Optimal. Leaf size=51 \[ \frac {e^{a+b x}}{b}+\frac {2 e^{a+b x}}{b \left (e^{2 a+2 b x}+1\right )}-\frac {2 \tan ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 51, 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, 203} \[ \frac {e^{a+b x}}{b}+\frac {2 e^{a+b x}}{b \left (e^{2 a+2 b x}+1\right )}-\frac {2 \tan ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 203
Rule 288
Rule 390
Rule 2282
Rubi steps
\begin {align*} \int e^{a+b x} \tanh ^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 \tan ^{-1}\left (e^{a+b x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 40, normalized size = 0.78 \[ \frac {e^{a+b x} \left (\frac {2}{e^{2 (a+b x)}+1}+1\right )-2 \tan ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 147, normalized size = 2.88 \[ \frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} - 2 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 3 \, {\left (\cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + 3 \, \cosh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 41, normalized size = 0.80 \[ \frac {\frac {2 \, e^{\left (b x + a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} + 1} - 2 \, \arctan \left (e^{\left (b x + a\right )}\right ) + e^{\left (b x + a\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 57, normalized size = 1.12 \[ \frac {\sinh ^{2}\left (b x +a \right )}{b \cosh \left (b x +a \right )}+\frac {2}{b \cosh \left (b x +a \right )}+\frac {\sinh \left (b x +a \right )}{b}-\frac {2 \arctan \left ({\mathrm e}^{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 47, normalized size = 0.92 \[ -\frac {2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} + \frac {e^{\left (b x + a\right )}}{b} + \frac {2 \, e^{\left (b x + a\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 58, normalized size = 1.14 \[ \frac {{\mathrm {e}}^{a+b\,x}}{b}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}}+\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
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 \[ e^{a} \int e^{b x} \tanh ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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