Optimal. Leaf size=16 \[ a x-\frac{b \cot (c+d x)}{d} \]
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Rubi [A] time = 0.0124905, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3767, 8} \[ a x-\frac{b \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \left (a+b \csc ^2(c+d x)\right ) \, dx &=a x+b \int \csc ^2(c+d x) \, dx\\ &=a x-\frac{b \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=a x-\frac{b \cot (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0082534, size = 16, normalized size = 1. \[ a x-\frac{b \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 17, normalized size = 1.1 \begin{align*} ax-{\frac{b\cot \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976349, size = 24, normalized size = 1.5 \begin{align*} a x - \frac{b}{d \tan \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.468637, size = 76, normalized size = 4.75 \begin{align*} \frac{a d x \sin \left (d x + c\right ) - b \cos \left (d x + c\right )}{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 \csc ^{2}{\left (c + d x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37975, size = 24, normalized size = 1.5 \begin{align*} a x - \frac{b}{d \tan \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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