Integrand size = 68, antiderivative size = 28 \[ \int \frac {4-4 x^2+6 x^3-8 x^5+e^x \left (-x^3+x^4+x^5-x^6\right )+\left (-x^3+x^5\right ) \log \left (-x+x^3\right )}{-x^5+x^7} \, dx=\frac {1}{x^4}+\frac {5-e^x-x-\log \left (-x+x^3\right )}{x} \]
[Out]
Time = 0.69 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {1607, 6857, 2228, 213, 272, 46, 331, 2605, 464} \[ \int \frac {4-4 x^2+6 x^3-8 x^5+e^x \left (-x^3+x^4+x^5-x^6\right )+\left (-x^3+x^5\right ) \log \left (-x+x^3\right )}{-x^5+x^7} \, dx=\frac {1}{x^4}-\frac {\log \left (-x \left (1-x^2\right )\right )}{x}-\frac {e^x}{x}+\frac {5}{x} \]
[In]
[Out]
Rule 46
Rule 213
Rule 272
Rule 331
Rule 464
Rule 1607
Rule 2228
Rule 2605
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \frac {4-4 x^2+6 x^3-8 x^5+e^x \left (-x^3+x^4+x^5-x^6\right )+\left (-x^3+x^5\right ) \log \left (-x+x^3\right )}{x^5 \left (-1+x^2\right )} \, dx \\ & = \int \left (-\frac {e^x (-1+x)}{x^2}-\frac {8}{-1+x^2}+\frac {4}{x^5 \left (-1+x^2\right )}-\frac {4}{x^3 \left (-1+x^2\right )}+\frac {6}{x^2 \left (-1+x^2\right )}+\frac {\log \left (x \left (-1+x^2\right )\right )}{x^2}\right ) \, dx \\ & = 4 \int \frac {1}{x^5 \left (-1+x^2\right )} \, dx-4 \int \frac {1}{x^3 \left (-1+x^2\right )} \, dx+6 \int \frac {1}{x^2 \left (-1+x^2\right )} \, dx-8 \int \frac {1}{-1+x^2} \, dx-\int \frac {e^x (-1+x)}{x^2} \, dx+\int \frac {\log \left (x \left (-1+x^2\right )\right )}{x^2} \, dx \\ & = \frac {6}{x}-\frac {e^x}{x}+8 \text {arctanh}(x)-\frac {\log \left (-x \left (1-x^2\right )\right )}{x}+2 \text {Subst}\left (\int \frac {1}{(-1+x) x^3} \, dx,x,x^2\right )-2 \text {Subst}\left (\int \frac {1}{(-1+x) x^2} \, dx,x,x^2\right )+6 \int \frac {1}{-1+x^2} \, dx+\int \frac {-1+3 x^2}{x^2 \left (-1+x^2\right )} \, dx \\ & = \frac {5}{x}-\frac {e^x}{x}+2 \text {arctanh}(x)-\frac {\log \left (-x \left (1-x^2\right )\right )}{x}+2 \int \frac {1}{-1+x^2} \, dx-2 \text {Subst}\left (\int \left (\frac {1}{-1+x}-\frac {1}{x^2}-\frac {1}{x}\right ) \, dx,x,x^2\right )+2 \text {Subst}\left (\int \left (\frac {1}{-1+x}-\frac {1}{x^3}-\frac {1}{x^2}-\frac {1}{x}\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{x^4}+\frac {5}{x}-\frac {e^x}{x}-\frac {\log \left (-x \left (1-x^2\right )\right )}{x} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.20 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {4-4 x^2+6 x^3-8 x^5+e^x \left (-x^3+x^4+x^5-x^6\right )+\left (-x^3+x^5\right ) \log \left (-x+x^3\right )}{-x^5+x^7} \, dx=\frac {1}{x^4}-\frac {1}{x}-\frac {e^x}{x}+6 \text {arctanh}(x)+\frac {6 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},x^2\right )}{x}-\frac {\log \left (x \left (-1+x^2\right )\right )}{x} \]
[In]
[Out]
Time = 0.42 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07
method | result | size |
default | \(-\frac {\ln \left (x^{3}-x \right )}{x}+\frac {5}{x}-\frac {{\mathrm e}^{x}}{x}+\frac {1}{x^{4}}\) | \(30\) |
parts | \(-\frac {\ln \left (x^{3}-x \right )}{x}+\frac {5}{x}-\frac {{\mathrm e}^{x}}{x}+\frac {1}{x^{4}}\) | \(30\) |
parallelrisch | \(\frac {2-2 \,{\mathrm e}^{x} x^{3}-2 \ln \left (x^{3}-x \right ) x^{3}+10 x^{3}}{2 x^{4}}\) | \(33\) |
risch | \(-\frac {\ln \left (x^{2}-1\right )}{x}-\frac {-i \pi \,x^{3} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (x^{2}-1\right )\right ) \operatorname {csgn}\left (i x \left (x^{2}-1\right )\right )+i \pi \,x^{3} \operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (i x \left (x^{2}-1\right )\right )}^{2}+i \pi \,x^{3} \operatorname {csgn}\left (i \left (x^{2}-1\right )\right ) {\operatorname {csgn}\left (i x \left (x^{2}-1\right )\right )}^{2}-i \pi \,x^{3} {\operatorname {csgn}\left (i x \left (x^{2}-1\right )\right )}^{3}+2 \,{\mathrm e}^{x} x^{3}+2 x^{3} \ln \left (x \right )-10 x^{3}-2}{2 x^{4}}\) | \(141\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {4-4 x^2+6 x^3-8 x^5+e^x \left (-x^3+x^4+x^5-x^6\right )+\left (-x^3+x^5\right ) \log \left (-x+x^3\right )}{-x^5+x^7} \, dx=-\frac {x^{3} e^{x} + x^{3} \log \left (x^{3} - x\right ) - 5 \, x^{3} - 1}{x^{4}} \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {4-4 x^2+6 x^3-8 x^5+e^x \left (-x^3+x^4+x^5-x^6\right )+\left (-x^3+x^5\right ) \log \left (-x+x^3\right )}{-x^5+x^7} \, dx=- \frac {e^{x}}{x} - \frac {\log {\left (x^{3} - x \right )}}{x} - \frac {- 5 x^{3} - 1}{x^{4}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (22) = 44\).
Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {4-4 x^2+6 x^3-8 x^5+e^x \left (-x^3+x^4+x^5-x^6\right )+\left (-x^3+x^5\right ) \log \left (-x+x^3\right )}{-x^5+x^7} \, dx=-\frac {{\left (x + 1\right )} \log \left (x + 1\right ) - {\left (x - 1\right )} \log \left (x - 1\right ) + e^{x} + \log \left (x\right ) + 1}{x} + \frac {6}{x} - \frac {2}{x^{2}} + \frac {2 \, x^{2} + 1}{x^{4}} + \log \left (x + 1\right ) - \log \left (x - 1\right ) \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {4-4 x^2+6 x^3-8 x^5+e^x \left (-x^3+x^4+x^5-x^6\right )+\left (-x^3+x^5\right ) \log \left (-x+x^3\right )}{-x^5+x^7} \, dx=-\frac {x^{3} e^{x} + x^{3} \log \left (x^{3} - x\right ) - 5 \, x^{3} - 1}{x^{4}} \]
[In]
[Out]
Time = 8.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {4-4 x^2+6 x^3-8 x^5+e^x \left (-x^3+x^4+x^5-x^6\right )+\left (-x^3+x^5\right ) \log \left (-x+x^3\right )}{-x^5+x^7} \, dx=-\frac {x^3\,{\mathrm {e}}^x+x^3\,\ln \left (x^3-x\right )-5\,x^3-1}{x^4} \]
[In]
[Out]