Optimal. Leaf size=11 \[ \frac {\sin ^{-1}(\tanh (a+b x))}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 216
Rule 4122
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*}
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Mathematica [B] time = 0.02, 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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fricas [A] time = 0.38, size = 19, normalized size = 1.73 \[ \frac {2 \, \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 12, normalized size = 1.09 \[ \frac {2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.43, size = 130, normalized size = 11.82 \[ \frac {i \ln \left ({\mathrm e}^{b x}+i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{b}-\frac {i \ln \left ({\mathrm e}^{b x}-i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 11, normalized size = 1.00 \[ \frac {\arctan \left (\sinh \left (b x + a\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.09 \[ \int \sqrt {\frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^2}} \,d x \]
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 \[ \int \sqrt {\operatorname {sech}^{2}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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