Optimal. Leaf size=22 \[ \frac {2 \sinh ^{-1}\left (\frac {\sqrt {-1+a+b x}}{\sqrt {2}}\right )}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {65, 221}
\begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {\sqrt {a+b x-1}}{\sqrt {2}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 221
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+a+b x} \sqrt {1+a+b x}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {2+x^2}} \, dx,x,\sqrt {-1+a+b x}\right )}{b}\\ &=\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-1+a+b x}}{\sqrt {2}}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 27, normalized size = 1.23 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {-1+a+b x}}{\sqrt {1+a+b x}}\right )}{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. \(93\) vs.
\(2(19)=38\).
time = 0.09, size = 94, normalized size = 4.27
method | result | size |
default | \(\frac {\sqrt {\left (b x +a -1\right ) \left (b x +a +1\right )}\, \ln \left (\frac {\frac {b \left (a -1\right )}{2}+\frac {b \left (1+a \right )}{2}+b^{2} x}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+\left (b \left (a -1\right )+b \left (1+a \right )\right ) x +\left (a -1\right ) \left (1+a \right )}\right )}{\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \sqrt {b^{2}}}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 38, normalized size = 1.73 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.53, size = 31, normalized size = 1.41 \begin {gather*} -\frac {\log \left (-b x + \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a\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 {a + b x - 1} \sqrt {a + b x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.67, size = 25, normalized size = 1.14 \begin {gather*} -\frac {2 \, \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.08, size = 53, normalized size = 2.41 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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