Optimal. Leaf size=27 \[ \frac {\log (\tanh (a+b x))}{b}-\frac {\tanh ^2(a+b x)}{2 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2700, 14}
\begin {gather*} \frac {\log (\tanh (a+b x))}{b}-\frac {\tanh ^2(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2700
Rubi steps
\begin {align*} \int \text {csch}(a+b x) \text {sech}^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {1+x^2}{x} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{x}+x\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac {\log (\tanh (a+b x))}{b}-\frac {\tanh ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 36, normalized size = 1.33 \begin {gather*} -\frac {2 \log (\cosh (a+b x))-2 \log (\sinh (a+b x))-\text {sech}^2(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 23, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {1}{2 \cosh \left (b x +a \right )^{2}}+\ln \left (\tanh \left (b x +a \right )\right )}{b}\) | \(23\) |
default | \(\frac {\frac {1}{2 \cosh \left (b x +a \right )^{2}}+\ln \left (\tanh \left (b x +a \right )\right )}{b}\) | \(23\) |
risch | \(\frac {2 \,{\mathrm e}^{2 b x +2 a}}{b \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\) | \(62\) |
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. 88 vs.
\(2 (25) = 50\).
time = 0.47, size = 88, normalized size = 3.26 \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} + \frac {2 \, e^{\left (-2 \, b x - 2 \, a\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} + 1\right )}} \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. 371 vs.
\(2 (25) = 50\).
time = 0.37, size = 371, normalized size = 13.74 \begin {gather*} \frac {2 \, \cosh \left (b x + a\right )^{2} - {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )^{2}}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\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}^{3}{\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. 93 vs.
\(2 (25) = 50\).
time = 0.38, size = 93, normalized size = 3.44 \begin {gather*} \frac {\frac {e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} + 6}{e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} + 2} - \log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} + 2\right ) + \log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} - 2\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.50, size = 78, normalized size = 2.89 \begin {gather*} \frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {2}{b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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