Optimal. Leaf size=13 \[ -\frac {1}{b (a+b \sinh (x))} \]
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Rubi [A]
time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2747, 32}
\begin {gather*} -\frac {1}{b (a+b \sinh (x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2747
Rubi steps
\begin {align*} \int \frac {\cosh (x)}{(a+b \sinh (x))^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{(a+x)^2} \, dx,x,b \sinh (x)\right )}{b}\\ &=-\frac {1}{b (a+b \sinh (x))}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{b (a+b \sinh (x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 14, normalized size = 1.08
method | result | size |
derivativedivides | \(-\frac {1}{b \left (a +b \sinh \left (x \right )\right )}\) | \(14\) |
default | \(-\frac {1}{b \left (a +b \sinh \left (x \right )\right )}\) | \(14\) |
risch | \(-\frac {2 \,{\mathrm e}^{x}}{b \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}-b \right )}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (b \sinh \left (x\right ) + a\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (13) = 26\).
time = 0.42, size = 51, normalized size = 3.92 \begin {gather*} -\frac {2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 19, normalized size = 1.46 \begin {gather*} \begin {cases} - \frac {1}{a b + b^{2} \sinh {\left (x \right )}} & \text {for}\: b \neq 0 \\\frac {\sinh {\left (x \right )}}{a^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 22, normalized size = 1.69 \begin {gather*} \frac {2}{{\left (b {\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.53, size = 14, normalized size = 1.08 \begin {gather*} \frac {\mathrm {sinh}\left (x\right )}{a\,\left (a+b\,\mathrm {sinh}\left (x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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