Optimal. Leaf size=81 \[ -\log \left (-x^2+\sqrt {x^4-12 x^3+14 x^2+4 x-7}+6 x-5\right )+\tan ^{-1}\left (\frac {4 x-4}{x^2-\sqrt {x^4-12 x^3+14 x^2+4 x-7}-2 x+1}\right )+\log (x-1) \]
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Rubi [F] time = 0.18, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx &=\int \left (\frac {1}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}}+\frac {x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}}\right ) \, dx\\ &=\int \frac {1}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.03, size = 76, normalized size = 0.94 \begin {gather*} \frac {(x-1) \sqrt {x^2-10 x-7} \left (\tan ^{-1}\left (\frac {-x-3}{\sqrt {x^2-10 x-7}}\right )+2 \tanh ^{-1}\left (\frac {x-5}{\sqrt {x^2-10 x-7}}\right )\right )}{2 \sqrt {(x-1)^2 \left (x^2-10 x-7\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 81, normalized size = 1.00 \begin {gather*} -\log \left (-x^2+\sqrt {x^4-12 x^3+14 x^2+4 x-7}+6 x-5\right )+\tan ^{-1}\left (\frac {4 x-4}{x^2-\sqrt {x^4-12 x^3+14 x^2+4 x-7}-2 x+1}\right )+\log (x-1) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 79, normalized size = 0.98 \begin {gather*} \arctan \left (-\frac {x^{2} - 2 \, x - \sqrt {x^{4} - 12 \, x^{3} + 14 \, x^{2} + 4 \, x - 7} + 1}{4 \, {\left (x - 1\right )}}\right ) - \log \left (-\frac {x^{2} - 6 \, x - \sqrt {x^{4} - 12 \, x^{3} + 14 \, x^{2} + 4 \, x - 7} + 5}{x - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 183, normalized size = 2.26 \begin {gather*} -\frac {\arctan \left (\frac {1}{7} \, \sqrt {7} {\left (\sqrt {2} + \frac {3 \, {\left (\sqrt {7} \sqrt {-\frac {10}{x} - \frac {7}{x^{2}} + 1} - 4 \, \sqrt {2}\right )}}{\frac {7}{x} + 5}\right )}\right )}{\mathrm {sgn}\left (-\frac {1}{x^{2}} + \frac {1}{x^{3}}\right )} - \frac {\log \left ({\left | 10 \, \sqrt {7} + 40 \, \sqrt {2} + \frac {50 \, {\left (\sqrt {7} \sqrt {-\frac {10}{x} - \frac {7}{x^{2}} + 1} - 4 \, \sqrt {2}\right )}}{\frac {7}{x} + 5} \right |}\right )}{\mathrm {sgn}\left (-\frac {1}{x^{2}} + \frac {1}{x^{3}}\right )} + \frac {\log \left ({\left | -2 \, \sqrt {7} + 8 \, \sqrt {2} + \frac {10 \, {\left (\sqrt {7} \sqrt {-\frac {10}{x} - \frac {7}{x^{2}} + 1} - 4 \, \sqrt {2}\right )}}{\frac {7}{x} + 5} \right |}\right )}{\mathrm {sgn}\left (-\frac {1}{x^{2}} + \frac {1}{x^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 70, normalized size = 0.86 \begin {gather*} \frac {\left (-1+x \right ) \sqrt {x^{2}-10 x -7}\, \left (2 \ln \left (x -5+\sqrt {x^{2}-10 x -7}\right )-\arctan \left (\frac {3+x}{\sqrt {x^{2}-10 x -7}}\right )\right )}{2 \sqrt {x^{4}-12 x^{3}+14 x^{2}+4 x -7}} \end {gather*}
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*} \int \frac {x + 1}{\sqrt {x^{4} - 12 \, x^{3} + 14 \, x^{2} + 4 \, x - 7}}\,{d x} \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 = -0.01 \begin {gather*} \int \frac {x+1}{\sqrt {x^4-12\,x^3+14\,x^2+4\,x-7}} \,d x \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 + 1}{\sqrt {\left (x - 1\right )^{2} \left (x^{2} - 10 x - 7\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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