Optimal. Leaf size=11 \[ \frac {\log (\tanh (a+b x))}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2700, 29}
\begin {gather*} \frac {\log (\tanh (a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2700
Rubi steps
\begin {align*} \int \text {csch}(a+b x) \text {sech}(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac {\log (\tanh (a+b x))}{b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(31\) vs. \(2(11)=22\).
time = 0.02, size = 31, normalized size = 2.82 \begin {gather*} 2 \left (-\frac {\log (\cosh (a+b x))}{2 b}+\frac {\log (\sinh (a+b x))}{2 b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.62, size = 12, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {\ln \left (\tanh \left (b x +a \right )\right )}{b}\) | \(12\) |
default | \(\frac {\ln \left (\tanh \left (b x +a \right )\right )}{b}\) | \(12\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 50 vs.
\(2 (11) = 22\).
time = 0.47, size = 50, normalized size = 4.55 \begin {gather*} \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} - \frac {\log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (11) = 22\).
time = 0.35, size = 60, normalized size = 5.45 \begin {gather*} -\frac {\log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) - \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \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 \operatorname {csch}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (11) = 22\).
time = 0.40, size = 41, normalized size = 3.73 \begin {gather*} -\frac {\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) - \log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 30, normalized size = 2.73 \begin {gather*} -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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