Optimal. Leaf size=49 \[ \frac {\log \left (1-e^{2 c (a+b x)}\right )}{b c}+\frac {e^{a c+b c x} \tanh ^{-1}(\text {sech}(c (a+b x)))}{b c} \]
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Rubi [A] time = 0.07, 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 = {2194, 6275, 2282, 12, 260} \[ \frac {\log \left (1-e^{2 c (a+b x)}\right )}{b c}+\frac {e^{a c+b c x} \tanh ^{-1}(\text {sech}(c (a+b x)))}{b c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 260
Rule 2194
Rule 2282
Rule 6275
Rubi steps
\begin {align*} \int e^{c (a+b x)} \tanh ^{-1}(\text {sech}(a c+b c x)) \, dx &=\frac {\operatorname {Subst}\left (\int e^x \tanh ^{-1}(\text {sech}(x)) \, dx,x,a c+b c x\right )}{b c}\\ &=\frac {e^{a c+b c x} \tanh ^{-1}(\text {sech}(c (a+b x)))}{b c}+\frac {\operatorname {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}(\text {sech}(c (a+b x)))}{b c}+\frac {\operatorname {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}(\text {sech}(c (a+b x)))}{b c}+\frac {2 \operatorname {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}(\text {sech}(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 = 0.08, size = 59, normalized size = 1.20 \[ \frac {\log \left (1-e^{2 c (a+b x)}\right )+e^{c (a+b x)} \tanh ^{-1}\left (\frac {2 e^{c (a+b x)}}{e^{2 c (a+b x)}+1}\right )}{b c} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 92, normalized size = 1.88 \[ \frac {{\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )} \log \left (\frac {\cosh \left (b c x + a c\right ) + 1}{\cosh \left (b c x + a c\right ) - 1}\right ) + 2 \, \log \left (\frac {2 \, \sinh \left (b c x + a c\right )}{\cosh \left (b c x + a c\right ) - \sinh \left (b c x + a c\right )}\right )}{2 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 98, normalized size = 2.00 \[ \frac {e^{\left ({\left (b x + a\right )} c\right )} \log \left (-\frac {\frac {2}{e^{\left (b c x + a c\right )} + e^{\left (-b c x - a c\right )}} + 1}{\frac {2}{e^{\left (b c x + a c\right )} + e^{\left (-b c x - a c\right )}} - 1}\right )}{2 \, b c} + \frac {\log \left ({\left | e^{\left (2 \, b c x + 2 \, a c\right )} - 1 \right |}\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.38, size = 872, normalized size = 17.80 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 64, normalized size = 1.31 \[ \frac {\operatorname {artanh}\left (\operatorname {sech}\left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} + \frac {\log \left (e^{\left (b c x + a c\right )} + 1\right )}{b c} + \frac {\log \left (e^{\left (b c x + a c\right )} - 1\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.71, size = 119, normalized size = 2.43 \[ \frac {\ln \left ({\mathrm {e}}^{2\,b\,c\,x}\,{\mathrm {e}}^{2\,a\,c}-1\right )}{b\,c}-\frac {{\mathrm {e}}^{a\,c+b\,c\,x}\,\ln \left (1-\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}+\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}\right )}{2\,b\,c}+\frac {\ln \left (\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}+\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}+1\right )\,{\mathrm {e}}^{a\,c+b\,c\,x}}{2\,b\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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