Optimal. Leaf size=48 \[ -\frac{a \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a \cot (e+f x) \csc (e+f x)}{2 f}-\frac{b \cot (e+f x)}{f} \]
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Rubi [A] time = 0.0470495, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {2748, 3768, 3770, 3767, 8} \[ -\frac{a \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a \cot (e+f x) \csc (e+f x)}{2 f}-\frac{b \cot (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \csc ^3(e+f x) (a+b \sin (e+f x)) \, dx &=a \int \csc ^3(e+f x) \, dx+b \int \csc ^2(e+f x) \, dx\\ &=-\frac{a \cot (e+f x) \csc (e+f x)}{2 f}+\frac{1}{2} a \int \csc (e+f x) \, dx-\frac{b \operatorname{Subst}(\int 1 \, dx,x,\cot (e+f x))}{f}\\ &=-\frac{a \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{b \cot (e+f x)}{f}-\frac{a \cot (e+f x) \csc (e+f x)}{2 f}\\ \end{align*}
Mathematica [A] time = 0.0279821, size = 91, normalized size = 1.9 \[ -\frac{a \csc ^2\left (\frac{1}{2} (e+f x)\right )}{8 f}+\frac{a \sec ^2\left (\frac{1}{2} (e+f x)\right )}{8 f}+\frac{a \log \left (\sin \left (\frac{1}{2} (e+f x)\right )\right )}{2 f}-\frac{a \log \left (\cos \left (\frac{1}{2} (e+f x)\right )\right )}{2 f}-\frac{b \cot (e+f x)}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 54, normalized size = 1.1 \begin{align*} -{\frac{\cot \left ( fx+e \right ) a\csc \left ( fx+e \right ) }{2\,f}}+{\frac{a\ln \left ( \csc \left ( fx+e \right ) -\cot \left ( fx+e \right ) \right ) }{2\,f}}-{\frac{b\cot \left ( fx+e \right ) }{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61466, size = 81, normalized size = 1.69 \begin{align*} \frac{a{\left (\frac{2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - \frac{4 \, b}{\tan \left (f x + e\right )}}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74508, size = 251, normalized size = 5.23 \begin{align*} \frac{4 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 2 \, a \cos \left (f x + e\right ) -{\left (a \cos \left (f x + e\right )^{2} - a\right )} \log \left (\frac{1}{2} \, \cos \left (f x + e\right ) + \frac{1}{2}\right ) +{\left (a \cos \left (f x + e\right )^{2} - a\right )} \log \left (-\frac{1}{2} \, \cos \left (f x + e\right ) + \frac{1}{2}\right )}{4 \,{\left (f \cos \left (f x + e\right )^{2} - f\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 \sin{\left (e + f x \right )}\right ) \csc ^{3}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.05635, size = 124, normalized size = 2.58 \begin{align*} \frac{a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 4 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) \right |}\right ) + 4 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - \frac{6 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 4 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + a}{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2}}}{8 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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