Optimal. Leaf size=53 \[ \frac {e^{a+b x}}{b}+\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2282, 390, 288, 206} \[ \frac {e^{a+b x}}{b}+\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 288
Rule 390
Rule 2282
Rubi steps
\begin {align*} \int e^{a+b x} \coth ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {4 x^2}{\left (1-x^2\right )^2}\right ) \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {e^{a+b x}}{b}+\frac {4 \operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {e^{a+b x}}{b}+\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {e^{a+b x}}{b}+\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end {align*}
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Mathematica [C] time = 1.91, size = 179, normalized size = 3.38 \[ \frac {e^{a+b x} \left (\frac {4}{105} \left (e^{a+b x}+e^{3 (a+b x)}\right )^2 \, _4F_3\left (\frac {3}{2},2,2,2;1,1,\frac {9}{2};e^{2 (a+b x)}\right )+\frac {1}{48} e^{-4 (a+b x)} \left (-713 e^{2 (a+b x)}-181 e^{4 (a+b x)}+61 e^{6 (a+b x)}+\frac {3 \left (196 e^{2 (a+b x)}-14 e^{4 (a+b x)}-52 e^{6 (a+b x)}+e^{8 (a+b x)}+125\right ) \tanh ^{-1}\left (\sqrt {e^{2 (a+b x)}}\right )}{\sqrt {e^{2 (a+b x)}}}-375\right )\right )}{b} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.47, size = 198, normalized size = 3.74 \[ \frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 3 \, {\left (\cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - 3 \, \cosh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 56, normalized size = 1.06 \[ -\frac {\frac {2 \, e^{\left (b x + a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - e^{\left (b x + a\right )} + \log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 48, normalized size = 0.91 \[ \frac {\cosh \left (b x +a \right )-2 \arctanh \left ({\mathrm e}^{b x +a}\right )+\frac {\cosh ^{2}\left (b x +a \right )}{\sinh \left (b x +a \right )}-\frac {2}{\sinh \left (b x +a \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 62, normalized size = 1.17 \[ \frac {e^{\left (b x + a\right )}}{b} - \frac {\log \left (e^{\left (b x + a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (b x + a\right )} - 1\right )}{b} - \frac {2 \, e^{\left (b x + a\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 62, normalized size = 1.17 \[ \frac {{\mathrm {e}}^{a+b\,x}}{b}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
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 \[ e^{a} \int e^{b x} \coth ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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