Optimal. Leaf size=39 \[ \frac{\sin ^{-1}(a+b x)}{b}-\frac{\sqrt{-a-b x+1} \sqrt{a+b x+1}}{b} \]
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Rubi [A] time = 0.0264667, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {6161, 50, 53, 619, 216} \[ \frac{\sin ^{-1}(a+b x)}{b}-\frac{\sqrt{-a-b x+1} \sqrt{a+b x+1}}{b} \]
Antiderivative was successfully verified.
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Rule 6161
Rule 50
Rule 53
Rule 619
Rule 216
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a+b x)} \, dx &=\int \frac{\sqrt{1+a+b x}}{\sqrt{1-a-b x}} \, dx\\ &=-\frac{\sqrt{1-a-b x} \sqrt{1+a+b x}}{b}+\int \frac{1}{\sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx\\ &=-\frac{\sqrt{1-a-b x} \sqrt{1+a+b x}}{b}+\int \frac{1}{\sqrt{(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a-b x} \sqrt{1+a+b x}}{b}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{2 b^2}\\ &=-\frac{\sqrt{1-a-b x} \sqrt{1+a+b x}}{b}+\frac{\sin ^{-1}(a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0161056, size = 28, normalized size = 0.72 \[ \frac{\sin ^{-1}(a+b x)-\sqrt{1-(a+b x)^2}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 71, normalized size = 1.8 \begin{align*} -{\frac{1}{b}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}+{\arctan \left ({\sqrt{{b}^{2}} \left ( x+{\frac{a}{b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.26897, size = 170, normalized size = 4.36 \begin{align*} -\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} + \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\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 \frac{a + b x + 1}{\sqrt{- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20283, size = 49, normalized size = 1.26 \begin{align*} -\frac{\arcsin \left (-b x - a\right ) \mathrm{sgn}\left (b\right )}{{\left | b \right |}} - \frac{\sqrt{-{\left (b x + a\right )}^{2} + 1}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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