Optimal. Leaf size=27 \[ \frac {\sqrt {1+a+b x}}{b \sqrt {1-a-b x}} \]
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Rubi [A]
time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {6299, 37}
\begin {gather*} \frac {\sqrt {a+b x+1}}{b \sqrt {-a-b x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 6299
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a+b x)}}{1-a^2-2 a b x-b^2 x^2} \, dx &=\int \frac {1}{(1-a-b x)^{3/2} \sqrt {1+a+b x}} \, dx\\ &=\frac {\sqrt {1+a+b x}}{b \sqrt {1-a-b x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 3 vs. order 2 in
optimal.
time = 0.06, size = 12, normalized size = 0.44 \begin {gather*} \frac {e^{\tanh ^{-1}(a+b x)}}{b} \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. \(209\) vs.
\(2(23)=46\).
time = 0.08, size = 210, normalized size = 7.78
method | result | size |
trager | \(-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b \left (b x +a -1\right )}\) | \(36\) |
gosper | \(-\frac {\left (b x +a -1\right ) \left (b x +a +1\right )^{2}}{b \left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}\) | \(42\) |
default | \(b \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )+\frac {2 a \left (-2 b^{2} x -2 b a \right )}{\left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {-4 b^{2} x -4 b a}{\left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\) | \(210\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (23) = 46\).
time = 0.48, size = 65, normalized size = 2.41 \begin {gather*} -\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{2}}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}} {\left (b^{3} x + a b^{2} - b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 37, normalized size = 1.37 \begin {gather*} -\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{2} x + {\left (a - 1\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}{a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} - \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, 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. 49 vs.
\(2 (23) = 46\).
time = 0.43, size = 49, normalized size = 1.81 \begin {gather*} \frac {2}{{\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} - 1\right )} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 26, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {1-{\left (a+b\,x\right )}^2}}{b\,\left (a+b\,x-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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