Integrand size = 6, antiderivative size = 44 \[ \int \text {sech}^{-1}(a+b x) \, dx=\frac {(a+b x) \text {sech}^{-1}(a+b x)}{b}-\frac {2 \arctan \left (\sqrt {\frac {1-a-b x}{1+a+b x}}\right )}{b} \]
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Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6448, 1983, 12, 209} \[ \int \text {sech}^{-1}(a+b x) \, dx=\frac {(a+b x) \text {sech}^{-1}(a+b x)}{b}-\frac {2 \arctan \left (\sqrt {\frac {-a-b x+1}{a+b x+1}}\right )}{b} \]
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Rule 12
Rule 209
Rule 1983
Rule 6448
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x) \text {sech}^{-1}(a+b x)}{b}+\int \frac {\sqrt {\frac {1-a-b x}{1+a+b x}}}{1-a-b x} \, dx \\ & = \frac {(a+b x) \text {sech}^{-1}(a+b x)}{b}-(4 b) \text {Subst}\left (\int \frac {1}{2 b^2 \left (1+x^2\right )} \, dx,x,\sqrt {\frac {1-a-b x}{1+a+b x}}\right ) \\ & = \frac {(a+b x) \text {sech}^{-1}(a+b x)}{b}-\frac {2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\frac {1-a-b x}{1+a+b x}}\right )}{b} \\ & = \frac {(a+b x) \text {sech}^{-1}(a+b x)}{b}-\frac {2 \arctan \left (\sqrt {\frac {1-a-b x}{1+a+b x}}\right )}{b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(97\) vs. \(2(44)=88\).
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.20 \[ \int \text {sech}^{-1}(a+b x) \, dx=x \text {sech}^{-1}(a+b x)-\frac {2 \sqrt {-\frac {-1+a+b x}{1+a+b x}} \left (-a \arctan \left (\sqrt {\frac {-1+a+b x}{1+a+b x}}\right )+\text {arctanh}\left (\sqrt {\frac {-1+a+b x}{1+a+b x}}\right )\right )}{b \sqrt {\frac {-1+a+b x}{1+a+b x}}} \]
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Time = 0.41 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\left (b x +a \right ) \operatorname {arcsech}\left (b x +a \right )-\arctan \left (\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )}{b}\) | \(44\) |
default | \(\frac {\left (b x +a \right ) \operatorname {arcsech}\left (b x +a \right )-\arctan \left (\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )}{b}\) | \(44\) |
parts | \(x \,\operatorname {arcsech}\left (b x +a \right )+\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (\operatorname {csgn}\left (b \right ) \operatorname {arctanh}\left (\frac {1}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a +\arctan \left (\frac {\operatorname {csgn}\left (b \right ) \left (b x +a \right )}{\sqrt {-\left (b x +a -1\right ) \left (b x +a +1\right )}}\right )\right ) \operatorname {csgn}\left (b \right )}{b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\) | \(129\) |
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Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (40) = 80\).
Time = 0.27 (sec) , antiderivative size = 253, normalized size of antiderivative = 5.75 \[ \int \text {sech}^{-1}(a+b x) \, dx=\frac {2 \, b x \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) + a \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) - a \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) - 2 \, \arctan \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{2 \, b} \]
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\[ \int \text {sech}^{-1}(a+b x) \, dx=\int \operatorname {asech}{\left (a + b x \right )}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.70 \[ \int \text {sech}^{-1}(a+b x) \, dx=\frac {{\left (b x + a\right )} \operatorname {arsech}\left (b x + a\right ) - \arctan \left (\sqrt {\frac {1}{{\left (b x + a\right )}^{2}} - 1}\right )}{b} \]
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\[ \int \text {sech}^{-1}(a+b x) \, dx=\int { \operatorname {arsech}\left (b x + a\right ) \,d x } \]
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Time = 4.98 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \text {sech}^{-1}(a+b x) \, dx=\frac {\mathrm {atan}\left (\frac {1}{\sqrt {\frac {1}{a+b\,x}-1}\,\sqrt {\frac {1}{a+b\,x}+1}}\right )+\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )\,\left (a+b\,x\right )}{b} \]
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