) + B}{d \sin \left (2 \, d x + 2 \, c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 39, normalized size = 2.29 \[ -\frac {2 \, {\left (d x + c\right )} B - B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {B}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 22, normalized size = 1.29 \[ \frac {B \left (-\cot \left (d x +c \right )-d x -c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 23, normalized size = 1.35 \[ -\frac {{\left (d x + c\right )} B + \frac {B}{\tan \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.25, size = 16, normalized size = 0.94 \[ -\frac {B\,\left (\mathrm {cot}\left (c+d\,x\right )+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.99, size = 37, normalized size = 2.18 \[ \begin {cases} - B x - \frac {B \cot {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\\frac {x \left (B a + B b \tan {\relax (c )}\right ) \cot ^{2}{\relax (c )}}{a + b \tan {\relax (c )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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