Optimal. Leaf size=67 \[ \frac {1}{2} \text {Int}\left (\frac {\text {csch}^2(a+b x) \sec (\coth (a+b x))}{\coth (a+b x)-1},x\right )-\frac {1}{2} \text {Int}\left (\frac {\text {csch}^2(a+b x) \sec (\coth (a+b x))}{\coth (a+b x)+1},x\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}
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Mathematica [A] time = 6.63, size = 0, normalized size = 0.00 \[ \int \sec (\coth (a+b x)) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sec \left (\coth \left (b x + a\right )\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec \left (\coth \left (b x + a\right )\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 0, normalized size = 0.00 \[ \int \sec \left (\coth \left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec \left (\coth \left (b x + a\right )\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\cos \left (\mathrm {coth}\left (a+b\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec {\left (\coth {\left (a + b x \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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