Integrand size = 15, antiderivative size = 219 \[ \int \frac {d+e x}{a+c x^4} \, dx=\frac {e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {c}}-\frac {d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {d \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{c}}-\frac {d \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {d \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{c}} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {1890, 217, 1179, 642, 1176, 631, 210, 281, 211} \[ \int \frac {d+e x}{a+c x^4} \, dx=-\frac {d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {d \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{c}}-\frac {d \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {d \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {c}} \]
[In]
[Out]
Rule 210
Rule 211
Rule 217
Rule 281
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1890
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d}{a+c x^4}+\frac {e x}{a+c x^4}\right ) \, dx \\ & = d \int \frac {1}{a+c x^4} \, dx+e \int \frac {x}{a+c x^4} \, dx \\ & = \frac {d \int \frac {\sqrt {a}-\sqrt {c} x^2}{a+c x^4} \, dx}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {a}+\sqrt {c} x^2}{a+c x^4} \, dx}{2 \sqrt {a}}+\frac {1}{2} e \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right ) \\ & = \frac {e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {c}}+\frac {d \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \sqrt {a} \sqrt {c}}+\frac {d \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \sqrt {a} \sqrt {c}}-\frac {d \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} \sqrt [4]{c}}-\frac {d \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} \sqrt [4]{c}} \\ & = \frac {e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {c}}-\frac {d \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {d \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {d \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} \sqrt [4]{c}}-\frac {d \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} \sqrt [4]{c}} \\ & = \frac {e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {c}}-\frac {d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {d \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{c}}-\frac {d \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {d \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{c}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.84 \[ \int \frac {d+e x}{a+c x^4} \, dx=\frac {-2 \left (\sqrt {2} \sqrt [4]{c} d+2 \sqrt [4]{a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \left (\sqrt {2} \sqrt [4]{c} d-2 \sqrt [4]{a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+\sqrt {2} \sqrt [4]{c} d \left (-\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )\right )}{8 a^{3/4} \sqrt {c}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.80 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.15
method | result | size |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (\textit {\_R} e +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 c}\) | \(32\) |
default | \(\frac {d \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {e \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{2 \sqrt {a c}}\) | \(124\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.23 (sec) , antiderivative size = 41851, normalized size of antiderivative = 191.10 \[ \int \frac {d+e x}{a+c x^4} \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.57 \[ \int \frac {d+e x}{a+c x^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{3} c^{2} + 32 t^{2} a^{2} c e^{2} - 16 t a c d^{2} e + a e^{4} + c d^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 128 t^{3} a^{3} c e^{2} - 16 t^{2} a^{2} c d^{2} e - 8 t a^{2} e^{4} - 4 t a c d^{4} + 5 a d^{2} e^{3}}{4 a d e^{4} - c d^{5}} \right )} \right )\right )} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.95 \[ \int \frac {d+e x}{a+c x^4} \, dx=\frac {\sqrt {2} d \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} d \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} c^{\frac {1}{4}}} + \frac {{\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} d - 2 \, \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{4 \, a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {1}{4}}} + \frac {{\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} d + 2 \, \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{4 \, a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {1}{4}}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.97 \[ \int \frac {d+e x}{a+c x^4} \, dx=\frac {\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} d \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c} - \frac {\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} d \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c} - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a c} c e - \left (a c^{3}\right )^{\frac {1}{4}} c d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{2}} - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a c} c e - \left (a c^{3}\right )^{\frac {1}{4}} c d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{2}} \]
[In]
[Out]
Time = 9.16 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.73 \[ \int \frac {d+e x}{a+c x^4} \, dx=\left \{\begin {array}{cl} -\frac {2\,d+3\,e\,x}{6\,c\,x^3} & \text {\ if\ \ }a=0\\ \frac {\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/4}\,x}{a^{1/4}}-1\right )\,\left (2\,a^{1/4}\,e+\sqrt {2}\,c^{1/4}\,d\right )}{4\,a^{3/4}\,\sqrt {c}}-\frac {\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/4}\,x}{a^{1/4}}+1\right )\,\left (4\,a^{1/4}\,e-2\,\sqrt {2}\,c^{1/4}\,d\right )}{8\,a^{3/4}\,\sqrt {c}}+\frac {\sqrt {2}\,d\,\ln \left (\frac {\sqrt {a}+\sqrt {c}\,x^2+\sqrt {2}\,a^{1/4}\,c^{1/4}\,x}{\sqrt {a}+\sqrt {c}\,x^2-\sqrt {2}\,a^{1/4}\,c^{1/4}\,x}\right )}{8\,a^{3/4}\,c^{1/4}} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
[In]
[Out]