Optimal. Leaf size=11 \[ -\frac {\coth (a+b x)}{b} \]
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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 = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3767, 8} \[ -\frac {\coth (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rubi steps
\begin {align*} \int \text {csch}^2(a+b x) \, dx &=-\frac {i \operatorname {Subst}(\int 1 \, dx,x,-i \coth (a+b x))}{b}\\ &=-\frac {\coth (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ -\frac {\coth (a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 43, normalized size = 3.91 \[ -\frac {2}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 18, normalized size = 1.64 \[ -\frac {2}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 12, normalized size = 1.09 \[ -\frac {\coth \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 18, normalized size = 1.64 \[ \frac {2}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 18, normalized size = 1.64 \[ -\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
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 \[ \int \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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