Optimal. Leaf size=27 \[ \frac {2}{d \sqrt {a \cosh (c+d x)-a \sinh (c+d x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {3150}
\begin {gather*} \frac {2}{d \sqrt {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}{\sqrt {a \cosh (c+d x)-a \sinh (c+d x)}} \, dx &=\frac {2}{d \sqrt {a \cosh (c+d x)-a \sinh (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 26, normalized size = 0.96 \begin {gather*} \frac {2}{d \sqrt {a (\cosh (c+d x)-\sinh (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.64, size = 26, normalized size = 0.96
method | result | size |
risch | \(\frac {2}{\sqrt {a \,{\mathrm e}^{-d x -c}}\, d}\) | \(19\) |
gosper | \(\frac {2}{d \sqrt {a \cosh \left (d x +c \right )-a \sinh \left (d x +c \right )}}\) | \(26\) |
derivativedivides | \(\frac {2}{d \sqrt {a \cosh \left (d x +c \right )-a \sinh \left (d x +c \right )}}\) | \(26\) |
default | \(\frac {2}{d \sqrt {a \cosh \left (d x +c \right )-a \sinh \left (d x +c \right )}}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 17, normalized size = 0.63 \begin {gather*} \frac {2 \, e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{\sqrt {a} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 40, normalized size = 1.48 \begin {gather*} \frac {2 \, \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{a d} \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 \frac {1}{\sqrt {- a \sinh {\left (c + d x \right )} + a \cosh {\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 17, normalized size = 0.63 \begin {gather*} \frac {2 \, e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{\sqrt {a} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 27, normalized size = 1.00 \begin {gather*} \frac {2\,{\mathrm {e}}^{c+d\,x}\,\sqrt {a\,{\mathrm {e}}^{-c-d\,x}}}{a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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