Optimal. Leaf size=37 \[ \frac {\text {ArcTan}\left (\frac {\cosh (x) (b+a \tanh (x))}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3590, 212}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\cosh (x) (a \tanh (x)+b)}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3590
Rubi steps
\begin {align*} \int \frac {\text {sech}(x)}{a+b \tanh (x)} \, dx &=i \text {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,\cosh (x) (-i b-i a \tanh (x))\right )\\ &=\frac {\tan ^{-1}\left (\frac {\cosh (x) (b+a \tanh (x))}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 46, normalized size = 1.24 \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 39, normalized size = 1.05
method | result | size |
default | \(\frac {2 \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}\) | \(39\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 148, normalized size = 4.00 \begin {gather*} \left [-\frac {\sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - a + b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} + a - b}\right )}{a^{2} - b^{2}}, -\frac {2 \, \arctan \left (\frac {\sqrt {a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )}\right )}{\sqrt {a^{2} - b^{2}}}\right ] \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*} \int \frac {\operatorname {sech}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 35, normalized size = 0.95 \begin {gather*} \frac {2 \, \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 35, normalized size = 0.95 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2-b^2}}{a-b}\right )}{\sqrt {a^2-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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