Optimal. Leaf size=47 \[ \frac {\log \left (e^{2 c (a+b x)}+1\right )}{b c}+\frac {e^{a c+b c x} \cot ^{-1}(\sinh (c (a+b x)))}{b c} \]
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Rubi [A] time = 0.07, antiderivative size = 47, 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, 5208, 2282, 12, 260} \[ \frac {\log \left (e^{2 c (a+b x)}+1\right )}{b c}+\frac {e^{a c+b c x} \cot ^{-1}(\sinh (c (a+b x)))}{b c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 260
Rule 2194
Rule 2282
Rule 5208
Rubi steps
\begin {align*} \int e^{c (a+b x)} \cot ^{-1}(\sinh (a c+b c x)) \, dx &=\frac {\operatorname {Subst}\left (\int e^x \cot ^{-1}(\sinh (x)) \, dx,x,a c+b c x\right )}{b c}\\ &=\frac {e^{a c+b c x} \cot ^{-1}(\sinh (c (a+b x)))}{b c}+\frac {\operatorname {Subst}\left (\int e^x \text {sech}(x) \, dx,x,a c+b c x\right )}{b c}\\ &=\frac {e^{a c+b c x} \cot ^{-1}(\sinh (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} \cot ^{-1}(\sinh (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} \cot ^{-1}(\sinh (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 = 0.11, size = 61, normalized size = 1.30 \[ \frac {\log \left (e^{2 c (a+b x)}+1\right )-e^{c (a+b x)} \cot ^{-1}\left (\frac {1}{2} e^{-c (a+b x)}-\frac {1}{2} e^{c (a+b x)}\right )}{b c} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.62, size = 131, normalized size = 2.79 \[ \frac {{\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )} \arctan \left (\frac {2 \, {\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )}}{\cosh \left (b c x + a c\right )^{2} + 2 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )^{2} - 1}\right ) + \log \left (\frac {2 \, \cosh \left (b c x + a c\right )}{\cosh \left (b c x + a c\right ) - \sinh \left (b c x + a c\right )}\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 66, normalized size = 1.40 \[ \frac {{\left (\arctan \left (\frac {2}{e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}}\right ) e^{\left (b c x\right )} + e^{\left (-a c\right )} \log \left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )\right )} e^{\left (a c\right )}}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.52, size = 1281, normalized size = 27.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 47, normalized size = 1.00 \[ \frac {\operatorname {arccot}\left (\sinh \left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} + \frac {\log \left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.73, size = 65, normalized size = 1.38 \[ \frac {\ln \left ({\mathrm {e}}^{2\,b\,c\,x}\,{\mathrm {e}}^{2\,a\,c}+1\right )}{b\,c}+\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}\,\mathrm {acot}\left (\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}-\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}\right )}{b\,c} \]
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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