Optimal. Leaf size=22 \[ \frac {\sinh (a+b x)}{b}-\frac {\text {csch}(a+b x)}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2590, 14} \[ \frac {\sinh (a+b x)}{b}-\frac {\text {csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2590
Rubi steps
\begin {align*} \int \cosh (a+b x) \coth ^2(a+b x) \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=-\frac {\text {csch}(a+b x)}{b}+\frac {\sinh (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ \frac {\sinh (a+b x)}{b}-\frac {\text {csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 31, normalized size = 1.41 \[ \frac {\cosh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{2} - 3}{2 \, b \sinh \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 45, normalized size = 2.05 \[ -\frac {\frac {4}{e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}} - e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 33, normalized size = 1.50 \[ \frac {\frac {\cosh ^{2}\left (b x +a \right )}{\sinh \left (b x +a \right )}-\frac {2}{\sinh \left (b x +a \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 56, normalized size = 2.55 \[ -\frac {e^{\left (-b x - a\right )}}{2 \, b} - \frac {5 \, e^{\left (-2 \, b x - 2 \, a\right )} - 1}{2 \, b {\left (e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 22, normalized size = 1.00 \[ \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2-1}{b\,\mathrm {sinh}\left (a+b\,x\right )} \]
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 \[ \int \cosh ^{3}{\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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