Optimal. Leaf size=25 \[ \frac{b \log (\tan (c+d x))}{d}-\frac{a \cot (c+d x)}{d} \]
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Rubi [A] time = 0.0783231, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {43} \[ \frac{b \log (\tan (c+d x))}{d}-\frac{a \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \csc ^2(c+d x) (a+b \tan (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b x}{x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^2}+\frac{b}{x}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{a \cot (c+d x)}{d}+\frac{b \log (\tan (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0594411, size = 36, normalized size = 1.44 \[ -\frac{a \cot (c+d x)}{d}-\frac{b (\log (\cos (c+d x))-\log (\sin (c+d x)))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 26, normalized size = 1. \begin{align*} -{\frac{\cot \left ( dx+c \right ) a}{d}}+{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38836, size = 34, normalized size = 1.36 \begin{align*} \frac{b \log \left (\tan \left (d x + c\right )\right ) - \frac{a}{\tan \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.49515, size = 171, normalized size = 6.84 \begin{align*} -\frac{b \log \left (\cos \left (d x + c\right )^{2}\right ) \sin \left (d x + c\right ) - b \log \left (-\frac{1}{4} \, \cos \left (d x + c\right )^{2} + \frac{1}{4}\right ) \sin \left (d x + c\right ) + 2 \, a \cos \left (d x + c\right )}{2 \, d \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\left (c + d x \right )}\right ) \csc ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36389, size = 47, normalized size = 1.88 \begin{align*} \frac{b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - \frac{b \tan \left (d x + c\right ) + a}{\tan \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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