Optimal. Leaf size=10 \[ \frac {\text {ArcSin}(1+b x)}{b} \]
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Rubi [A]
time = 0.02, 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 = {6298, 55, 633,
222} \begin {gather*} \frac {\text {ArcSin}(b x+1)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 55
Rule 222
Rule 633
Rule 6298
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 {\text {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] Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(10)=20\).
time = 0.01, size = 37, normalized size = 3.70 \begin {gather*} -\frac {2 \sqrt {-b x} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2} \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(33\) vs.
\(2(10)=20\).
time = 0.05, size = 34, normalized size = 3.40
method | result | size |
default | \(\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1}{b}\right )}{\sqrt {-b^{2} x^{2}-2 b x}}\right )}{\sqrt {b^{2}}}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 18, normalized size = 1.80 \begin {gather*} -\frac {\arcsin \left (-\frac {b^{2} x + b}{b}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs.
\(2 (10) = 20\).
time = 0.37, size = 28, normalized size = 2.80 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, b x}}{b x}\right )}{b} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {1 - \left (b x + 1\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 38 vs.
\(2 (10) = 20\).
time = 0.41, size = 38, normalized size = 3.80 \begin {gather*} \frac {1}{2} \, \sqrt {-b^{2} x^{2} - 2 \, b x} {\left (x + \frac {1}{b}\right )} - \frac {\arcsin \left (-b x - 1\right ) \mathrm {sgn}\left (b\right )}{2 \, {\left | b \right |}} \end {gather*}
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 \begin {gather*} \frac {\mathrm {asin}\left (b\,x+1\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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