Optimal. Leaf size=26 \[ -\frac {1}{2 d (a \cosh (c+d x)+a \sinh (c+d x))^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3150}
\begin {gather*} -\frac {1}{2 d (a \sinh (c+d x)+a \cosh (c+d x))^2} \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))^2} \, dx &=-\frac {1}{2 d (a \cosh (c+d x)+a \sinh (c+d x))^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 26, normalized size = 1.00 \begin {gather*} -\frac {1}{2 d (a \cosh (c+d x)+a \sinh (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.79, size = 25, normalized size = 0.96
method | result | size |
risch | \(-\frac {{\mathrm e}^{-2 d x -2 c}}{2 a^{2} d}\) | \(18\) |
gosper | \(-\frac {1}{2 d \,a^{2} \left (\cosh \left (d x +c \right )+\sinh \left (d x +c \right )\right )^{2}}\) | \(24\) |
derivativedivides | \(-\frac {1}{2 d \left (a \cosh \left (d x +c \right )+a \sinh \left (d x +c \right )\right )^{2}}\) | \(25\) |
default | \(-\frac {1}{2 d \left (a \cosh \left (d x +c \right )+a \sinh \left (d x +c \right )\right )^{2}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 17, normalized size = 0.65 \begin {gather*} -\frac {e^{\left (-2 \, d x - 2 \, c\right )}}{2 \, a^{2} d} \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. 49 vs.
\(2 (24) = 48\).
time = 0.38, size = 49, normalized size = 1.88 \begin {gather*} -\frac {1}{2 \, {\left (a^{2} d \cosh \left (d x + c\right )^{2} + 2 \, a^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} d \sinh \left (d x + c\right )^{2}\right )}} \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. 66 vs.
\(2 (24) = 48\).
time = 0.41, size = 66, normalized size = 2.54 \begin {gather*} \begin {cases} - \frac {1}{2 a^{2} d \sinh ^{2}{\left (c + d x \right )} + 4 a^{2} d \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )} + 2 a^{2} d \cosh ^{2}{\left (c + d x \right )}} & \text {for}\: d \neq 0 \\\frac {x}{\left (a \sinh {\left (c \right )} + a \cosh {\left (c \right )}\right )^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 17, normalized size = 0.65 \begin {gather*} -\frac {e^{\left (-2 \, d x - 2 \, c\right )}}{2 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 17, normalized size = 0.65 \begin {gather*} -\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{2\,a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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