Optimal. Leaf size=35 \[ \frac {\text {ArcTan}(\sinh (c+b x)) \cosh (a-c)}{b}-\frac {\text {sech}(c+b x) \sinh (a-c)}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5746, 2686, 8,
3855} \begin {gather*} \frac {\cosh (a-c) \text {ArcTan}(\sinh (b x+c))}{b}-\frac {\sinh (a-c) \text {sech}(b x+c)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2686
Rule 3855
Rule 5746
Rubi steps
\begin {align*} \int \cosh (a+b x) \text {sech}^2(c+b x) \, dx &=\cosh (a-c) \int \text {sech}(c+b x) \, dx+\sinh (a-c) \int \text {sech}(c+b x) \tanh (c+b x) \, dx\\ &=\frac {\tan ^{-1}(\sinh (c+b x)) \cosh (a-c)}{b}-\frac {\sinh (a-c) \text {Subst}(\int 1 \, dx,x,\text {sech}(c+b x))}{b}\\ &=\frac {\tan ^{-1}(\sinh (c+b x)) \cosh (a-c)}{b}-\frac {\text {sech}(c+b x) \sinh (a-c)}{b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(35)=70\).
time = 0.06, size = 83, normalized size = 2.37 \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {(\cosh (c)-\sinh (c)) \left (\cosh \left (\frac {b x}{2}\right ) \sinh (c)+\cosh (c) \sinh \left (\frac {b x}{2}\right )\right )}{\cosh (c) \cosh \left (\frac {b x}{2}\right )-\cosh \left (\frac {b x}{2}\right ) \sinh (c)}\right ) \cosh (a-c)}{b}-\frac {\text {sech}(c+b x) \sinh (a-c)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.25, size = 183, normalized size = 5.23
method | result | size |
risch | \(-\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 a}-{\mathrm e}^{2 c}\right )}{b \left ({\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )}+\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}+\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}\) | \(183\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 70, normalized size = 2.00 \begin {gather*} -\frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (-b x - c\right )}\right ) e^{\left (-a - c\right )}}{b} - \frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-b x - a\right )}}{b {\left (e^{\left (-2 \, b x\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. 405 vs.
\(2 (35) = 70\).
time = 0.35, size = 405, normalized size = 11.57 \begin {gather*} \frac {2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} + {\left ({\left (\cosh \left (-a + c\right )^{2} + 1\right )} \cosh \left (b x + c\right )^{2} + {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} + 1\right )} \sinh \left (b x + c\right )^{2} + {\left (\cosh \left (b x + c\right )^{2} + 1\right )} \sinh \left (-a + c\right )^{2} + \cosh \left (-a + c\right )^{2} - 2 \, {\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - {\left (\cosh \left (-a + c\right )^{2} + 1\right )} \cosh \left (b x + c\right )\right )} \sinh \left (b x + c\right ) - 2 \, {\left (\cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) + \cosh \left (-a + c\right )\right )} \sinh \left (-a + c\right ) + 1\right )} \arctan \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right )\right ) - {\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right ) - {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} - 1\right )} \sinh \left (b x + c\right )}{b \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) + {\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right )^{2} + b \cosh \left (-a + c\right ) + 2 \, {\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right ) - {\left (b \cosh \left (b x + c\right )^{2} + b\right )} \sinh \left (-a + c\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 \cosh {\left (a + b x \right )} \operatorname {sech}^{2}{\left (b x + c \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 68, normalized size = 1.94 \begin {gather*} \frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (b x + c\right )}\right ) e^{\left (-a - c\right )} - \frac {{\left (e^{\left (b x + 2 \, a\right )} - e^{\left (b x + 2 \, c\right )}\right )} e^{\left (-a\right )}}{e^{\left (2 \, b x + 2 \, c\right )} + 1}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.55, size = 148, normalized size = 4.23 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{b\,x}\,\left (\sqrt {b^2}+{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\,\sqrt {b^2}\right )}{b\,\sqrt {{\mathrm {e}}^{-2\,a}\,{\mathrm {e}}^{2\,c}\,\left (2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}+{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}+1\right )}}\right )\,\sqrt {{\mathrm {e}}^{2\,c-2\,a}\,\left (2\,{\mathrm {e}}^{2\,a-2\,c}+{\mathrm {e}}^{4\,a-4\,c}+1\right )}}{\sqrt {b^2}}-\frac {{\mathrm {e}}^{a+b\,x}\,\left ({\mathrm {e}}^{2\,a-2\,c}-1\right )}{b\,\left ({\mathrm {e}}^{2\,a-2\,c}+{\mathrm {e}}^{2\,a+2\,b\,x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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