Optimal. Leaf size=43 \[ \frac {3 x}{2 a}-\frac {2 \sinh (x)}{a}+\frac {3 \cosh (x) \sinh (x)}{2 a}-\frac {\cosh ^2(x) \sinh (x)}{a+a \cosh (x)} \]
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Rubi [A]
time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2846, 2813}
\begin {gather*} \frac {3 x}{2 a}-\frac {2 \sinh (x)}{a}-\frac {\sinh (x) \cosh ^2(x)}{a \cosh (x)+a}+\frac {3 \sinh (x) \cosh (x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2813
Rule 2846
Rubi steps
\begin {align*} \int \frac {\cosh ^3(x)}{a+a \cosh (x)} \, dx &=-\frac {\cosh ^2(x) \sinh (x)}{a+a \cosh (x)}-\frac {\int \cosh (x) (2 a-3 a \cosh (x)) \, dx}{a^2}\\ &=\frac {3 x}{2 a}-\frac {2 \sinh (x)}{a}+\frac {3 \cosh (x) \sinh (x)}{2 a}-\frac {\cosh ^2(x) \sinh (x)}{a+a \cosh (x)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 45, normalized size = 1.05 \begin {gather*} \frac {\text {sech}\left (\frac {x}{2}\right ) \left (12 x \cosh \left (\frac {x}{2}\right )-12 \sinh \left (\frac {x}{2}\right )-3 \sinh \left (\frac {3 x}{2}\right )+\sinh \left (\frac {5 x}{2}\right )\right )}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 70, normalized size = 1.63
method | result | size |
risch | \(\frac {3 x}{2 a}+\frac {{\mathrm e}^{2 x}}{8 a}-\frac {{\mathrm e}^{x}}{2 a}+\frac {{\mathrm e}^{-x}}{2 a}-\frac {{\mathrm e}^{-2 x}}{8 a}+\frac {2}{\left ({\mathrm e}^{x}+1\right ) a}\) | \(53\) |
default | \(\frac {-\tanh \left (\frac {x}{2}\right )+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}}{a}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 56, normalized size = 1.30 \begin {gather*} \frac {3 \, x}{2 \, a} + \frac {4 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )}}{8 \, a} - \frac {3 \, e^{\left (-x\right )} + 20 \, e^{\left (-2 \, x\right )} - 1}{8 \, {\left (a e^{\left (-2 \, x\right )} + a e^{\left (-3 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 70, normalized size = 1.63 \begin {gather*} \frac {\cosh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right ) - 4\right )} \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (12 \, x - 1\right )} \cosh \left (x\right ) - 4 \, \cosh \left (x\right )^{2} + {\left (3 \, \cosh \left (x\right )^{2} + 12 \, x - 4 \, \cosh \left (x\right ) - 7\right )} \sinh \left (x\right ) + 12 \, x + 20}{8 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + a\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. 189 vs.
\(2 (39) = 78\).
time = 0.41, size = 189, normalized size = 4.40 \begin {gather*} \frac {3 x \tanh ^{4}{\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} - \frac {6 x \tanh ^{2}{\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} + \frac {3 x}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} - \frac {2 \tanh ^{5}{\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} + \frac {10 \tanh ^{3}{\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} - \frac {4 \tanh {\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 51, normalized size = 1.19 \begin {gather*} \frac {3 \, x}{2 \, a} + \frac {{\left (20 \, e^{\left (2 \, x\right )} + 3 \, e^{x} - 1\right )} e^{\left (-2 \, x\right )}}{8 \, a {\left (e^{x} + 1\right )}} + \frac {a e^{\left (2 \, x\right )} - 4 \, a e^{x}}{8 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.92, size = 52, normalized size = 1.21 \begin {gather*} \frac {{\mathrm {e}}^{-x}}{2\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}+\frac {{\mathrm {e}}^{2\,x}}{8\,a}+\frac {3\,x}{2\,a}+\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}-\frac {{\mathrm {e}}^x}{2\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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