Optimal. Leaf size=36 \[ \frac {\text {csch}(a-c) \log (\cosh (a+b x))}{b}-\frac {\text {csch}(a-c) \log (\cosh (c+b x))}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5763, 3556}
\begin {gather*} \frac {\text {csch}(a-c) \log (\cosh (a+b x))}{b}-\frac {\text {csch}(a-c) \log (\cosh (b x+c))}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 5763
Rubi steps
\begin {align*} \int \text {sech}(a+b x) \text {sech}(c+b x) \, dx &=\text {csch}(a-c) \int \tanh (a+b x) \, dx-\text {csch}(a-c) \int \tanh (c+b x) \, dx\\ &=\frac {\text {csch}(a-c) \log (\cosh (a+b x))}{b}-\frac {\text {csch}(a-c) \log (\cosh (c+b x))}{b}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 27, normalized size = 0.75 \begin {gather*} \frac {\text {csch}(a-c) (\log (\cosh (a+b x))-\log (\cosh (c+b x)))}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs.
\(2(36)=72\).
time = 0.63, size = 77, normalized size = 2.14
method | result | size |
risch | \(-\frac {2 \ln \left ({\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 a -2 c}\right ) {\mathrm e}^{a +c}}{\left ({\mathrm e}^{2 a}-{\mathrm e}^{2 c}\right ) b}+\frac {2 \ln \left ({\mathrm e}^{2 b x +2 a}+1\right ) {\mathrm e}^{a +c}}{\left ({\mathrm e}^{2 a}-{\mathrm e}^{2 c}\right ) b}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 68, normalized size = 1.89 \begin {gather*} \frac {2 \, e^{\left (a + c\right )} \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} - \frac {2 \, e^{\left (a + c\right )} \log \left (e^{\left (-2 \, b x\right )} + e^{\left (2 \, c\right )}\right )}{b {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\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. 184 vs.
\(2 (36) = 72\).
time = 0.37, size = 184, normalized size = 5.11 \begin {gather*} \frac {2 \, {\left ({\left (\cosh \left (-a + c\right ) - \sinh \left (-a + c\right )\right )} \log \left (\frac {2 \, {\left (\cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - \sinh \left (b x + c\right ) \sinh \left (-a + c\right )\right )}}{\cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - {\left (\cosh \left (-a + c\right ) + \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right ) + \cosh \left (b x + c\right ) \sinh \left (-a + c\right )}\right ) - {\left (\cosh \left (-a + c\right ) - \sinh \left (-a + c\right )\right )} \log \left (\frac {2 \, \cosh \left (b x + c\right )}{\cosh \left (b x + c\right ) - \sinh \left (b x + c\right )}\right )\right )}}{b \cosh \left (-a + c\right )^{2} - 2 \, b \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + b \sinh \left (-a + c\right )^{2} - 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 {sech}{\left (a + b x \right )} \operatorname {sech}{\left (b x + c \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. 79 vs.
\(2 (36) = 72\).
time = 0.39, size = 79, normalized size = 2.19 \begin {gather*} \frac {2 \, {\left (\frac {e^{\left (3 \, a + c\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{e^{\left (4 \, a\right )} - e^{\left (2 \, a + 2 \, c\right )}} - \frac {e^{\left (a + 3 \, c\right )} \log \left (e^{\left (2 \, b x + 2 \, c\right )} + 1\right )}{e^{\left (2 \, a + 2 \, c\right )} - e^{\left (4 \, c\right )}}\right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.39, size = 266, normalized size = 7.39 \begin {gather*} \frac {4\,\sqrt {{\mathrm {e}}^{2\,a-2\,c}}\,\mathrm {atan}\left (\frac {b\,\left ({\mathrm {e}}^{-a}\,{\mathrm {e}}^c+{\mathrm {e}}^{-3\,a}\,{\mathrm {e}}^{3\,c}\right )\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\right )}^{3/2}}{\sqrt {-b^2\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}-1\right )}^2}}+\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\left (\frac {2\,{\mathrm {e}}^{-c}\,{\mathrm {e}}^a}{b\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\right )}^{3/2}}+\frac {2\,\left ({\mathrm {e}}^{-a}\,{\mathrm {e}}^c+{\mathrm {e}}^{-3\,a}\,{\mathrm {e}}^{3\,c}\right )\,\left (b\,\sqrt {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}}+b\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\right )}^{3/2}\right )}{\sqrt {-b^2\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}-1\right )}^2}\,\sqrt {2\,b^2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}-b^2-b^2\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}}}\right )\,\sqrt {2\,b^2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}-b^2-b^2\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}}}{4}\right )}{\sqrt {2\,b^2\,{\mathrm {e}}^{2\,a-2\,c}-b^2\,{\mathrm {e}}^{4\,a-4\,c}-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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