Optimal. Leaf size=11 \[ \frac{\sin ^{-1}(\tanh (a+b x))}{b} \]
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Rubi [A] time = 0.0109753, antiderivative size = 11, 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, 216} \[ \frac{\sin ^{-1}(\tanh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 4122
Rule 216
Rubi steps
\begin{align*} \int \sqrt{\text{sech}^2(a+b x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac{\sin ^{-1}(\tanh (a+b x))}{b}\\ \end{align*}
Mathematica [B] time = 0.01561, size = 29, normalized size = 2.64 \[ \frac{\cosh (a+b x) \sqrt{\text{sech}^2(a+b x)} \tan ^{-1}(\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.132, size = 130, normalized size = 11.8 \begin{align*}{\frac{i \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) \ln \left ({{\rm e}^{bx}}+i{{\rm e}^{-a}} \right ){{\rm e}^{-bx-a}}}{b}\sqrt{{\frac{{{\rm e}^{2\,bx+2\,a}}}{ \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) ^{2}}}}}-{\frac{i \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) \ln \left ({{\rm e}^{bx}}-i{{\rm e}^{-a}} \right ){{\rm e}^{-bx-a}}}{b}\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 = 0.995687, size = 15, normalized size = 1.36 \begin{align*} \frac{\arctan \left (\sinh \left (b x + a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00202, size = 58, normalized size = 5.27 \begin{align*} \frac{2 \, \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\operatorname{sech}^{2}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14072, size = 16, normalized size = 1.45 \begin{align*} \frac{2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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