Integrand size = 13, antiderivative size = 222 \[ \int \frac {x^4}{\left (a+c x^4\right )^3} \, dx=-\frac {x}{8 c \left (a+c x^4\right )^2}+\frac {x}{32 a c \left (a+c x^4\right )}-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} c^{5/4}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} c^{5/4}}-\frac {3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{7/4} c^{5/4}}+\frac {3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{7/4} c^{5/4}} \]
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Time = 0.10 (sec) , antiderivative size = 222, 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 = {294, 205, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^4}{\left (a+c x^4\right )^3} \, dx=-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} c^{5/4}}+\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{7/4} c^{5/4}}-\frac {3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{7/4} c^{5/4}}+\frac {3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{7/4} c^{5/4}}+\frac {x}{32 a c \left (a+c x^4\right )}-\frac {x}{8 c \left (a+c x^4\right )^2} \]
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Rule 205
Rule 210
Rule 217
Rule 294
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\frac {x}{8 c \left (a+c x^4\right )^2}+\frac {\int \frac {1}{\left (a+c x^4\right )^2} \, dx}{8 c} \\ & = -\frac {x}{8 c \left (a+c x^4\right )^2}+\frac {x}{32 a c \left (a+c x^4\right )}+\frac {3 \int \frac {1}{a+c x^4} \, dx}{32 a c} \\ & = -\frac {x}{8 c \left (a+c x^4\right )^2}+\frac {x}{32 a c \left (a+c x^4\right )}+\frac {3 \int \frac {\sqrt {a}-\sqrt {c} x^2}{a+c x^4} \, dx}{64 a^{3/2} c}+\frac {3 \int \frac {\sqrt {a}+\sqrt {c} x^2}{a+c x^4} \, dx}{64 a^{3/2} c} \\ & = -\frac {x}{8 c \left (a+c x^4\right )^2}+\frac {x}{32 a c \left (a+c x^4\right )}+\frac {3 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^{3/2} c^{3/2}}+\frac {3 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^{3/2} c^{3/2}}-\frac {3 \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}{128 \sqrt {2} a^{7/4} c^{5/4}}-\frac {3 \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}{128 \sqrt {2} a^{7/4} c^{5/4}} \\ & = -\frac {x}{8 c \left (a+c x^4\right )^2}+\frac {x}{32 a c \left (a+c x^4\right )}-\frac {3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{7/4} c^{5/4}}+\frac {3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{7/4} c^{5/4}}+\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} c^{5/4}}-\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} c^{5/4}} \\ & = -\frac {x}{8 c \left (a+c x^4\right )^2}+\frac {x}{32 a c \left (a+c x^4\right )}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} c^{5/4}}+\frac {3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} c^{5/4}}-\frac {3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{7/4} c^{5/4}}+\frac {3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{7/4} c^{5/4}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.91 \[ \int \frac {x^4}{\left (a+c x^4\right )^3} \, dx=\frac {-\frac {32 \sqrt [4]{c} x}{\left (a+c x^4\right )^2}+\frac {8 \sqrt [4]{c} x}{a^2+a c x^4}-\frac {6 \sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {6 \sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}-\frac {3 \sqrt {2} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}+\frac {3 \sqrt {2} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}}{256 c^{5/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.93 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.25
method | result | size |
risch | \(\frac {\frac {x^{5}}{32 a}-\frac {3 x}{32 c}}{\left (x^{4} c +a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} c +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{128 a \,c^{2}}\) | \(56\) |
default | \(\frac {\frac {x^{5}}{32 a}-\frac {3 x}{32 c}}{\left (x^{4} c +a \right )^{2}}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a^{2} c}\) | \(131\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.20 \[ \int \frac {x^4}{\left (a+c x^4\right )^3} \, dx=\frac {4 \, c x^{5} + 3 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (a^{2} c \left (-\frac {1}{a^{7} c^{5}}\right )^{\frac {1}{4}} + x\right ) - 3 \, {\left (-i \, a c^{3} x^{8} - 2 i \, a^{2} c^{2} x^{4} - i \, a^{3} c\right )} \left (-\frac {1}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (i \, a^{2} c \left (-\frac {1}{a^{7} c^{5}}\right )^{\frac {1}{4}} + x\right ) - 3 \, {\left (i \, a c^{3} x^{8} + 2 i \, a^{2} c^{2} x^{4} + i \, a^{3} c\right )} \left (-\frac {1}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (-i \, a^{2} c \left (-\frac {1}{a^{7} c^{5}}\right )^{\frac {1}{4}} + x\right ) - 3 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (-a^{2} c \left (-\frac {1}{a^{7} c^{5}}\right )^{\frac {1}{4}} + x\right ) - 12 \, a x}{128 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} \]
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Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.30 \[ \int \frac {x^4}{\left (a+c x^4\right )^3} \, dx=\frac {- 3 a x + c x^{5}}{32 a^{3} c + 64 a^{2} c^{2} x^{4} + 32 a c^{3} x^{8}} + \operatorname {RootSum} {\left (268435456 t^{4} a^{7} c^{5} + 81, \left ( t \mapsto t \log {\left (\frac {128 t a^{2} c}{3} + x \right )} \right )\right )} \]
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Time = 0.30 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.97 \[ \int \frac {x^4}{\left (a+c x^4\right )^3} \, dx=\frac {c x^{5} - 3 \, a x}{32 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}}} + \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {1}{4}}}\right )}}{256 \, a c} \]
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Time = 0.28 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.93 \[ \int \frac {x^4}{\left (a+c x^4\right )^3} \, dx=\frac {3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{2} c^{2}} + \frac {3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{2} c^{2}} + \frac {3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{2} c^{2}} - \frac {3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{2} c^{2}} + \frac {c x^{5} - 3 \, a x}{32 \, {\left (c x^{4} + a\right )}^{2} a c} \]
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Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.36 \[ \int \frac {x^4}{\left (a+c x^4\right )^3} \, dx=\frac {3\,\mathrm {atan}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,c^{5/4}}-\frac {\frac {3\,x}{32\,c}-\frac {x^5}{32\,a}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\frac {3\,\mathrm {atanh}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,c^{5/4}} \]
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