Optimal. Leaf size=84 \[ -\frac{\sqrt{-a-b x+1} (a+b x+1)^{3/2}}{2 b^2}-\frac{(1-2 a) \sqrt{-a-b x+1} \sqrt{a+b x+1}}{2 b^2}+\frac{(1-2 a) \sin ^{-1}(a+b x)}{2 b^2} \]
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Rubi [A] time = 0.0639097, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6163, 80, 50, 53, 619, 216} \[ -\frac{\sqrt{-a-b x+1} (a+b x+1)^{3/2}}{2 b^2}-\frac{(1-2 a) \sqrt{-a-b x+1} \sqrt{a+b x+1}}{2 b^2}+\frac{(1-2 a) \sin ^{-1}(a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 6163
Rule 80
Rule 50
Rule 53
Rule 619
Rule 216
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a+b x)} x \, dx &=\int \frac{x \sqrt{1+a+b x}}{\sqrt{1-a-b x}} \, dx\\ &=-\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac{(1-2 a) \int \frac{\sqrt{1+a+b x}}{\sqrt{1-a-b x}} \, dx}{2 b}\\ &=-\frac{(1-2 a) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^2}-\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac{(1-2 a) \int \frac{1}{\sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{2 b}\\ &=-\frac{(1-2 a) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^2}-\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac{(1-2 a) \int \frac{1}{\sqrt{(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{2 b}\\ &=-\frac{(1-2 a) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^2}-\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 b^2}-\frac{(1-2 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{4 b^3}\\ &=-\frac{(1-2 a) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^2}-\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac{(1-2 a) \sin ^{-1}(a+b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0909827, size = 130, normalized size = 1.55 \[ \frac{\sqrt{b} \sqrt{-a^2-2 a b x-b^2 x^2+1} (a-b x-2)+2 \sqrt{-b} \sinh ^{-1}\left (\frac{\sqrt{-b} \sqrt{-a-b x+1}}{\sqrt{2} \sqrt{b}}\right )+4 a \sqrt{-b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{-a-b x+1}}{\sqrt{2} \sqrt{-b}}\right )}{2 b^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.058, size = 178, normalized size = 2.1 \begin{align*} -{\frac{x}{2\,b}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}+{\frac{a}{2\,{b}^{2}}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}+{\frac{1}{2\,b}\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}}}}}-{\frac{1}{{b}^{2}}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}-{\frac{a}{b}\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 [A] time = 2.15044, size = 209, normalized size = 2.49 \begin{align*} \frac{{\left (2 \, a - 1\right )} \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 ) - \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x - a + 2\right )}}{2 \, b^{2}} \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{x \left (a + b x + 1\right )}{\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 = 1.20743, size = 80, normalized size = 0.95 \begin{align*} -\frac{1}{2} \, \sqrt{-{\left (b x + a\right )}^{2} + 1}{\left (\frac{x}{b} - \frac{a b - 2 \, b}{b^{3}}\right )} + \frac{{\left (2 \, a - 1\right )} \arcsin \left (-b x - a\right ) \mathrm{sgn}\left (b\right )}{2 \, b{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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