Optimal. Leaf size=38 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {35} (1-x)}{2 \sqrt {13-22 x+10 x^2}}\right )}{2 \sqrt {35}} \]
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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1043, 212}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {35} (1-x)}{2 \sqrt {10 x^2-22 x+13}}\right )}{2 \sqrt {35}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 1043
Rubi steps
\begin {align*} \int \frac {-2+x}{\left (17-18 x+5 x^2\right ) \sqrt {13-22 x+10 x^2}} \, dx &=8 \text {Subst}\left (\int \frac {1}{64-140 x^2} \, dx,x,\frac {2-2 x}{\sqrt {13-22 x+10 x^2}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {35} (1-x)}{2 \sqrt {13-22 x+10 x^2}}\right )}{2 \sqrt {35}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(38)=76\).
time = 0.38, size = 85, normalized size = 2.24 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {-135+145 x-50 x^2+\sqrt {10} (-9+5 x) \sqrt {13-22 x+10 x^2}}{-20 \sqrt {14}+10 \sqrt {14} x-2 \sqrt {35} \sqrt {13-22 x+10 x^2}}\right )}{2 \sqrt {35}} \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.
\(2(28)=56\).
time = 0.18, size = 94, normalized size = 2.47
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-35\right ) \ln \left (-\frac {-75 \RootOf \left (\textit {\_Z}^{2}-35\right ) x^{2}+140 \sqrt {10 x^{2}-22 x +13}\, x +158 \RootOf \left (\textit {\_Z}^{2}-35\right ) x -140 \sqrt {10 x^{2}-22 x +13}-87 \RootOf \left (\textit {\_Z}^{2}-35\right )}{5 x^{2}-18 x +17}\right )}{140}\) | \(82\) |
default | \(-\frac {\sqrt {\frac {\left (-2+x \right )^{2}}{\left (1-x \right )^{2}}+9}\, \sqrt {35}\, \arctanh \left (\frac {2 \sqrt {\frac {\left (-2+x \right )^{2}}{\left (1-x \right )^{2}}+9}\, \sqrt {35}}{35}\right )}{70 \sqrt {\frac {\frac {\left (-2+x \right )^{2}}{\left (1-x \right )^{2}}+9}{\left (\frac {-2+x}{1-x}+1\right )^{2}}}\, \left (\frac {-2+x}{1-x}+1\right )}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 81 vs.
\(2 (26) = 52\).
time = 0.49, size = 81, normalized size = 2.13 \begin {gather*} \frac {1}{280} \, \sqrt {35} \log \left (\frac {11225 \, x^{4} - 47220 \, x^{3} - 8 \, \sqrt {35} {\left (75 \, x^{3} - 233 \, x^{2} + 245 \, x - 87\right )} \sqrt {10 \, x^{2} - 22 \, x + 13} + 75534 \, x^{2} - 54372 \, x + 14849}{25 \, x^{4} - 180 \, x^{3} + 494 \, x^{2} - 612 \, x + 289}\right ) \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 {x - 2}{\left (5 x^{2} - 18 x + 17\right ) \sqrt {10 x^{2} - 22 x + 13}}\, 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. 231 vs.
\(2 (26) = 52\).
time = 34.67, size = 231, normalized size = 6.08 \begin {gather*} \frac {1}{140} \, \sqrt {35} \log \left ({\left | 21875000000 \, \sqrt {14} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )}^{2} + 82031250000 \, {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )}^{2} - 91875000000 \, \sqrt {35} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )} - 172812500000 \, \sqrt {10} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )} + 240625000000 \, \sqrt {14} + 913281250000 \right |}\right ) - \frac {1}{140} \, \sqrt {35} \log \left ({\left | -21875000000 \, \sqrt {14} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )}^{2} + 82031250000 \, {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )}^{2} + 91875000000 \, \sqrt {35} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )} - 172812500000 \, \sqrt {10} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )} - 240625000000 \, \sqrt {14} + 913281250000 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x-2}{\left (5\,x^2-18\,x+17\right )\,\sqrt {10\,x^2-22\,x+13}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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