Optimal. Leaf size=35 \[ \frac {\sinh ((1-m) x)}{2 (1-m)}+\frac {\sinh ((1+m) x)}{2 (1+m)} \]
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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5733, 2717}
\begin {gather*} \frac {\sinh ((1-m) x)}{2 (1-m)}+\frac {\sinh ((m+1) x)}{2 (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 5733
Rubi steps
\begin {align*} \int \cosh (x) \cosh (m x) \, dx &=\int \left (\frac {1}{2} \cosh ((1-m) x)+\frac {1}{2} \cosh ((1+m) x)\right ) \, dx\\ &=\frac {1}{2} \int \cosh ((1-m) x) \, dx+\frac {1}{2} \int \cosh ((1+m) x) \, dx\\ &=\frac {\sinh ((1-m) x)}{2 (1-m)}+\frac {\sinh ((1+m) x)}{2 (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 25, normalized size = 0.71 \begin {gather*} \frac {-\cosh (m x) \sinh (x)+m \cosh (x) \sinh (m x)}{-1+m^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.24, size = 28, normalized size = 0.80
method | result | size |
default | \(\frac {\sinh \left (\left (-1+m \right ) x \right )}{-2+2 m}+\frac {\sinh \left (\left (1+m \right ) x \right )}{2+2 m}\) | \(28\) |
risch | \(\frac {\left (m \,{\mathrm e}^{2 x}-{\mathrm e}^{2 x}+m +1\right ) {\mathrm e}^{\left (-1+m \right ) x}}{4 \left (1+m \right ) \left (-1+m \right )}-\frac {\left (m \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x}+m -1\right ) {\mathrm e}^{-\left (1+m \right ) x}}{4 \left (1+m \right ) \left (-1+m \right )}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 42, normalized size = 1.20 \begin {gather*} \frac {m \cosh \left (x\right ) \sinh \left (m x\right ) - \cosh \left (m x\right ) \sinh \left (x\right )}{{\left (m^{2} - 1\right )} \cosh \left (x\right )^{2} - {\left (m^{2} - 1\right )} \sinh \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (22) = 44\).
time = 0.20, size = 56, normalized size = 1.60 \begin {gather*} \begin {cases} - \frac {x \sinh ^{2}{\left (x \right )}}{2} + \frac {x \cosh ^{2}{\left (x \right )}}{2} + \frac {\sinh {\left (x \right )} \cosh {\left (x \right )}}{2} & \text {for}\: m = -1 \vee m = 1 \\\frac {m \sinh {\left (m x \right )} \cosh {\left (x \right )}}{m^{2} - 1} - \frac {\sinh {\left (x \right )} \cosh {\left (m x \right )}}{m^{2} - 1} & \text {otherwise} \end {cases} \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. 59 vs.
\(2 (28) = 56\).
time = 0.39, size = 59, normalized size = 1.69 \begin {gather*} \frac {e^{\left (m x + x\right )}}{4 \, {\left (m + 1\right )}} + \frac {e^{\left (m x - x\right )}}{4 \, {\left (m - 1\right )}} - \frac {e^{\left (-m x + x\right )}}{4 \, {\left (m - 1\right )}} - \frac {e^{\left (-m x - x\right )}}{4 \, {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 26, normalized size = 0.74 \begin {gather*} -\frac {\mathrm {cosh}\left (m\,x\right )\,\mathrm {sinh}\left (x\right )-m\,\mathrm {sinh}\left (m\,x\right )\,\mathrm {cosh}\left (x\right )}{m^2-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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