Optimal. Leaf size=20 \[ \frac {x}{2 a}+\frac {\sinh (x) \cosh (x)}{2 a} \]
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Rubi [A] time = 0.05, 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 = {3175, 2635, 8} \[ \frac {x}{2 a}+\frac {\sinh (x) \cosh (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3175
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 \[ \frac {\frac {x}{2}+\frac {1}{4} \sinh (2 x)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 12, normalized size = 0.60 \[ \frac {\cosh \relax (x) \sinh \relax (x) + x}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 28, normalized size = 1.40 \[ -\frac {{\left (2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} - 4 \, x - e^{\left (2 \, x\right )}}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 78, normalized size = 3.90 \[ \frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a}-\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 25, normalized size = 1.25 \[ \frac {x}{2 \, a} + \frac {e^{\left (2 \, x\right )}}{8 \, a} - \frac {e^{\left (-2 \, x\right )}}{8 \, a} \]
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 \[ \frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}+\frac {x}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.69, size = 153, normalized size = 7.65 \[ \frac {x \tanh ^{4}{\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} - \frac {2 x \tanh ^{2}{\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} + \frac {x}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} + \frac {2 \tanh ^{3}{\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} + \frac {2 \tanh {\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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