Optimal. Leaf size=77 \[ \frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 (a-b) \sqrt {b}}+\frac {\log (x)}{a-b}-\frac {\log \left (a-b+2 a x^2+a x^4\right )}{4 (a-b)} \]
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Rubi [A]
time = 0.06, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1128, 719, 29,
648, 632, 212, 642} \begin {gather*} \frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 \sqrt {b} (a-b)}-\frac {\log \left (a x^4+2 a x^2+a-b\right )}{4 (a-b)}+\frac {\log (x)}{a-b} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 212
Rule 632
Rule 642
Rule 648
Rule 719
Rule 1128
Rubi steps
\begin {align*} \int \frac {1}{x \left (a-b+2 a x^2+a x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \left (a-b+2 a x+a x^2\right )} \, dx,x,x^2\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 (a-b)}+\frac {\text {Subst}\left (\int \frac {-2 a-a x}{a-b+2 a x+a x^2} \, dx,x,x^2\right )}{2 (a-b)}\\ &=\frac {\log (x)}{a-b}-\frac {\text {Subst}\left (\int \frac {2 a+2 a x}{a-b+2 a x+a x^2} \, dx,x,x^2\right )}{4 (a-b)}-\frac {a \text {Subst}\left (\int \frac {1}{a-b+2 a x+a x^2} \, dx,x,x^2\right )}{2 (a-b)}\\ &=\frac {\log (x)}{a-b}-\frac {\log \left (a-b+2 a x^2+a x^4\right )}{4 (a-b)}+\frac {a \text {Subst}\left (\int \frac {1}{4 a b-x^2} \, dx,x,2 a \left (1+x^2\right )\right )}{a-b}\\ &=\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 (a-b) \sqrt {b}}+\frac {\log (x)}{a-b}-\frac {\log \left (a-b+2 a x^2+a x^4\right )}{4 (a-b)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 90, normalized size = 1.17 \begin {gather*} \frac {-4 \sqrt {b} \log (x)+\left (\sqrt {a}+\sqrt {b}\right ) \log \left (-\sqrt {b}+\sqrt {a} \left (1+x^2\right )\right )+\left (-\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt {b}+\sqrt {a} \left (1+x^2\right )\right )}{4 \sqrt {b} (-a+b)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 70, normalized size = 0.91
method | result | size |
risch | \(\frac {\ln \left (x \right )}{a -b}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (-1+\left (a b -b^{2}\right ) \textit {\_Z}^{2}+2 b \textit {\_Z} \right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a -5 b \right ) \textit {\_R} +5\right ) x^{2}+\left (-a +b \right ) \textit {\_R} +4\right )\right )}{4}\) | \(64\) |
default | \(-\frac {a \left (\frac {\ln \left (a \,x^{4}+2 a \,x^{2}+a -b \right )}{2 a}-\frac {\arctanh \left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{\sqrt {a b}}\right )}{2 \left (a -b \right )}+\frac {\ln \left (x \right )}{a -b}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 85, normalized size = 1.10 \begin {gather*} -\frac {a \log \left (\frac {a x^{2} + a - \sqrt {a b}}{a x^{2} + a + \sqrt {a b}}\right )}{4 \, \sqrt {a b} {\left (a - b\right )}} - \frac {\log \left (a x^{4} + 2 \, a x^{2} + a - b\right )}{4 \, {\left (a - b\right )}} + \frac {\log \left (x^{2}\right )}{2 \, {\left (a - b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 151, normalized size = 1.96 \begin {gather*} \left [-\frac {\sqrt {\frac {a}{b}} \log \left (\frac {a x^{4} + 2 \, a x^{2} - 2 \, {\left (b x^{2} + b\right )} \sqrt {\frac {a}{b}} + a + b}{a x^{4} + 2 \, a x^{2} + a - b}\right ) + \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) - 4 \, \log \left (x\right )}{4 \, {\left (a - b\right )}}, -\frac {2 \, \sqrt {-\frac {a}{b}} \arctan \left (\frac {b \sqrt {-\frac {a}{b}}}{a x^{2} + a}\right ) + \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) - 4 \, \log \left (x\right )}{4 \, {\left (a - b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 184 vs.
\(2 (61) = 122\).
time = 2.59, size = 184, normalized size = 2.39 \begin {gather*} \left (- \frac {1}{4 \left (a - b\right )} - \frac {\sqrt {a b}}{4 b \left (a - b\right )}\right ) \log {\left (x^{2} + \frac {4 a b \left (- \frac {1}{4 \left (a - b\right )} - \frac {\sqrt {a b}}{4 b \left (a - b\right )}\right ) + a - 4 b^{2} \left (- \frac {1}{4 \left (a - b\right )} - \frac {\sqrt {a b}}{4 b \left (a - b\right )}\right ) + b}{a} \right )} + \left (- \frac {1}{4 \left (a - b\right )} + \frac {\sqrt {a b}}{4 b \left (a - b\right )}\right ) \log {\left (x^{2} + \frac {4 a b \left (- \frac {1}{4 \left (a - b\right )} + \frac {\sqrt {a b}}{4 b \left (a - b\right )}\right ) + a - 4 b^{2} \left (- \frac {1}{4 \left (a - b\right )} + \frac {\sqrt {a b}}{4 b \left (a - b\right )}\right ) + b}{a} \right )} + \frac {\log {\left (x \right )}}{a - b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.75, size = 71, normalized size = 0.92 \begin {gather*} -\frac {a \arctan \left (\frac {a x^{2} + a}{\sqrt {-a b}}\right )}{2 \, \sqrt {-a b} {\left (a - b\right )}} - \frac {\log \left (a x^{4} + 2 \, a x^{2} + a - b\right )}{4 \, {\left (a - b\right )}} + \frac {\log \left (x^{2}\right )}{2 \, {\left (a - b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.56, size = 183, normalized size = 2.38 \begin {gather*} \frac {\ln \left (x\right )}{a-b}-\frac {\ln \left (16\,a^4+20\,a^4\,x^2+\frac {\left (b-\sqrt {a\,b}\right )\,\left (x^2\,\left (16\,a^5+80\,b\,a^4\right )-16\,a^4\,b+16\,a^5\right )}{4\,\left (a\,b-b^2\right )}\right )\,\left (b-\sqrt {a\,b}\right )}{4\,\left (a\,b-b^2\right )}-\frac {\ln \left (16\,a^4+20\,a^4\,x^2+\frac {\left (b+\sqrt {a\,b}\right )\,\left (x^2\,\left (16\,a^5+80\,b\,a^4\right )-16\,a^4\,b+16\,a^5\right )}{4\,\left (a\,b-b^2\right )}\right )\,\left (b+\sqrt {a\,b}\right )}{4\,\left (a\,b-b^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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