Optimal. Leaf size=19 \[ \frac {\text {sech}^2(x)}{2 a}-\frac {\text {sech}(x)}{a} \]
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Rubi [A] time = 0.07, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2706, 2606, 30, 8} \[ \frac {\text {sech}^2(x)}{2 a}-\frac {\text {sech}(x)}{a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2606
Rule 2706
Rubi steps
\begin {align*} \int \frac {\tanh ^3(x)}{a+a \cosh (x)} \, dx &=\frac {\int \text {sech}(x) \tanh (x) \, dx}{a}-\frac {\int \text {sech}^2(x) \tanh (x) \, dx}{a}\\ &=-\frac {\operatorname {Subst}(\int 1 \, dx,x,\text {sech}(x))}{a}+\frac {\operatorname {Subst}(\int x \, dx,x,\text {sech}(x))}{a}\\ &=-\frac {\text {sech}(x)}{a}+\frac {\text {sech}^2(x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 17, normalized size = 0.89 \[ \frac {2 \sinh ^4\left (\frac {x}{2}\right ) \text {sech}^2(x)}{a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 66, normalized size = 3.47 \[ -\frac {2 \, {\left (\cosh \relax (x)^{2} + {\left (2 \, \cosh \relax (x) - 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} - \cosh \relax (x) + 1\right )}}{a \cosh \relax (x)^{3} + 3 \, a \cosh \relax (x) \sinh \relax (x)^{2} + a \sinh \relax (x)^{3} + 3 \, a \cosh \relax (x) + {\left (3 \, a \cosh \relax (x)^{2} + a\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 22, normalized size = 1.16 \[ -\frac {2 \, {\left (e^{\left (-x\right )} + e^{x} - 1\right )}}{a {\left (e^{\left (-x\right )} + e^{x}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 18, normalized size = 0.95 \[ \frac {-\frac {1}{\cosh \relax (x )}+\frac {1}{2 \cosh \relax (x )^{2}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 70, normalized size = 3.68 \[ -\frac {2 \, e^{\left (-x\right )}}{2 \, a e^{\left (-2 \, x\right )} + a e^{\left (-4 \, x\right )} + a} + \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, a e^{\left (-2 \, x\right )} + a e^{\left (-4 \, x\right )} + a} - \frac {2 \, e^{\left (-3 \, x\right )}}{2 \, a e^{\left (-2 \, x\right )} + a e^{\left (-4 \, x\right )} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 25, normalized size = 1.32 \[ -\frac {2\,{\mathrm {e}}^x\,\left ({\mathrm {e}}^{2\,x}-{\mathrm {e}}^x+1\right )}{a\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^2} \]
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 \[ \frac {\int \frac {\tanh ^{3}{\relax (x )}}{\cosh {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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