Integrand size = 18, antiderivative size = 86 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}} \]
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Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1181, 211, 214} \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} c-\sqrt {a} d\right )}{2 a^{3/4} b^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} d+\sqrt {b} c\right )}{2 a^{3/4} b^{3/4}} \]
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Rule 211
Rule 214
Rule 1181
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \left (-\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx+\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx \\ & = \frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.10 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {2 \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-\left (\sqrt {b} c+\sqrt {a} d\right ) \left (\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )\right )}{4 a^{3/4} b^{3/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.42
method | result | size |
risch | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (\textit {\_R}^{2} d +c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b}\) | \(36\) |
default | \(\frac {c \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}-\frac {d \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(104\) |
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Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (58) = 116\).
Time = 0.26 (sec) , antiderivative size = 755, normalized size of antiderivative = 8.78 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {1}{4} \, \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x + {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - a b^{2} c^{3} - a^{2} b c d^{2}\right )} \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}}\right ) - \frac {1}{4} \, \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x - {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - a b^{2} c^{3} - a^{2} b c d^{2}\right )} \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}}\right ) - \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x + {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + a b^{2} c^{3} + a^{2} b c d^{2}\right )} \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}}\right ) + \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x - {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + a b^{2} c^{3} + a^{2} b c d^{2}\right )} \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}}\right ) \]
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Time = 0.36 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.28 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=- \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{3} - 64 t^{2} a^{2} b^{2} c d - a^{2} d^{4} + 2 a b c^{2} d^{2} - b^{2} c^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{3} b^{2} d + 12 t a^{2} b c d^{2} + 4 t a b^{2} c^{3}}{a^{2} d^{4} - b^{2} c^{4}} \right )} \right )\right )} \]
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Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.27 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {{\left (\sqrt {b} c - \sqrt {a} d\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (\sqrt {b} c + \sqrt {a} d\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (58) = 116\).
Time = 0.27 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.67 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=-\frac {\sqrt {2} {\left (b^{2} c + \sqrt {-a b} b d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b d\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b d\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} \]
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Time = 0.33 (sec) , antiderivative size = 579, normalized size of antiderivative = 6.73 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=2\,\mathrm {atanh}\left (\frac {8\,b^3\,c^2\,x\,\sqrt {\frac {c\,d}{8\,a\,b}-\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}-\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3-\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}-\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}+\frac {8\,a\,b^2\,d^2\,x\,\sqrt {\frac {c\,d}{8\,a\,b}-\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}-\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3-\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}-\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}\right )\,\sqrt {-\frac {a\,d^2\,\sqrt {a^3\,b^3}+b\,c^2\,\sqrt {a^3\,b^3}-2\,a^2\,b^2\,c\,d}{16\,a^3\,b^3}}+2\,\mathrm {atanh}\left (\frac {8\,b^3\,c^2\,x\,\sqrt {\frac {c\,d}{8\,a\,b}+\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}+\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3+\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}+\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}+\frac {8\,a\,b^2\,d^2\,x\,\sqrt {\frac {c\,d}{8\,a\,b}+\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}+\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3+\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}+\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}\right )\,\sqrt {\frac {a\,d^2\,\sqrt {a^3\,b^3}+b\,c^2\,\sqrt {a^3\,b^3}+2\,a^2\,b^2\,c\,d}{16\,a^3\,b^3}} \]
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