Optimal. Leaf size=65 \[ \frac{1}{2} \text{Unintegrable}\left (\frac{\text{csch}^2(a+b x) \sec (\coth (a+b x))}{\coth (a+b x)-1},x\right )-\frac{1}{2} \text{Unintegrable}\left (\frac{\text{csch}^2(a+b x) \sec (\coth (a+b x))}{\coth (a+b x)+1},x\right ) \]
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Rubi [A] time = 0.0770718, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \sec (\coth (a+b x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \sec (\coth (a+b x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sec (x)}{1-x^2} \, dx,x,\coth (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{\sec (x)}{2 (-1+x)}+\frac{\sec (x)}{2 (1+x)}\right ) \, dx,x,\coth (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sec (x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{\sec (x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 5.48446, size = 0, normalized size = 0. \[ \int \sec (\coth (a+b x)) \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.067, size = 0, normalized size = 0. \begin{align*} \int \sec \left ({\rm coth} \left (bx+a\right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sec \left (\coth \left (b x + a\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sec \left (\coth \left (b x + a\right )\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sec{\left (\coth{\left (a + b x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sec \left (\coth \left (b x + a\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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