Optimal. Leaf size=11 \[ -\frac {\text {csch}(a+b x)}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2606, 8} \[ -\frac {\text {csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2606
Rubi steps
\begin {align*} \int \coth (a+b x) \text {csch}(a+b x) \, dx &=-\frac {i \operatorname {Subst}(\int 1 \, dx,x,-i \text {csch}(a+b x))}{b}\\ &=-\frac {\text {csch}(a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ -\frac {\text {csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 56, normalized size = 5.09 \[ -\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 25, normalized size = 2.27 \[ -\frac {2}{b {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 12, normalized size = 1.09 \[ -\frac {\mathrm {csch}\left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 25, normalized size = 2.27 \[ -\frac {2}{b {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.42, size = 13, normalized size = 1.18 \[ -\frac {1}{b\,\mathrm {sinh}\left (a+b\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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