Optimal. Leaf size=17 \[ b x-\frac{a \tanh ^{-1}(\cos (e+f x))}{f} \]
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Rubi [A] time = 0.0215194, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2735, 3770} \[ b x-\frac{a \tanh ^{-1}(\cos (e+f x))}{f} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int \csc (e+f x) (a+b \sin (e+f x)) \, dx &=b x+a \int \csc (e+f x) \, dx\\ &=b x-\frac{a \tanh ^{-1}(\cos (e+f x))}{f}\\ \end{align*}
Mathematica [B] time = 0.0204587, size = 43, normalized size = 2.53 \[ \frac{a \log \left (\sin \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{f}-\frac{a \log \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{f}+b x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 32, normalized size = 1.9 \begin{align*} bx+{\frac{a\ln \left ( \csc \left ( fx+e \right ) -\cot \left ( fx+e \right ) \right ) }{f}}+{\frac{be}{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.74925, size = 39, normalized size = 2.29 \begin{align*} \frac{{\left (f x + e\right )} b - a \log \left (\cot \left (f x + e\right ) + \csc \left (f x + e\right )\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75978, size = 111, normalized size = 6.53 \begin{align*} \frac{2 \, b f x - a \log \left (\frac{1}{2} \, \cos \left (f x + e\right ) + \frac{1}{2}\right ) + a \log \left (-\frac{1}{2} \, \cos \left (f x + e\right ) + \frac{1}{2}\right )}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.32409, size = 51, normalized size = 3. \begin{align*} a \left (\begin{cases} \frac{x \cot{\left (e \right )} \csc{\left (e \right )}}{\cot{\left (e \right )} + \csc{\left (e \right )}} + \frac{x \csc ^{2}{\left (e \right )}}{\cot{\left (e \right )} + \csc{\left (e \right )}} & \text{for}\: f = 0 \\- \frac{\log{\left (\cot{\left (e + f x \right )} + \csc{\left (e + f x \right )} \right )}}{f} & \text{otherwise} \end{cases}\right ) + b x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.15185, size = 36, normalized size = 2.12 \begin{align*} \frac{{\left (f x + e\right )} b + a \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) \right |}\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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