Optimal. Leaf size=49 \[ \frac {e^{a c+b c x} \tanh ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\log \left (1-e^{2 c (a+b x)}\right )}{b c} \]
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Rubi [A]
time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2225, 6410,
2320, 12, 266} \begin {gather*} \frac {\log \left (1-e^{2 c (a+b x)}\right )}{b c}+\frac {e^{a c+b c x} \tanh ^{-1}(\cosh (c (a+b x)))}{b c} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 266
Rule 2225
Rule 2320
Rule 6410
Rubi steps
\begin {align*} \int e^{c (a+b x)} \tanh ^{-1}(\cosh (a c+b c x)) \, dx &=\frac {\text {Subst}\left (\int e^x \tanh ^{-1}(\cosh (x)) \, dx,x,a c+b c x\right )}{b c}\\ &=\frac {e^{a c+b c x} \tanh ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\text {Subst}\left (\int e^x \text {csch}(x) \, dx,x,a c+b c x\right )}{b c}\\ &=\frac {e^{a c+b c x} \tanh ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\text {Subst}\left (\int \frac {2 x}{-1+x^2} \, dx,x,e^{a c+b c x}\right )}{b c}\\ &=\frac {e^{a c+b c x} \tanh ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {2 \text {Subst}\left (\int \frac {x}{-1+x^2} \, dx,x,e^{a c+b c x}\right )}{b c}\\ &=\frac {e^{a c+b c x} \tanh ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\log \left (1-e^{2 c (a+b x)}\right )}{b c}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 60, normalized size = 1.22 \begin {gather*} \frac {e^{c (a+b x)} \tanh ^{-1}\left (\frac {1}{2} e^{-c (a+b x)} \left (1+e^{2 c (a+b x)}\right )\right )+\log \left (1-e^{2 c (a+b x)}\right )}{b c} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.27, size = 887, normalized size = 18.10
method | result | size |
risch | \(\text {Expression too large to display}\) | \(887\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 64, normalized size = 1.31 \begin {gather*} \frac {\operatorname {artanh}\left (\cosh \left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} + \frac {\log \left (e^{\left (b c x + a c\right )} + 1\right )}{b c} + \frac {\log \left (e^{\left (b c x + a c\right )} - 1\right )}{b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 93, normalized size = 1.90 \begin {gather*} \frac {{\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )} \log \left (-\frac {\cosh \left (b c x + a c\right ) + 1}{\cosh \left (b c x + a c\right ) - 1}\right ) + 2 \, \log \left (\frac {2 \, \sinh \left (b c x + a c\right )}{\cosh \left (b c x + a c\right ) - \sinh \left (b c x + a c\right )}\right )}{2 \, b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 147 vs.
\(2 (47) = 94\).
time = 0.46, size = 147, normalized size = 3.00 \begin {gather*} \frac {{\left (e^{\left (b c x\right )} \log \left (-\frac {e^{\left (2 \, b c x + 2 \, a c\right )}}{e^{\left (2 \, b c x + 2 \, a c\right )} - 2 \, e^{\left (b c x + a c\right )} + 1} - \frac {2 \, e^{\left (b c x + a c\right )}}{e^{\left (2 \, b c x + 2 \, a c\right )} - 2 \, e^{\left (b c x + a c\right )} + 1} - \frac {1}{e^{\left (2 \, b c x + 2 \, a c\right )} - 2 \, e^{\left (b c x + a c\right )} + 1}\right ) + 2 \, e^{\left (-a c\right )} \log \left ({\left | e^{\left (2 \, b c x + 2 \, a c\right )} - 1 \right |}\right )\right )} e^{\left (a c\right )}}{2 \, b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.52, size = 111, normalized size = 2.27 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^{2\,b\,c\,x}\,{\mathrm {e}}^{2\,a\,c}-1\right )}{b\,c}-\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}\,\ln \left (1-\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}-\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}\right )}{2\,b\,c}+\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}\,\ln \left (\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}+\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}+1\right )}{2\,b\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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