Integrand size = 9, antiderivative size = 219 \[ \int \frac {1}{\left (a+c x^4\right )^3} \, dx=\frac {x}{8 a \left (a+c x^4\right )^2}+\frac {7 x}{32 a^2 \left (a+c x^4\right )}-\frac {21 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}-\frac {21 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}} \]
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Time = 0.09 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {205, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{\left (a+c x^4\right )^3} \, dx=-\frac {21 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}-\frac {21 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {7 x}{32 a^2 \left (a+c x^4\right )}+\frac {x}{8 a \left (a+c x^4\right )^2} \]
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Rule 205
Rule 210
Rule 217
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {x}{8 a \left (a+c x^4\right )^2}+\frac {7 \int \frac {1}{\left (a+c x^4\right )^2} \, dx}{8 a} \\ & = \frac {x}{8 a \left (a+c x^4\right )^2}+\frac {7 x}{32 a^2 \left (a+c x^4\right )}+\frac {21 \int \frac {1}{a+c x^4} \, dx}{32 a^2} \\ & = \frac {x}{8 a \left (a+c x^4\right )^2}+\frac {7 x}{32 a^2 \left (a+c x^4\right )}+\frac {21 \int \frac {\sqrt {a}-\sqrt {c} x^2}{a+c x^4} \, dx}{64 a^{5/2}}+\frac {21 \int \frac {\sqrt {a}+\sqrt {c} x^2}{a+c x^4} \, dx}{64 a^{5/2}} \\ & = \frac {x}{8 a \left (a+c x^4\right )^2}+\frac {7 x}{32 a^2 \left (a+c x^4\right )}+\frac {21 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^{5/2} \sqrt {c}}+\frac {21 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^{5/2} \sqrt {c}}-\frac {21 \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^{11/4} \sqrt [4]{c}}-\frac {21 \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^{11/4} \sqrt [4]{c}} \\ & = \frac {x}{8 a \left (a+c x^4\right )^2}+\frac {7 x}{32 a^2 \left (a+c x^4\right )}-\frac {21 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 \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^{11/4} \sqrt [4]{c}}-\frac {21 \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^{11/4} \sqrt [4]{c}} \\ & = \frac {x}{8 a \left (a+c x^4\right )^2}+\frac {7 x}{32 a^2 \left (a+c x^4\right )}-\frac {21 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}-\frac {21 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {32 a^{7/4} x}{\left (a+c x^4\right )^2}+\frac {56 a^{3/4} x}{a+c x^4}-\frac {42 \sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac {42 \sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}-\frac {21 \sqrt {2} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{\sqrt [4]{c}}+\frac {21 \sqrt {2} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{\sqrt [4]{c}}}{256 a^{11/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.80 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.26
method | result | size |
risch | \(\frac {\frac {7 c \,x^{5}}{32 a^{2}}+\frac {11 x}{32 a}}{\left (c \,x^{4}+a \right )^{2}}+\frac {21 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{128 a^{2} c}\) | \(57\) |
default | \(\frac {x}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {\frac {7 x}{32 a \left (c \,x^{4}+a \right )}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a^{2}}}{a}\) | \(139\) |
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Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\left (a+c x^4\right )^3} \, dx=\frac {28 \, c x^{5} + 21 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{11} c}\right )^{\frac {1}{4}} \log \left (a^{3} \left (-\frac {1}{a^{11} c}\right )^{\frac {1}{4}} + x\right ) - 21 \, {\left (-i \, a^{2} c^{2} x^{8} - 2 i \, a^{3} c x^{4} - i \, a^{4}\right )} \left (-\frac {1}{a^{11} c}\right )^{\frac {1}{4}} \log \left (i \, a^{3} \left (-\frac {1}{a^{11} c}\right )^{\frac {1}{4}} + x\right ) - 21 \, {\left (i \, a^{2} c^{2} x^{8} + 2 i \, a^{3} c x^{4} + i \, a^{4}\right )} \left (-\frac {1}{a^{11} c}\right )^{\frac {1}{4}} \log \left (-i \, a^{3} \left (-\frac {1}{a^{11} c}\right )^{\frac {1}{4}} + x\right ) - 21 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{11} c}\right )^{\frac {1}{4}} \log \left (-a^{3} \left (-\frac {1}{a^{11} c}\right )^{\frac {1}{4}} + x\right ) + 44 \, a x}{128 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} \]
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Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\left (a+c x^4\right )^3} \, dx=\frac {11 a x + 7 c x^{5}}{32 a^{4} + 64 a^{3} c x^{4} + 32 a^{2} c^{2} x^{8}} + \operatorname {RootSum} {\left (268435456 t^{4} a^{11} c + 194481, \left ( t \mapsto t \log {\left (\frac {128 t a^{3}}{21} + x \right )} \right )\right )} \]
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Time = 0.27 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+c x^4\right )^3} \, dx=\frac {7 \, c x^{5} + 11 \, a x}{32 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} + \frac {21 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}}} + \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {1}{4}}}\right )}}{256 \, a^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+c x^4\right )^3} \, dx=\frac {21 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c} + \frac {21 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c} + \frac {21 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c} - \frac {21 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c} + \frac {7 \, c x^{5} + 11 \, a x}{32 \, {\left (c x^{4} + a\right )}^{2} a^{2}} \]
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Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.37 \[ \int \frac {1}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {11\,x}{32\,a}+\frac {7\,c\,x^5}{32\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {21\,\mathrm {atan}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{64\,{\left (-a\right )}^{11/4}\,c^{1/4}}-\frac {21\,\mathrm {atanh}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{64\,{\left (-a\right )}^{11/4}\,c^{1/4}} \]
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