Optimal. Leaf size=20 \[ \frac {\text {ArcTan}(\sinh (x))}{a}-\frac {\tanh (x)}{a+a \text {sech}(x)} \]
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Rubi [A]
time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3874, 3855,
3879} \begin {gather*} \frac {\text {ArcTan}(\sinh (x))}{a}-\frac {\tanh (x)}{a \text {sech}(x)+a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rule 3874
Rule 3879
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(x)}{a+a \text {sech}(x)} \, dx &=\frac {\int \text {sech}(x) \, dx}{a}-\int \frac {\text {sech}(x)}{a+a \text {sech}(x)} \, dx\\ &=\frac {\tan ^{-1}(\sinh (x))}{a}-\frac {\tanh (x)}{a+a \text {sech}(x)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 1.10 \begin {gather*} \frac {2 \text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )-\tanh \left (\frac {x}{2}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 19, normalized size = 0.95
method | result | size |
default | \(\frac {-\tanh \left (\frac {x}{2}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a}\) | \(19\) |
risch | \(\frac {2}{\left ({\mathrm e}^{x}+1\right ) a}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{a}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{a}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 23, normalized size = 1.15 \begin {gather*} -\frac {2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} - \frac {2}{a e^{\left (-x\right )} + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 29, normalized size = 1.45 \begin {gather*} \frac {2 \, {\left ({\left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 1\right )}}{a \cosh \left (x\right ) + a \sinh \left (x\right ) + a} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\operatorname {sech}^{2}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 \, \arctan \left (e^{x}\right )}{a} + \frac {2}{a {\left (e^{x} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.30, size = 31, normalized size = 1.55 \begin {gather*} \frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}+\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{\sqrt {a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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