Integrand size = 13, antiderivative size = 46 \[ \int \frac {x^{11}}{\left (a+c x^4\right )^2} \, dx=\frac {x^4}{4 c^2}-\frac {a^2}{4 c^3 \left (a+c x^4\right )}-\frac {a \log \left (a+c x^4\right )}{2 c^3} \]
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Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {x^{11}}{\left (a+c x^4\right )^2} \, dx=-\frac {a^2}{4 c^3 \left (a+c x^4\right )}-\frac {a \log \left (a+c x^4\right )}{2 c^3}+\frac {x^4}{4 c^2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {x^2}{(a+c x)^2} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (\frac {1}{c^2}+\frac {a^2}{c^2 (a+c x)^2}-\frac {2 a}{c^2 (a+c x)}\right ) \, dx,x,x^4\right ) \\ & = \frac {x^4}{4 c^2}-\frac {a^2}{4 c^3 \left (a+c x^4\right )}-\frac {a \log \left (a+c x^4\right )}{2 c^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.83 \[ \int \frac {x^{11}}{\left (a+c x^4\right )^2} \, dx=\frac {c x^4-\frac {a^2}{a+c x^4}-2 a \log \left (a+c x^4\right )}{4 c^3} \]
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Time = 4.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {x^{4}}{4 c^{2}}-\frac {a^{2}}{4 c^{3} \left (x^{4} c +a \right )}-\frac {a \ln \left (x^{4} c +a \right )}{2 c^{3}}\) | \(41\) |
norman | \(\frac {\frac {x^{8}}{4 c}-\frac {a^{2}}{2 c^{3}}}{x^{4} c +a}-\frac {a \ln \left (x^{4} c +a \right )}{2 c^{3}}\) | \(43\) |
default | \(\frac {x^{4}}{4 c^{2}}-\frac {a \left (\frac {a}{2 c \left (x^{4} c +a \right )}+\frac {\ln \left (x^{4} c +a \right )}{c}\right )}{2 c^{2}}\) | \(44\) |
parallelrisch | \(-\frac {-c^{2} x^{8}+2 \ln \left (x^{4} c +a \right ) x^{4} a c +2 \ln \left (x^{4} c +a \right ) a^{2}+2 a^{2}}{4 c^{3} \left (x^{4} c +a \right )}\) | \(57\) |
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none
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.22 \[ \int \frac {x^{11}}{\left (a+c x^4\right )^2} \, dx=\frac {c^{2} x^{8} + a c x^{4} - a^{2} - 2 \, {\left (a c x^{4} + a^{2}\right )} \log \left (c x^{4} + a\right )}{4 \, {\left (c^{4} x^{4} + a c^{3}\right )}} \]
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Time = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89 \[ \int \frac {x^{11}}{\left (a+c x^4\right )^2} \, dx=- \frac {a^{2}}{4 a c^{3} + 4 c^{4} x^{4}} - \frac {a \log {\left (a + c x^{4} \right )}}{2 c^{3}} + \frac {x^{4}}{4 c^{2}} \]
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none
Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93 \[ \int \frac {x^{11}}{\left (a+c x^4\right )^2} \, dx=\frac {x^{4}}{4 \, c^{2}} - \frac {a^{2}}{4 \, {\left (c^{4} x^{4} + a c^{3}\right )}} - \frac {a \log \left (c x^{4} + a\right )}{2 \, c^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07 \[ \int \frac {x^{11}}{\left (a+c x^4\right )^2} \, dx=\frac {x^{4}}{4 \, c^{2}} - \frac {a \log \left ({\left | c x^{4} + a \right |}\right )}{2 \, c^{3}} + \frac {2 \, a c x^{4} + a^{2}}{4 \, {\left (c x^{4} + a\right )} c^{3}} \]
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Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \frac {x^{11}}{\left (a+c x^4\right )^2} \, dx=\frac {x^4}{4\,c^2}-\frac {a^2}{4\,\left (c^4\,x^4+a\,c^3\right )}-\frac {a\,\ln \left (c\,x^4+a\right )}{2\,c^3} \]
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