Optimal. Leaf size=83 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{1+a}}\right )}{2 \sqrt {1+a} \sqrt [4]{1-a^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{1+a}}\right )}{2 \sqrt {1+a} \sqrt [4]{1-a^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {218, 214, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt {a+1} \sqrt [4]{1-a^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt {a+1} \sqrt [4]{1-a^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 218
Rubi steps
\begin {align*} \int \frac {1}{1+a+(-1+a) x^4} \, dx &=\frac {\int \frac {1}{\sqrt {1+a}-\sqrt {1-a} x^2} \, dx}{2 \sqrt {1+a}}+\frac {\int \frac {1}{\sqrt {1+a}+\sqrt {1-a} x^2} \, dx}{2 \sqrt {1+a}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{1+a}}\right )}{2 \sqrt {1+a} \sqrt [4]{1-a^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{1+a}}\right )}{2 \sqrt {1+a} \sqrt [4]{1-a^2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 160, normalized size = 1.93 \begin {gather*} \frac {-2 \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{\frac {-1+a}{1+a}} x\right )+2 \tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{\frac {-1+a}{1+a}} x\right )-\log \left (\sqrt {1+a}-\sqrt {2} \sqrt [4]{-1+a} \sqrt [4]{1+a} x+\sqrt {-1+a} x^2\right )+\log \left (\sqrt {1+a}+\sqrt {2} \sqrt [4]{-1+a} \sqrt [4]{1+a} x+\sqrt {-1+a} x^2\right )}{4 \sqrt {2} \sqrt [4]{-1+a} (1+a)^{3/4}} \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. \(131\) vs.
\(2(63)=126\).
time = 0.16, size = 132, normalized size = 1.59
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (-1+a \right ) \textit {\_Z}^{4}+1+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3} a -\textit {\_R}^{3}}\right )}{4}\) | \(37\) |
default | \(\frac {\left (\frac {1+a}{-1+a}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {1+a}{-1+a}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1+a}{-1+a}}}{x^{2}-\left (\frac {1+a}{-1+a}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1+a}{-1+a}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1+a}{-1+a}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1+a}{-1+a}\right )^{\frac {1}{4}}}-1\right )\right )}{8+8 a}\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 318 vs.
\(2 (63) = 126\).
time = 0.50, size = 318, normalized size = 3.83 \begin {gather*} \frac {\sqrt {2} \log \left (\sqrt {a - 1} x^{2} + \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}} x + \sqrt {a + 1}\right )}{8 \, {\left (a + 1\right )}^{\frac {3}{4}} {\left (a - 1\right )}^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {a - 1} x^{2} - \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}} x + \sqrt {a + 1}\right )}{8 \, {\left (a + 1\right )}^{\frac {3}{4}} {\left (a - 1\right )}^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (\frac {2 \, \sqrt {a - 1} x - \sqrt {2} \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} + \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}}}{2 \, \sqrt {a - 1} x + \sqrt {2} \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} + \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}}}\right )}{8 \, \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} \sqrt {a + 1}} + \frac {\sqrt {2} \log \left (\frac {2 \, \sqrt {a - 1} x - \sqrt {2} \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} - \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}}}{2 \, \sqrt {a - 1} x + \sqrt {2} \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} - \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}}}\right )}{8 \, \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} \sqrt {a + 1}} \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. 216 vs.
\(2 (63) = 126\).
time = 0.37, size = 216, normalized size = 2.60 \begin {gather*} \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} \arctan \left (-{\left (a^{3} + a^{2} - a - 1\right )} x \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {3}{4}} + {\left (a^{3} + a^{2} - a - 1\right )} \sqrt {x^{2} + {\left (a^{2} + 2 \, a + 1\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}}} \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {3}{4}}\right ) + \frac {1}{4} \, \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} \log \left ({\left (a + 1\right )} \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} + x\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} \log \left (-{\left (a + 1\right )} \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} + x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 32, normalized size = 0.39 \begin {gather*} \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{4} + 512 a^{3} - 512 a - 256\right ) + 1, \left ( t \mapsto t \log {\left (4 t a + 4 t + x \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. 267 vs.
\(2 (63) = 126\).
time = 0.48, size = 267, normalized size = 3.22 \begin {gather*} \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{2} - \sqrt {2}\right )}} + \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{2} - \sqrt {2}\right )}} + \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a + 1}{a - 1}}\right )}{4 \, {\left (\sqrt {2} a^{2} - \sqrt {2}\right )}} - \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a + 1}{a - 1}}\right )}{4 \, {\left (\sqrt {2} a^{2} - \sqrt {2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 543, normalized size = 6.54 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}-\frac {4\,a^4-8\,a^3+8\,a-4}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}\right )\,1{}\mathrm {i}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}+\frac {\left (\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}+\frac {4\,a^4-8\,a^3+8\,a-4}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}\right )\,1{}\mathrm {i}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}{\frac {\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}-\frac {4\,a^4-8\,a^3+8\,a-4}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}-\frac {\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}+\frac {4\,a^4-8\,a^3+8\,a-4}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}\right )\,1{}\mathrm {i}}{2\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}+\frac {\mathrm {atan}\left (\frac {\frac {\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}-\frac {\left (4\,a^4-8\,a^3+8\,a-4\right )\,1{}\mathrm {i}}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}+\frac {\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}+\frac {\left (4\,a^4-8\,a^3+8\,a-4\right )\,1{}\mathrm {i}}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}{\frac {\left (\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}-\frac {\left (4\,a^4-8\,a^3+8\,a-4\right )\,1{}\mathrm {i}}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}\right )\,1{}\mathrm {i}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}-\frac {\left (\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}+\frac {\left (4\,a^4-8\,a^3+8\,a-4\right )\,1{}\mathrm {i}}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}\right )\,1{}\mathrm {i}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}\right )}{2\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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