Optimal. Leaf size=44 \[ \frac {(a+b x) \text {sech}^{-1}(a+b x)}{b}-\frac {2 \tan ^{-1}\left (\sqrt {\frac {-a-b x+1}{a+b x+1}}\right )}{b} \]
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Rubi [A] time = 0.06, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6313, 1961, 12, 203} \[ \frac {(a+b x) \text {sech}^{-1}(a+b x)}{b}-\frac {2 \tan ^{-1}\left (\sqrt {\frac {-a-b x+1}{a+b x+1}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 1961
Rule 6313
Rubi steps
\begin {align*} \int \text {sech}^{-1}(a+b x) \, dx &=\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) \operatorname {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 \operatorname {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 \tan ^{-1}\left (\sqrt {\frac {1-a-b x}{1+a+b x}}\right )}{b}\\ \end {align*}
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Mathematica [B] time = 0.31, size = 125, normalized size = 2.84 \[ x \text {sech}^{-1}(a+b x)-\frac {2 b \sqrt {-\frac {a+b x-1}{a+b x+1}} \left (\sqrt {-b} \sinh ^{-1}\left (\frac {\sqrt {a+b x-1}}{\sqrt {2}}\right )-a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {-b} \sqrt {\frac {a+b x-1}{a+b x+1}}}{\sqrt {b}}\right )\right )}{(-b)^{5/2} \sqrt {\frac {a+b x-1}{a+b x+1}}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.76, size = 253, normalized size = 5.75 \[ \frac {2 \, b x \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) + a \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) - a \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) - 2 \, \arctan \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arsech}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 50, normalized size = 1.14 \[ x \,\mathrm {arcsech}\left (b x +a \right )+\frac {\mathrm {arcsech}\left (b x +a \right ) a}{b}-\frac {\arctan \left (\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 31, normalized size = 0.70 \[ \frac {{\left (b x + a\right )} \operatorname {arsech}\left (b x + a\right ) - \arctan \left (\sqrt {\frac {1}{{\left (b x + a\right )}^{2}} - 1}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.16, size = 43, normalized size = 0.98 \[ \frac {\mathrm {atan}\left (\frac {1}{\sqrt {\frac {1}{a+b\,x}-1}\,\sqrt {\frac {1}{a+b\,x}+1}}\right )+\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )\,\left (a+b\,x\right )}{b} \]
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 \[ \int \operatorname {asech}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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