Optimal. Leaf size=10 \[ \frac{\sin ^{-1}(b x+1)}{b} \]
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Rubi [A] time = 0.0353483, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6163, 53, 619, 216} \[ \frac{\sin ^{-1}(b x+1)}{b} \]
Antiderivative was successfully verified.
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Rule 6163
Rule 53
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(1+b x)}}{2+b x} \, dx &=\int \frac{1}{\sqrt{-b x} \sqrt{2+b x}} \, dx\\ &=\int \frac{1}{\sqrt{-2 b x-b^2 x^2}} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4 b^2}}} \, dx,x,-2 b-2 b^2 x\right )}{2 b^2}\\ &=\frac{\sin ^{-1}(1+b x)}{b}\\ \end{align*}
Mathematica [B] time = 0.0120201, size = 37, normalized size = 3.7 \[ -\frac{2 \sqrt{-b x} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2} \sqrt{x}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.032, size = 34, normalized size = 3.4 \begin{align*}{\arctan \left ({(x+{b}^{-1})\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,bx}}}} \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 = 1.4805, size = 58, normalized size = 5.8 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, b x}}{b x}\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{1}{\sqrt{1 - \left (b x + 1\right )^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20334, size = 20, normalized size = 2. \begin{align*} -\frac{\arcsin \left (-b x - 1\right ) \mathrm{sgn}\left (b\right )}{{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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