3.21.64 \(\int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} (1+2 x^2) (3-7 x+7 x^2-6 x^3+2 x^4)} \, dx\) [2064]

3.21.64.1 Optimal result

Integrand size = 71, antiderivative size = 149 \[ \int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}}{2-2 x+\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}}\right )+\log \left (-1+x+\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}\right )-\frac {1}{2} \log \left (1-2 x+x^2+(1-x) \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}+\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3}\right ) \]

output

3^(1/2)*arctan(3^(1/2)*((-2*x^2+1)/(2*x^2+1))^(1/3)/(2-2*x+((-2*x^2+1)/(2* 
x^2+1))^(1/3)))+ln(-1+x+((-2*x^2+1)/(2*x^2+1))^(1/3))-1/2*ln(1-2*x+x^2+(1- 
x)*((-2*x^2+1)/(2*x^2+1))^(1/3)+((-2*x^2+1)/(2*x^2+1))^(2/3))
 

3.21.64.2 Mathematica [A] (verified)

Time = 10.53 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00 \[ \int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}}{2-2 x+\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}}\right )+\log \left (-1+x+\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}\right )-\frac {1}{2} \log \left (1-2 x+x^2+(1-x) \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}+\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3}\right ) \]

input

Integrate[(3 - 8*x + 8*x^2 - 12*x^4)/(x*((1 - 2*x^2)/(1 + 2*x^2))^(1/3)*(1 
 + 2*x^2)*(3 - 7*x + 7*x^2 - 6*x^3 + 2*x^4)),x]
 

output

Sqrt[3]*ArcTan[(Sqrt[3]*((1 - 2*x^2)/(1 + 2*x^2))^(1/3))/(2 - 2*x + ((1 - 
2*x^2)/(1 + 2*x^2))^(1/3))] + Log[-1 + x + ((1 - 2*x^2)/(1 + 2*x^2))^(1/3) 
] - Log[1 - 2*x + x^2 + (1 - x)*((1 - 2*x^2)/(1 + 2*x^2))^(1/3) + ((1 - 2* 
x^2)/(1 + 2*x^2))^(2/3)]/2
 

3.21.64.3 Rubi [F]

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.

\(\displaystyle \int \frac {-12 x^4+8 x^2-8 x+3}{x \sqrt [3]{\frac {1-2 x^2}{2 x^2+1}} \left (2 x^2+1\right ) \left (2 x^4-6 x^3+7 x^2-7 x+3\right )} \, dx\)

\(\Big \downarrow \) 2058

\(\displaystyle \frac {\sqrt [3]{1-2 x^2} \int \frac {-12 x^4+8 x^2-8 x+3}{x \sqrt [3]{1-2 x^2} \left (2 x^2+1\right )^{2/3} \left (2 x^4-6 x^3+7 x^2-7 x+3\right )}dx}{\sqrt [3]{\frac {1-2 x^2}{2 x^2+1}} \sqrt [3]{2 x^2+1}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {\sqrt [3]{1-2 x^2} \int \left (\frac {-14 x^3+6 x^2+x-1}{\sqrt [3]{1-2 x^2} \left (2 x^2+1\right )^{2/3} \left (2 x^4-6 x^3+7 x^2-7 x+3\right )}+\frac {1}{x \sqrt [3]{1-2 x^2} \left (2 x^2+1\right )^{2/3}}\right )dx}{\sqrt [3]{\frac {1-2 x^2}{2 x^2+1}} \sqrt [3]{2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt [3]{1-2 x^2} \left (\int \frac {1}{\sqrt [3]{1-2 x^2} \left (2 x^2+1\right )^{2/3} \left (-2 x^4+6 x^3-7 x^2+7 x-3\right )}dx+\int \frac {x}{\sqrt [3]{1-2 x^2} \left (2 x^2+1\right )^{2/3} \left (2 x^4-6 x^3+7 x^2-7 x+3\right )}dx+6 \int \frac {x^2}{\sqrt [3]{1-2 x^2} \left (2 x^2+1\right )^{2/3} \left (2 x^4-6 x^3+7 x^2-7 x+3\right )}dx-14 \int \frac {x^3}{\sqrt [3]{1-2 x^2} \left (2 x^2+1\right )^{2/3} \left (2 x^4-6 x^3+7 x^2-7 x+3\right )}dx+\frac {1}{2} \sqrt {3} \arctan \left (\frac {2 \sqrt [3]{1-2 x^2}}{\sqrt {3} \sqrt [3]{2 x^2+1}}+\frac {1}{\sqrt {3}}\right )+\frac {3}{4} \log \left (\sqrt [3]{1-2 x^2}-\sqrt [3]{2 x^2+1}\right )-\frac {\log (x)}{2}\right )}{\sqrt [3]{\frac {1-2 x^2}{2 x^2+1}} \sqrt [3]{2 x^2+1}}\)

input

Int[(3 - 8*x + 8*x^2 - 12*x^4)/(x*((1 - 2*x^2)/(1 + 2*x^2))^(1/3)*(1 + 2*x 
^2)*(3 - 7*x + 7*x^2 - 6*x^3 + 2*x^4)),x]
 

output

$Aborted
 

3.21.64.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.21.64.4 Maple [C] (verified)

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

Time = 2.90 (sec) , antiderivative size = 2146, normalized size of antiderivative = 14.40

method	result	size
trager	\(\text {Expression too large to display}\)	\(2146\)

input

int((-12*x^4+8*x^2-8*x+3)/x/((-2*x^2+1)/(2*x^2+1))^(1/3)/(2*x^2+1)/(2*x^4- 
6*x^3+7*x^2-7*x+3),x,method=_RETURNVERBOSE)
 

output

-2*RootOf(4*_Z^2+2*_Z+1)*ln((8-12*x-28*RootOf(4*_Z^2+2*_Z+1)^2-20*RootOf(4 
*_Z^2+2*_Z+1)+15*(-(2*x^2-1)/(2*x^2+1))^(2/3)+15*(-(2*x^2-1)/(2*x^2+1))^(1 
/3)-84*RootOf(4*_Z^2+2*_Z+1)*x^4+56*RootOf(4*_Z^2+2*_Z+1)^2*x^2+98*RootOf( 
4*_Z^2+2*_Z+1)*x^3-58*RootOf(4*_Z^2+2*_Z+1)*x^2+42*RootOf(4*_Z^2+2*_Z+1)*x 
-30*x*(-(2*x^2-1)/(2*x^2+1))^(1/3)-8*x^5+12*x^2-28*x^3+24*x^4+28*RootOf(4* 
_Z^2+2*_Z+1)*x^5+12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^3 
-12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^4-12*RootOf(4*_Z^ 
2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^2+24*RootOf(4*_Z^2+2*_Z+1)*(-(2*x 
^2-1)/(2*x^2+1))^(1/3)*x^3-18*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1)) 
^(1/3)*x^2+12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x+6*x*Roo 
tOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)-6*RootOf(4*_Z^2+2*_Z+1)*(- 
(2*x^2-1)/(2*x^2+1))^(2/3)-6*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^ 
(1/3)+30*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^4-60*(-(2*x^2-1)/(2*x^2+1))^(1/3)* 
x^3+30*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^2+45*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^ 
2-30*x^3*(-(2*x^2-1)/(2*x^2+1))^(2/3)-15*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x)/x 
/(2*x^4-6*x^3+7*x^2-7*x+3))+2*RootOf(4*_Z^2+2*_Z+1)*ln(-(-11+33*x+28*RootO 
f(4*_Z^2+2*_Z+1)^2+8*RootOf(4*_Z^2+2*_Z+1)-18*(-(2*x^2-1)/(2*x^2+1))^(2/3) 
-18*(-(2*x^2-1)/(2*x^2+1))^(1/3)-84*RootOf(4*_Z^2+2*_Z+1)*x^4-56*RootOf(4* 
_Z^2+2*_Z+1)^2*x^2+98*RootOf(4*_Z^2+2*_Z+1)*x^3-114*RootOf(4*_Z^2+2*_Z+1)* 
x^2+42*RootOf(4*_Z^2+2*_Z+1)*x+36*x*(-(2*x^2-1)/(2*x^2+1))^(1/3)+22*x^5...
 

3.21.64.5 Fricas [A] (verification not implemented)

Time = 2.53 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.87 \[ \int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=-\sqrt {3} \arctan \left (\frac {434 \, \sqrt {3} {\left (2 \, x^{3} - 2 \, x^{2} + x - 1\right )} \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {2}{3}} + 682 \, \sqrt {3} {\left (2 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {1}{3}} + \sqrt {3} {\left (242 \, x^{5} - 726 \, x^{4} + 847 \, x^{3} - 1095 \, x^{2} + 363 \, x + 124\right )}}{2662 \, x^{5} - 7986 \, x^{4} + 9317 \, x^{3} - 5969 \, x^{2} + 3993 \, x - 1674}\right ) + \frac {1}{2} \, \log \left (\frac {2 \, x^{5} - 6 \, x^{4} + 7 \, x^{3} - 7 \, x^{2} + 3 \, {\left (2 \, x^{3} - 2 \, x^{2} + x - 1\right )} \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {2}{3}} + 3 \, {\left (2 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {1}{3}} + 3 \, x}{2 \, x^{5} - 6 \, x^{4} + 7 \, x^{3} - 7 \, x^{2} + 3 \, x}\right ) \]

input

integrate((-12*x^4+8*x^2-8*x+3)/x/((-2*x^2+1)/(2*x^2+1))^(1/3)/(2*x^2+1)/( 
2*x^4-6*x^3+7*x^2-7*x+3),x, algorithm="fricas")
 

output

-sqrt(3)*arctan((434*sqrt(3)*(2*x^3 - 2*x^2 + x - 1)*(-(2*x^2 - 1)/(2*x^2 
+ 1))^(2/3) + 682*sqrt(3)*(2*x^4 - 4*x^3 + 3*x^2 - 2*x + 1)*(-(2*x^2 - 1)/ 
(2*x^2 + 1))^(1/3) + sqrt(3)*(242*x^5 - 726*x^4 + 847*x^3 - 1095*x^2 + 363 
*x + 124))/(2662*x^5 - 7986*x^4 + 9317*x^3 - 5969*x^2 + 3993*x - 1674)) + 
1/2*log((2*x^5 - 6*x^4 + 7*x^3 - 7*x^2 + 3*(2*x^3 - 2*x^2 + x - 1)*(-(2*x^ 
2 - 1)/(2*x^2 + 1))^(2/3) + 3*(2*x^4 - 4*x^3 + 3*x^2 - 2*x + 1)*(-(2*x^2 - 
 1)/(2*x^2 + 1))^(1/3) + 3*x)/(2*x^5 - 6*x^4 + 7*x^3 - 7*x^2 + 3*x))
 

3.21.64.6 Sympy [F(-1)]

Timed out. \[ \int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=\text {Timed out} \]

input

integrate((-12*x**4+8*x**2-8*x+3)/x/((-2*x**2+1)/(2*x**2+1))**(1/3)/(2*x** 
2+1)/(2*x**4-6*x**3+7*x**2-7*x+3),x)
 

output

Timed out
 

3.21.64.7 Maxima [F]

\[ \int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=\int { -\frac {12 \, x^{4} - 8 \, x^{2} + 8 \, x - 3}{{\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2} - 7 \, x + 3\right )} {\left (2 \, x^{2} + 1\right )} x \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {1}{3}}} \,d x } \]

input

integrate((-12*x^4+8*x^2-8*x+3)/x/((-2*x^2+1)/(2*x^2+1))^(1/3)/(2*x^2+1)/( 
2*x^4-6*x^3+7*x^2-7*x+3),x, algorithm="maxima")
 

output

-integrate((12*x^4 - 8*x^2 + 8*x - 3)/((2*x^4 - 6*x^3 + 7*x^2 - 7*x + 3)*( 
2*x^2 + 1)*x*(-(2*x^2 - 1)/(2*x^2 + 1))^(1/3)), x)
 

3.21.64.8 Giac [F]

\[ \int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=\int { -\frac {12 \, x^{4} - 8 \, x^{2} + 8 \, x - 3}{{\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2} - 7 \, x + 3\right )} {\left (2 \, x^{2} + 1\right )} x \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {1}{3}}} \,d x } \]

input

integrate((-12*x^4+8*x^2-8*x+3)/x/((-2*x^2+1)/(2*x^2+1))^(1/3)/(2*x^2+1)/( 
2*x^4-6*x^3+7*x^2-7*x+3),x, algorithm="giac")
 

output

integrate(-(12*x^4 - 8*x^2 + 8*x - 3)/((2*x^4 - 6*x^3 + 7*x^2 - 7*x + 3)*( 
2*x^2 + 1)*x*(-(2*x^2 - 1)/(2*x^2 + 1))^(1/3)), x)
 

3.21.64.9 Mupad [F(-1)]

Timed out. \[ \int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=-\int \frac {12\,x^4-8\,x^2+8\,x-3}{x\,\left (2\,x^2+1\right )\,{\left (-\frac {2\,x^2-1}{2\,x^2+1}\right )}^{1/3}\,\left (2\,x^4-6\,x^3+7\,x^2-7\,x+3\right )} \,d x \]

input

int(-(8*x - 8*x^2 + 12*x^4 - 3)/(x*(2*x^2 + 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^ 
(1/3)*(7*x^2 - 7*x - 6*x^3 + 2*x^4 + 3)),x)
 

output

-int((8*x - 8*x^2 + 12*x^4 - 3)/(x*(2*x^2 + 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^ 
(1/3)*(7*x^2 - 7*x - 6*x^3 + 2*x^4 + 3)), x)