Integrand size = 13, antiderivative size = 214 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=-\frac {7}{12 a^2 x^3}+\frac {1}{4 a x^3 \left (a+c x^4\right )}+\frac {7 c^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{11/4}}-\frac {7 c^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{11/4}}+\frac {7 c^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{11/4}}-\frac {7 c^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{11/4}} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {296, 331, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=\frac {7 c^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{11/4}}-\frac {7 c^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{11/4}}+\frac {7 c^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{11/4}}-\frac {7 c^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{11/4}}-\frac {7}{12 a^2 x^3}+\frac {1}{4 a x^3 \left (a+c x^4\right )} \]
[In]
[Out]
Rule 210
Rule 217
Rule 296
Rule 331
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4 a x^3 \left (a+c x^4\right )}+\frac {7 \int \frac {1}{x^4 \left (a+c x^4\right )} \, dx}{4 a} \\ & = -\frac {7}{12 a^2 x^3}+\frac {1}{4 a x^3 \left (a+c x^4\right )}-\frac {(7 c) \int \frac {1}{a+c x^4} \, dx}{4 a^2} \\ & = -\frac {7}{12 a^2 x^3}+\frac {1}{4 a x^3 \left (a+c x^4\right )}-\frac {(7 c) \int \frac {\sqrt {a}-\sqrt {c} x^2}{a+c x^4} \, dx}{8 a^{5/2}}-\frac {(7 c) \int \frac {\sqrt {a}+\sqrt {c} x^2}{a+c x^4} \, dx}{8 a^{5/2}} \\ & = -\frac {7}{12 a^2 x^3}+\frac {1}{4 a x^3 \left (a+c x^4\right )}-\frac {\left (7 \sqrt {c}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{5/2}}-\frac {\left (7 \sqrt {c}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{5/2}}+\frac {\left (7 c^{3/4}\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}{16 \sqrt {2} a^{11/4}}+\frac {\left (7 c^{3/4}\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}{16 \sqrt {2} a^{11/4}} \\ & = -\frac {7}{12 a^2 x^3}+\frac {1}{4 a x^3 \left (a+c x^4\right )}+\frac {7 c^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{11/4}}-\frac {7 c^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{11/4}}-\frac {\left (7 c^{3/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{11/4}}+\frac {\left (7 c^{3/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{11/4}} \\ & = -\frac {7}{12 a^2 x^3}+\frac {1}{4 a x^3 \left (a+c x^4\right )}+\frac {7 c^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{11/4}}-\frac {7 c^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{11/4}}+\frac {7 c^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{11/4}}-\frac {7 c^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{11/4}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=\frac {-\frac {32 a^{3/4}}{x^3}-\frac {24 a^{3/4} c x}{a+c x^4}+42 \sqrt {2} c^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-42 \sqrt {2} c^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+21 \sqrt {2} c^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )-21 \sqrt {2} c^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{96 a^{11/4}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.91 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.36
method | result | size |
risch | \(\frac {-\frac {7 c \,x^{4}}{12 a^{2}}-\frac {1}{3 a}}{x^{3} \left (x^{4} c +a \right )}+\frac {7 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} \textit {\_Z}^{4}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-5 \textit {\_R}^{4} a^{11}-4 c^{3}\right ) x -a^{3} c^{2} \textit {\_R} \right )\right )}{16}\) | \(76\) |
default | \(-\frac {1}{3 a^{2} x^{3}}-\frac {c \left (\frac {x}{4 x^{4} c +4 a}+\frac {7 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2}}\) | \(130\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=-\frac {28 \, c x^{4} + 21 \, {\left (a^{2} c x^{7} + a^{3} x^{3}\right )} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (7 \, a^{3} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} + 7 \, c x\right ) + 21 \, {\left (i \, a^{2} c x^{7} + i \, a^{3} x^{3}\right )} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (7 i \, a^{3} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} + 7 \, c x\right ) + 21 \, {\left (-i \, a^{2} c x^{7} - i \, a^{3} x^{3}\right )} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-7 i \, a^{3} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} + 7 \, c x\right ) - 21 \, {\left (a^{2} c x^{7} + a^{3} x^{3}\right )} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-7 \, a^{3} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} + 7 \, c x\right ) + 16 \, a}{48 \, {\left (a^{2} c x^{7} + a^{3} x^{3}\right )}} \]
[In]
[Out]
Time = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.27 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=\frac {- 4 a - 7 c x^{4}}{12 a^{3} x^{3} + 12 a^{2} c x^{7}} + \operatorname {RootSum} {\left (65536 t^{4} a^{11} + 2401 c^{3}, \left ( t \mapsto t \log {\left (- \frac {16 t a^{3}}{7 c} + x \right )} \right )\right )} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=-\frac {7 \, c x^{4} + 4 \, a}{12 \, {\left (a^{2} c x^{7} + a^{3} x^{3}\right )}} - \frac {7 \, {\left (\frac {2 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}}} + \frac {2 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}}} + \frac {\sqrt {2} c^{\frac {3}{4}} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} c^{\frac {3}{4}} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}\right )}}{32 \, a^{2}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=-\frac {c x}{4 \, {\left (c x^{4} + a\right )} a^{2}} - \frac {7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{3}} - \frac {7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{3}} - \frac {7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{3}} + \frac {7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{3}} - \frac {1}{3 \, a^{2} x^{3}} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.34 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=\frac {7\,{\left (-c\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,x}{a^{1/4}}\right )}{8\,a^{11/4}}-\frac {\frac {1}{3\,a}+\frac {7\,c\,x^4}{12\,a^2}}{c\,x^7+a\,x^3}+\frac {7\,{\left (-c\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,x}{a^{1/4}}\right )}{8\,a^{11/4}} \]
[In]
[Out]