Optimal. Leaf size=22 \[ -\frac {\text {csch}(a+b x)}{b}+\frac {\sinh (a+b x)}{b} \]
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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 = {2670, 14}
\begin {gather*} \frac {\sinh (a+b x)}{b}-\frac {\text {csch}(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2670
Rubi steps
\begin {align*} \int \cosh (a+b x) \coth ^2(a+b x) \, dx &=-\frac {i \text {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=-\frac {i \text {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 \begin {gather*} -\frac {\text {csch}(a+b x)}{b}+\frac {\sinh (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.88, size = 33, normalized size = 1.50
method | result | size |
derivativedivides | \(\frac {\frac {\cosh ^{2}\left (b x +a \right )}{\sinh \left (b x +a \right )}-\frac {2}{\sinh \left (b x +a \right )}}{b}\) | \(33\) |
default | \(\frac {\frac {\cosh ^{2}\left (b x +a \right )}{\sinh \left (b x +a \right )}-\frac {2}{\sinh \left (b x +a \right )}}{b}\) | \(33\) |
risch | \(\frac {{\mathrm e}^{b x +a}}{2 b}-\frac {{\mathrm e}^{-b x -a}}{2 b}-\frac {2 \,{\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (22) = 44\).
time = 0.26, size = 56, normalized size = 2.55 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 31, normalized size = 1.41 \begin {gather*} \frac {\cosh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{2} - 3}{2 \, b \sinh \left (b x + a\right )} \end {gather*}
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 \begin {gather*} \int \cosh {\left (a + b x \right )} \coth ^{2}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs.
\(2 (22) = 44\).
time = 0.40, size = 45, normalized size = 2.05 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.43, size = 49, normalized size = 2.23 \begin {gather*} \frac {{\mathrm {e}}^{-a-b\,x}\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-6\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}{2\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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