\(\int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx\) [2736]

Optimal result

Integrand size = 26, antiderivative size = 253 \[ \int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\frac {(-1+x)^{4/5} \left (-\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\sqrt {\frac {5}{10-2 \sqrt {5}}}+\frac {1}{\sqrt {10-2 \sqrt {5}}}-\frac {4 \sqrt [5]{-1+x}}{\sqrt {10-2 \sqrt {5}}}\right )+\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {5}{10+2 \sqrt {5}}}-\frac {1}{\sqrt {10+2 \sqrt {5}}}+\frac {4 \sqrt [5]{-1+x}}{\sqrt {10+2 \sqrt {5}}}\right )+\log \left (1+\sqrt [5]{-1+x}\right )+\frac {1}{4} \left (-1-\sqrt {5}\right ) \log \left (2+\left (-1-\sqrt {5}\right ) \sqrt [5]{-1+x}+2 (-1+x)^{2/5}\right )+\frac {1}{4} \left (-1+\sqrt {5}\right ) \log \left (2+\left (-1+\sqrt {5}\right ) \sqrt [5]{-1+x}+2 (-1+x)^{2/5}\right )\right )}{\sqrt [5]{(-1+x)^4}} \]

[Out]

(-1+x)^(4/5)*(-1/2*(10-2*5^(1/2))^(1/2)*arctan(5^(1/2)/(10-2*5^(1/2))^(1/2)+1/(10-2*5^(1/2))^(1/2)-4*(-1+x)^(1
/5)/(10-2*5^(1/2))^(1/2))+1/2*(10+2*5^(1/2))^(1/2)*arctan(5^(1/2)/(10+2*5^(1/2))^(1/2)-1/(10+2*5^(1/2))^(1/2)+
4*(-1+x)^(1/5)/(10+2*5^(1/2))^(1/2))+ln(1+(-1+x)^(1/5))+1/4*(-5^(1/2)-1)*ln(2+(-5^(1/2)-1)*(-1+x)^(1/5)+2*(-1+
x)^(2/5))+1/4*(5^(1/2)-1)*ln(2+(5^(1/2)-1)*(-1+x)^(1/5)+2*(-1+x)^(2/5)))/((-1+x)^4)^(1/5)

Rubi [F]

\[ \int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx \]

[In]

Int[1/(x*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5)),x]

[Out]

Defer[Int][1/(x*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.05 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\frac {(-1+x)^{4/5} \left (\log \left (1+\sqrt [5]{-1+x}\right )-\text {RootSum}\left [1-\text {$\#$1}+\text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-4 \log \left (\sqrt [5]{-1+x}-\text {$\#$1}\right )+3 \log \left (\sqrt [5]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (\sqrt [5]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt [5]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{-1+2 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]\right )}{\sqrt [5]{(-1+x)^4}} \]

[In]

Integrate[1/(x*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5)),x]

[Out]

((-1 + x)^(4/5)*(Log[1 + (-1 + x)^(1/5)] - RootSum[1 - #1 + #1^2 - #1^3 + #1^4 & , (-4*Log[(-1 + x)^(1/5) - #1
] + 3*Log[(-1 + x)^(1/5) - #1]*#1 - 2*Log[(-1 + x)^(1/5) - #1]*#1^2 + Log[(-1 + x)^(1/5) - #1]*#1^3)/(-1 + 2*#
1 - 3*#1^2 + 4*#1^3) & ]))/((-1 + x)^4)^(1/5)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 6.93 (sec) , antiderivative size = 10924, normalized size of antiderivative = 43.18

method	result	size
trager	\(\text {Expression too large to display}\)	\(10924\)

[In]

int(1/x/(x^4-4*x^3+6*x^2-4*x+1)^(1/5),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1084 vs. \(2 (185) = 370\).

Time = 0.89 (sec) , antiderivative size = 1084, normalized size of antiderivative = 4.28 \[ \int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\text {Too large to display} \]

[In]

integrate(1/x/(x^4-4*x^3+6*x^2-4*x+1)^(1/5),x, algorithm="fricas")

[Out]

-1/4*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1)*log(-1/64*((x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^3
+ (x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 - 4*(x - 1)*(s
qrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 + 16*(x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1) + ((x - 1)*(s
qrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 - 4*(x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1))*(sqrt(5) - 2*
sqrt(1/2*sqrt(5) - 5/2) + 1) - 64*(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)^(1/5))/(x - 1)) - 1/4*(sqrt(5) + 2*sqrt(1/2*
sqrt(5) - 5/2) + 1)*log(1/64*((x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^3 - 4*(x - 1)*(sqrt(5) + 2*sqr
t(1/2*sqrt(5) - 5/2) + 1)^2 + 16*(x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1) - 64*x + 64*(x^4 - 4*x^3 +
6*x^2 - 4*x + 1)^(1/5) + 64)/(x - 1)) + 1/4*(sqrt(5) - 2*sqrt(-3/16*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^
2 - 1/8*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) - 3)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1) - 3/16*(sqrt(5) -
2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 + 1/2*sqrt(5) + sqrt(1/2*sqrt(5) - 5/2) - 5/2) - 1)*log(1/64*((x - 1)*(sqrt(5
) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 + 4*sqrt(-3/16*(sqrt(5) + 2*sqr
t(1/2*sqrt(5) - 5/2) + 1)^2 - 1/8*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) - 3)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/
2) + 1) - 3/16*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 + 1/2*sqrt(5) + sqrt(1/2*sqrt(5) - 5/2) - 5/2)*(x -
1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1) + ((x - 1)*(sqrt(5) +
2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 - 4*(x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1))*(sqrt(5) - 2*sqrt(1/2*
sqrt(5) - 5/2) + 1) + 128*(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)^(1/5))/(x - 1)) + 1/4*(sqrt(5) + 2*sqrt(-3/16*(sqrt(
5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 - 1/8*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) - 3)*(sqrt(5) - 2*sqrt(1/2*sq
rt(5) - 5/2) + 1) - 3/16*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 + 1/2*sqrt(5) + sqrt(1/2*sqrt(5) - 5/2) -
5/2) - 1)*log(1/64*((x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) +
1)^2 - 4*sqrt(-3/16*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 - 1/8*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) - 3
)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1) - 3/16*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 + 1/2*sqrt(5) +
sqrt(1/2*sqrt(5) - 5/2) - 5/2)*(x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)*(sqrt(5) - 2*sqrt(1/2*sqrt(5
) - 5/2) + 1) + ((x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 - 4*(x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5)
- 5/2) + 1))*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1) + 128*(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)^(1/5))/(x - 1))
+ log((x + (x^4 - 4*x^3 + 6*x^2 - 4*x + 1)^(1/5) - 1)/(x - 1))

Sympy [F]

\[ \int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\int \frac {1}{x \sqrt [5]{\left (x - 1\right )^{4}}}\, dx \]

[In]

integrate(1/x/(x**4-4*x**3+6*x**2-4*x+1)**(1/5),x)

[Out]

Integral(1/(x*((x - 1)**4)**(1/5)), x)

Maxima [F]

\[ \int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}^{\frac {1}{5}} x} \,d x } \]

[In]

integrate(1/x/(x^4-4*x^3+6*x^2-4*x+1)^(1/5),x, algorithm="maxima")

[Out]

integrate(1/((x^4 - 4*x^3 + 6*x^2 - 4*x + 1)^(1/5)*x), x)

Giac [F]

\[ \int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}^{\frac {1}{5}} x} \,d x } \]

[In]

integrate(1/x/(x^4-4*x^3+6*x^2-4*x+1)^(1/5),x, algorithm="giac")

[Out]

integrate(1/((x^4 - 4*x^3 + 6*x^2 - 4*x + 1)^(1/5)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\int \frac {1}{x\,{\left (x^4-4\,x^3+6\,x^2-4\,x+1\right )}^{1/5}} \,d x \]

[In]

int(1/(x*(6*x^2 - 4*x - 4*x^3 + x^4 + 1)^(1/5)),x)

[Out]

int(1/(x*(6*x^2 - 4*x - 4*x^3 + x^4 + 1)^(1/5)), x)