Optimal. Leaf size=20 \[ \frac {x}{2 a}+\frac {\cosh (x) \sinh (x)}{2 a} \]
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Rubi [A]
time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3254, 2715, 8}
\begin {gather*} \frac {x}{2 a}+\frac {\sinh (x) \cosh (x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 3254
Rubi steps
\begin {align*} \int \frac {\cosh ^4(x)}{a+a \sinh ^2(x)} \, dx &=\frac {\int \cosh ^2(x) \, dx}{a}\\ &=\frac {\cosh (x) \sinh (x)}{2 a}+\frac {\int 1 \, dx}{2 a}\\ &=\frac {x}{2 a}+\frac {\cosh (x) \sinh (x)}{2 a}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 0.90 \begin {gather*} \frac {\frac {x}{2}+\frac {1}{4} \sinh (2 x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs.
\(2(16)=32\).
time = 0.43, size = 65, normalized size = 3.25
method | result | size |
risch | \(\frac {x}{2 a}+\frac {{\mathrm e}^{2 x}}{8 a}-\frac {{\mathrm e}^{-2 x}}{8 a}\) | \(26\) |
default | \(\frac {-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2}{4 \tanh \left (\frac {x}{2}\right )+4}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {2}{4 \tanh \left (\frac {x}{2}\right )-4}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}}{a}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 25, normalized size = 1.25 \begin {gather*} \frac {x}{2 \, a} + \frac {e^{\left (2 \, x\right )}}{8 \, a} - \frac {e^{\left (-2 \, x\right )}}{8 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 12, normalized size = 0.60 \begin {gather*} \frac {\cosh \left (x\right ) \sinh \left (x\right ) + x}{2 \, a} \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. 153 vs.
\(2 (14) = 28\).
time = 1.56, size = 153, normalized size = 7.65 \begin {gather*} \frac {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 {2 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 {x}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} + \frac {2 \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 {2 \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 = 0.42, size = 28, normalized size = 1.40 \begin {gather*} -\frac {{\left (2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} - 4 \, x - e^{\left (2 \, x\right )}}{8 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.33, size = 25, normalized size = 1.25 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}+\frac {x}{2\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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