Optimal. Leaf size=22 \[ \frac{\tanh (a+b x)}{b \sqrt{\text{sech}^2(a+b x)}} \]
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Rubi [A] time = 0.0164002, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4122, 191} \[ \frac{\tanh (a+b x)}{b \sqrt{\text{sech}^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 4122
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\text{sech}^2(a+b x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^{3/2}} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac{\tanh (a+b x)}{b \sqrt{\text{sech}^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0255771, size = 22, normalized size = 1. \[ \frac{\tanh (a+b x)}{b \sqrt{\text{sech}^2(a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.106, size = 97, normalized size = 4.4 \begin{align*}{\frac{{{\rm e}^{2\,bx+2\,a}}}{ \left ( 2+2\,{{\rm e}^{2\,bx+2\,a}} \right ) b}{\frac{1}{\sqrt{{\frac{{{\rm e}^{2\,bx+2\,a}}}{ \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) ^{2}}}}}}}-{\frac{1}{ \left ( 2+2\,{{\rm e}^{2\,bx+2\,a}} \right ) b}{\frac{1}{\sqrt{{\frac{{{\rm e}^{2\,bx+2\,a}}}{ \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976915, size = 35, normalized size = 1.59 \begin{align*} \frac{e^{\left (b x + a\right )}}{2 \, b} - \frac{e^{\left (-b x - a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97329, size = 23, normalized size = 1.05 \begin{align*} \frac{\sinh \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.0804, size = 29, normalized size = 1.32 \begin{align*} \begin{cases} \frac{\tanh{\left (a + b x \right )}}{b \sqrt{\operatorname{sech}^{2}{\left (a + b x \right )}}} & \text{for}\: b \neq 0 \\\frac{x}{\sqrt{\operatorname{sech}^{2}{\left (a \right )}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15026, size = 31, normalized size = 1.41 \begin{align*} \frac{e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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