Integrand size = 33, antiderivative size = 26 \[ \int \frac {1+3 x^4}{\left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx=-\frac {2 \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {-x+x^5}}\right )}{\sqrt {a}} \] Output:
-2*arctanh(x/a^(1/2)/(x^5-x)^(1/2))/a^(1/2)
Time = 11.51 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1+3 x^4}{\left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx=-\frac {2 \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {-x+x^5}}\right )}{\sqrt {a}} \] Input:
Integrate[(1 + 3*x^4)/((-a - x + a*x^4)*Sqrt[-x + x^5]),x]
Output:
(-2*ArcTanh[x/(Sqrt[a]*Sqrt[-x + x^5])])/Sqrt[a]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {3 x^4+1}{\sqrt {x^5-x} \left (a x^4-a-x\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^4-1} \int -\frac {3 x^4+1}{\sqrt {x} \sqrt {x^4-1} \left (-a x^4+x+a\right )}dx}{\sqrt {x^5-x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {x^4-1} \int \frac {3 x^4+1}{\sqrt {x} \sqrt {x^4-1} \left (-a x^4+x+a\right )}dx}{\sqrt {x^5-x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^4-1} \int \frac {3 x^4+1}{\sqrt {x^4-1} \left (-a x^4+x+a\right )}d\sqrt {x}}{\sqrt {x^5-x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^4-1} \int \left (\frac {4 a+3 x}{a \sqrt {x^4-1} \left (-a x^4+x+a\right )}-\frac {3}{a \sqrt {x^4-1}}\right )d\sqrt {x}}{\sqrt {x^5-x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^4-1} \left (-4 \int \frac {1}{\sqrt {x^4-1} \left (a x^4-x-a\right )}d\sqrt {x}-\frac {3 \int \frac {x}{\sqrt {x^4-1} \left (a x^4-x-a\right )}d\sqrt {x}}{a}-\frac {\left (\frac {3}{2}+\frac {3 i}{2}\right ) \sqrt {-\frac {(-1)^{3/4} \left (\sqrt [4]{-1} x+1\right )^2}{x}} \sqrt {\frac {i \left (1-x^4\right )}{x^2}} x^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {(-1)^{3/4} \left (i \sqrt {2} x^2-2 \sqrt [4]{-1} x+\sqrt {2}\right )}{x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{\sqrt {2 \left (2+\sqrt {2}\right )} a \left (\sqrt [4]{-1} x+1\right ) \sqrt {x^4-1}}+\frac {\left (\frac {3}{2}+\frac {3 i}{2}\right ) \sqrt {\frac {(-1)^{3/4} \left (1-\sqrt [4]{-1} x\right )^2}{x}} \sqrt {\frac {i \left (1-x^4\right )}{x^2}} x^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {(-1)^{3/4} \left (i \sqrt {2} x^2+2 \sqrt [4]{-1} x+\sqrt {2}\right )}{x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{\sqrt {2 \left (2+\sqrt {2}\right )} a \left (1-\sqrt [4]{-1} x\right ) \sqrt {x^4-1}}\right )}{\sqrt {x^5-x}}\) |
Input:
Int[(1 + 3*x^4)/((-a - x + a*x^4)*Sqrt[-x + x^5]),x]
Output:
$Aborted
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a}\, \sqrt {x^{5}-x}}{x}\right )}{\sqrt {a}}\) | \(23\) |
Input:
int((3*x^4+1)/(a*x^4-a-x)/(x^5-x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/a^(1/2)*arctanh(a^(1/2)*(x^5-x)^(1/2)/x)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.62 \[ \int \frac {1+3 x^4}{\left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx=\left [\frac {\log \left (\frac {a^{2} x^{8} - 2 \, a^{2} x^{4} + 6 \, a x^{5} - 4 \, {\left (a x^{4} - a + x\right )} \sqrt {x^{5} - x} \sqrt {a} + a^{2} - 6 \, a x + x^{2}}{a^{2} x^{8} - 2 \, a^{2} x^{4} - 2 \, a x^{5} + a^{2} + 2 \, a x + x^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (a x^{4} - a + x\right )} \sqrt {x^{5} - x} \sqrt {-a}}{2 \, {\left (a x^{5} - a x\right )}}\right )}{a}\right ] \] Input:
integrate((3*x^4+1)/(a*x^4-a-x)/(x^5-x)^(1/2),x, algorithm="fricas")
Output:
[1/2*log((a^2*x^8 - 2*a^2*x^4 + 6*a*x^5 - 4*(a*x^4 - a + x)*sqrt(x^5 - x)* sqrt(a) + a^2 - 6*a*x + x^2)/(a^2*x^8 - 2*a^2*x^4 - 2*a*x^5 + a^2 + 2*a*x + x^2))/sqrt(a), sqrt(-a)*arctan(1/2*(a*x^4 - a + x)*sqrt(x^5 - x)*sqrt(-a )/(a*x^5 - a*x))/a]
Timed out. \[ \int \frac {1+3 x^4}{\left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx=\text {Timed out} \] Input:
integrate((3*x**4+1)/(a*x**4-a-x)/(x**5-x)**(1/2),x)
Output:
Timed out
\[ \int \frac {1+3 x^4}{\left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx=\int { \frac {3 \, x^{4} + 1}{{\left (a x^{4} - a - x\right )} \sqrt {x^{5} - x}} \,d x } \] Input:
integrate((3*x^4+1)/(a*x^4-a-x)/(x^5-x)^(1/2),x, algorithm="maxima")
Output:
integrate((3*x^4 + 1)/((a*x^4 - a - x)*sqrt(x^5 - x)), x)
\[ \int \frac {1+3 x^4}{\left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx=\int { \frac {3 \, x^{4} + 1}{{\left (a x^{4} - a - x\right )} \sqrt {x^{5} - x}} \,d x } \] Input:
integrate((3*x^4+1)/(a*x^4-a-x)/(x^5-x)^(1/2),x, algorithm="giac")
Output:
integrate((3*x^4 + 1)/((a*x^4 - a - x)*sqrt(x^5 - x)), x)
Time = 7.54 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {1+3 x^4}{\left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx=\frac {\ln \left (\frac {a-x+2\,\sqrt {a}\,\sqrt {x^5-x}-a\,x^4}{-a\,x^4+x+a}\right )}{\sqrt {a}} \] Input:
int(-(3*x^4 + 1)/((x^5 - x)^(1/2)*(a + x - a*x^4)),x)
Output:
log((a - x + 2*a^(1/2)*(x^5 - x)^(1/2) - a*x^4)/(a + x - a*x^4))/a^(1/2)
\[ \int \frac {1+3 x^4}{\left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx=3 \left (\int \frac {\sqrt {x}\, \sqrt {x^{4}-1}\, x^{3}}{a \,x^{8}-2 a \,x^{4}-x^{5}+a +x}d x \right )+\int \frac {\sqrt {x}\, \sqrt {x^{4}-1}}{a \,x^{9}-2 a \,x^{5}-x^{6}+a x +x^{2}}d x \] Input:
int((3*x^4+1)/(a*x^4-a-x)/(x^5-x)^(1/2),x)
Output:
3*int((sqrt(x)*sqrt(x**4 - 1)*x**3)/(a*x**8 - 2*a*x**4 + a - x**5 + x),x) + int((sqrt(x)*sqrt(x**4 - 1))/(a*x**9 - 2*a*x**5 + a*x - x**6 + x**2),x)