Integrand size = 17, antiderivative size = 247 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}} \]
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Time = 0.10 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1182, 1176, 631, 210, 1179, 642} \[ \int \frac {d+e x^2}{a+c x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} e+\sqrt {c} d\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}} \]
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 c}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 c} \\ & = \frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} c^{3/4}} \\ & = -\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}} \\ & = -\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.74 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=\frac {-2 \left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )-\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )\right )}{4 \sqrt {2} a^{3/4} c^{3/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.14
method | result | size |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (\textit {\_R}^{2} e +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 c}\) | \(34\) |
default | \(\frac {d \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\) | \(206\) |
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Leaf count of result is larger than twice the leaf count of optimal. 767 vs. \(2 (166) = 332\).
Time = 0.30 (sec) , antiderivative size = 767, normalized size of antiderivative = 3.11 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=-\frac {1}{4} \, \sqrt {-\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + 2 \, d e}{a c}} \log \left (-{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x + {\left (a^{3} c^{2} e \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + a c^{2} d^{3} - a^{2} c d e^{2}\right )} \sqrt {-\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + 2 \, d e}{a c}}\right ) + \frac {1}{4} \, \sqrt {-\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + 2 \, d e}{a c}} \log \left (-{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x - {\left (a^{3} c^{2} e \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + a c^{2} d^{3} - a^{2} c d e^{2}\right )} \sqrt {-\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + 2 \, d e}{a c}}\right ) + \frac {1}{4} \, \sqrt {\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - 2 \, d e}{a c}} \log \left (-{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x + {\left (a^{3} c^{2} e \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - 2 \, d e}{a c}}\right ) - \frac {1}{4} \, \sqrt {\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - 2 \, d e}{a c}} \log \left (-{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x - {\left (a^{3} c^{2} e \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - 2 \, d e}{a c}}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.44 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{3} c^{3} + 64 t^{2} a^{2} c^{2} d e + a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}, \left ( t \mapsto t \log {\left (x + \frac {64 t^{3} a^{3} c^{2} e + 12 t a^{2} c d e^{2} - 4 t a c^{2} d^{3}}{a^{2} e^{4} - c^{2} d^{4}} \right )} \right )\right )} \]
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Time = 0.29 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.89 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=\frac {\sqrt {2} {\left (\sqrt {c} d + \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (\sqrt {c} d + \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} c^{\frac {3}{4}}} \]
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Time = 0.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.98 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=\frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d + \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d + \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d - \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d - \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} \]
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Time = 0.35 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.43 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=-2\,\mathrm {atanh}\left (\frac {8\,c^3\,d^2\,x\,\sqrt {\frac {e^2\,\sqrt {-a^3\,c^3}}{16\,a^2\,c^3}-\frac {d^2\,\sqrt {-a^3\,c^3}}{16\,a^3\,c^2}-\frac {d\,e}{8\,a\,c}}}{2\,c^2\,d^2\,e-2\,a\,c\,e^3+\frac {2\,c\,d^3\,\sqrt {-a^3\,c^3}}{a^2}-\frac {2\,d\,e^2\,\sqrt {-a^3\,c^3}}{a}}-\frac {8\,a\,c^2\,e^2\,x\,\sqrt {\frac {e^2\,\sqrt {-a^3\,c^3}}{16\,a^2\,c^3}-\frac {d^2\,\sqrt {-a^3\,c^3}}{16\,a^3\,c^2}-\frac {d\,e}{8\,a\,c}}}{2\,c^2\,d^2\,e-2\,a\,c\,e^3+\frac {2\,c\,d^3\,\sqrt {-a^3\,c^3}}{a^2}-\frac {2\,d\,e^2\,\sqrt {-a^3\,c^3}}{a}}\right )\,\sqrt {-\frac {c\,d^2\,\sqrt {-a^3\,c^3}-a\,e^2\,\sqrt {-a^3\,c^3}+2\,a^2\,c^2\,d\,e}{16\,a^3\,c^3}}-2\,\mathrm {atanh}\left (\frac {8\,c^3\,d^2\,x\,\sqrt {\frac {d^2\,\sqrt {-a^3\,c^3}}{16\,a^3\,c^2}-\frac {d\,e}{8\,a\,c}-\frac {e^2\,\sqrt {-a^3\,c^3}}{16\,a^2\,c^3}}}{2\,c^2\,d^2\,e-2\,a\,c\,e^3-\frac {2\,c\,d^3\,\sqrt {-a^3\,c^3}}{a^2}+\frac {2\,d\,e^2\,\sqrt {-a^3\,c^3}}{a}}-\frac {8\,a\,c^2\,e^2\,x\,\sqrt {\frac {d^2\,\sqrt {-a^3\,c^3}}{16\,a^3\,c^2}-\frac {d\,e}{8\,a\,c}-\frac {e^2\,\sqrt {-a^3\,c^3}}{16\,a^2\,c^3}}}{2\,c^2\,d^2\,e-2\,a\,c\,e^3-\frac {2\,c\,d^3\,\sqrt {-a^3\,c^3}}{a^2}+\frac {2\,d\,e^2\,\sqrt {-a^3\,c^3}}{a}}\right )\,\sqrt {-\frac {a\,e^2\,\sqrt {-a^3\,c^3}-c\,d^2\,\sqrt {-a^3\,c^3}+2\,a^2\,c^2\,d\,e}{16\,a^3\,c^3}} \]
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