Optimal. Leaf size=24 \[ \frac {1}{d (a \cosh (c+d x)-a \sinh (c+d x))} \]
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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3150}
\begin {gather*} \frac {1}{d (a \cosh (c+d x)-a \sinh (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 3150
Rubi steps
\begin {align*} \int \frac {1}{a \cosh (c+d x)-a \sinh (c+d x)} \, dx &=\frac {1}{d (a \cosh (c+d x)-a \sinh (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.92 \begin {gather*} \frac {1}{a d \cosh (c+d x)-a d \sinh (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 6.76, size = 22, normalized size = 0.92
method | result | size |
risch | \(\frac {{\mathrm e}^{d x +c}}{a d}\) | \(14\) |
derivativedivides | \(-\frac {2}{d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(22\) |
default | \(-\frac {2}{d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(22\) |
gosper | \(-\frac {1}{a d \left (\sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 13, normalized size = 0.54 \begin {gather*} \frac {e^{\left (d x + c\right )}}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 20, normalized size = 0.83 \begin {gather*} \frac {\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 32, normalized size = 1.33 \begin {gather*} \begin {cases} \frac {1}{- a d \sinh {\left (c + d x \right )} + a d \cosh {\left (c + d x \right )}} & \text {for}\: d \neq 0 \\\frac {x}{- a \sinh {\left (c \right )} + a \cosh {\left (c \right )}} & \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 = 13, normalized size = 0.54 \begin {gather*} \frac {e^{\left (d x + c\right )}}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.53, size = 13, normalized size = 0.54 \begin {gather*} \frac {{\mathrm {e}}^{c+d\,x}}{a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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