Optimal. Leaf size=10 \[ \frac {\sin ^{-1}(b x+1)}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6163, 53, 619, 216} \[ \frac {\sin ^{-1}(b x+1)}{b} \]
Antiderivative was successfully verified.
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Rule 53
Rule 216
Rule 619
Rule 6163
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*}
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Mathematica [B] time = 0.01, size = 37, normalized size = 3.70 \[ -\frac {2 \sqrt {-b x} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2} \sqrt {x}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.64, size = 28, normalized size = 2.80 \[ -\frac {2 \, \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, b x}}{b x}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 15, normalized size = 1.50 \[ -\frac {\arcsin \left (-b x - 1\right ) \mathrm {sgn}\relax (b)}{{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 34, normalized size = 3.40 \[ \frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1}{b}\right )}{\sqrt {-b^{2} x^{2}-2 b x}}\right )}{\sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 18, normalized size = 1.80 \[ -\frac {\arcsin \left (-\frac {b^{2} x + b}{b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 10, normalized size = 1.00 \[ \frac {\mathrm {asin}\left (b\,x+1\right )}{b} \]
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 \[ \int \frac {1}{\sqrt {1 - \left (b x + 1\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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