Integrand size = 11, antiderivative size = 68 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\frac {x^2}{8 a \left (a+c x^4\right )^2}+\frac {3 x^2}{16 a^2 \left (a+c x^4\right )}+\frac {3 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}} \]
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Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {281, 205, 211} \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\frac {3 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}+\frac {3 x^2}{16 a^2 \left (a+c x^4\right )}+\frac {x^2}{8 a \left (a+c x^4\right )^2} \]
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Rule 205
Rule 211
Rule 281
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (a+c x^2\right )^3} \, dx,x,x^2\right ) \\ & = \frac {x^2}{8 a \left (a+c x^4\right )^2}+\frac {3 \text {Subst}\left (\int \frac {1}{\left (a+c x^2\right )^2} \, dx,x,x^2\right )}{8 a} \\ & = \frac {x^2}{8 a \left (a+c x^4\right )^2}+\frac {3 x^2}{16 a^2 \left (a+c x^4\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{16 a^2} \\ & = \frac {x^2}{8 a \left (a+c x^4\right )^2}+\frac {3 x^2}{16 a^2 \left (a+c x^4\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.85 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\frac {1}{16} \left (\frac {5 a x^2+3 c x^6}{a^2 \left (a+c x^4\right )^2}+\frac {3 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{a^{5/2} \sqrt {c}}\right ) \]
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Time = 4.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {x^{2}}{8 a \left (x^{4} c +a \right )^{2}}+\frac {\frac {3 x^{2}}{16 a \left (x^{4} c +a \right )}+\frac {3 \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{16 a \sqrt {a c}}}{a}\) | \(63\) |
risch | \(\frac {\frac {3 c \,x^{6}}{16 a^{2}}+\frac {5 x^{2}}{16 a}}{\left (x^{4} c +a \right )^{2}}-\frac {3 \ln \left (x^{2} \sqrt {-a c}-a \right )}{32 \sqrt {-a c}\, a^{2}}+\frac {3 \ln \left (x^{2} \sqrt {-a c}+a \right )}{32 \sqrt {-a c}\, a^{2}}\) | \(80\) |
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Time = 0.29 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.88 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\left [\frac {6 \, a c^{2} x^{6} + 10 \, a^{2} c x^{2} - 3 \, {\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{4} - 2 \, \sqrt {-a c} x^{2} - a}{c x^{4} + a}\right )}{32 \, {\left (a^{3} c^{3} x^{8} + 2 \, a^{4} c^{2} x^{4} + a^{5} c\right )}}, \frac {3 \, a c^{2} x^{6} + 5 \, a^{2} c x^{2} - 3 \, {\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c}}{c x^{2}}\right )}{16 \, {\left (a^{3} c^{3} x^{8} + 2 \, a^{4} c^{2} x^{4} + a^{5} c\right )}}\right ] \]
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Time = 0.24 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.62 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=- \frac {3 \sqrt {- \frac {1}{a^{5} c}} \log {\left (- a^{3} \sqrt {- \frac {1}{a^{5} c}} + x^{2} \right )}}{32} + \frac {3 \sqrt {- \frac {1}{a^{5} c}} \log {\left (a^{3} \sqrt {- \frac {1}{a^{5} c}} + x^{2} \right )}}{32} + \frac {5 a x^{2} + 3 c x^{6}}{16 a^{4} + 32 a^{3} c x^{4} + 16 a^{2} c^{2} x^{8}} \]
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Time = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\frac {3 \, c x^{6} + 5 \, a x^{2}}{16 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} + \frac {3 \, \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\frac {3 \, \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{2}} + \frac {3 \, c x^{6} + 5 \, a x^{2}}{16 \, {\left (c x^{4} + a\right )}^{2} a^{2}} \]
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Time = 5.45 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {5\,x^2}{16\,a}+\frac {3\,c\,x^6}{16\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{16\,a^{5/2}\,\sqrt {c}} \]
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