Optimal. Leaf size=150 \[ \frac {\sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}} \]
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Rubi [A]
time = 0.06, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1107, 211}
\begin {gather*} \frac {\sqrt {2} \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 1107
Rubi steps
\begin {align*} \int \frac {1}{a+b x^2+c x^4} \, dx &=\frac {c \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {\sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 129, normalized size = 0.86 \begin {gather*} \frac {\sqrt {2} \sqrt {c} \left (\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 117, normalized size = 0.78
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b}\right )}{2}\) | \(38\) |
default | \(4 c \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )\) | \(117\) |
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. 613 vs.
\(2 (114) = 228\).
time = 0.36, size = 613, normalized size = 4.09 \begin {gather*} -\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (2 \, c x + \sqrt {\frac {1}{2}} {\left (b^{2} - 4 \, a c - \frac {a b^{3} - 4 \, a^{2} b c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (2 \, c x - \sqrt {\frac {1}{2}} {\left (b^{2} - 4 \, a c - \frac {a b^{3} - 4 \, a^{2} b c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (2 \, c x + \sqrt {\frac {1}{2}} {\left (b^{2} - 4 \, a c + \frac {a b^{3} - 4 \, a^{2} b c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (2 \, c x - \sqrt {\frac {1}{2}} {\left (b^{2} - 4 \, a c + \frac {a b^{3} - 4 \, a^{2} b c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.63, size = 87, normalized size = 0.58 \begin {gather*} \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{3} c^{2} - 128 a^{2} b^{2} c + 16 a b^{4}\right ) + t^{2} \left (- 16 a b c + 4 b^{3}\right ) + c, \left ( t \mapsto t \log {\left (x + \frac {32 t^{3} a^{2} b c - 8 t^{3} a b^{3} + 4 t a c - 2 t b^{2}}{c} \right )} \right )\right )} \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. 1024 vs.
\(2 (114) = 228\).
time = 4.23, size = 1024, normalized size = 6.83 \begin {gather*} \frac {{\left (\sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{4} - 8 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b^{2} c - 2 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{3} c - 2 \, b^{4} c + 16 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{2} c^{2} + 8 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b c^{2} + \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{2} c^{2} + 16 \, a b^{2} c^{2} - 2 \, b^{3} c^{2} - 4 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a c^{3} - 32 \, a^{2} c^{3} + 8 \, a b c^{3} + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{3} - 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b c - 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{2} c + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c - 8 \, {\left (b^{2} - 4 \, a c\right )} a c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {b + \sqrt {b^{2} - 4 \, a c}}{c}}}\right )}{4 \, {\left (a b^{4} - 8 \, a^{2} b^{2} c - 2 \, a b^{3} c + 16 \, a^{3} c^{2} + 8 \, a^{2} b c^{2} + a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} {\left | c \right |}} + \frac {{\left (\sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{4} - 8 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b^{2} c - 2 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{3} c + 2 \, b^{4} c + 16 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{2} c^{2} + 8 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b c^{2} + \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{2} c^{2} - 16 \, a b^{2} c^{2} - 2 \, b^{3} c^{2} - 4 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a c^{3} + 32 \, a^{2} c^{3} + 8 \, a b c^{3} + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{3} - 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b c - 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{2} c + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c + 8 \, {\left (b^{2} - 4 \, a c\right )} a c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {b - \sqrt {b^{2} - 4 \, a c}}{c}}}\right )}{4 \, {\left (a b^{4} - 8 \, a^{2} b^{2} c - 2 \, a b^{3} c + 16 \, a^{3} c^{2} + 8 \, a^{2} b c^{2} + a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} {\left | c \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.61, size = 763, normalized size = 5.09 \begin {gather*} -\mathrm {atan}\left (\frac {b^4\,x\,1{}\mathrm {i}+b\,x\,\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}\,1{}\mathrm {i}+a^2\,c^2\,x\,16{}\mathrm {i}-a\,b^2\,c\,x\,8{}\mathrm {i}}{4\,a\,b^4\,\sqrt {-\frac {b^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}+64\,a^3\,c^2\,\sqrt {-\frac {b^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}-32\,a^2\,b^2\,c\,\sqrt {-\frac {b^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}}\right )\,\sqrt {-\frac {b^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {b^4\,x\,1{}\mathrm {i}-b\,x\,\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}\,1{}\mathrm {i}+a^2\,c^2\,x\,16{}\mathrm {i}-a\,b^2\,c\,x\,8{}\mathrm {i}}{4\,a\,b^4\,\sqrt {\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-b^3+4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}+64\,a^3\,c^2\,\sqrt {\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-b^3+4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}-32\,a^2\,b^2\,c\,\sqrt {\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-b^3+4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}}\right )\,\sqrt {\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-b^3+4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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