\(\int \frac {8 (15-2 \sqrt {3}) x+96 \sqrt {3} x^3+24 \sqrt {3} x^5}{-4+(-20+60 \sqrt {3}) x^2+(-100+10 \sqrt {3}) x^4+(12+60 \sqrt {3}) x^6-36 x^8} \, dx\) [55]

Optimal result

Integrand size = 82, antiderivative size = 67 \[ \int \frac {8 \left (15-2 \sqrt {3}\right ) x+96 \sqrt {3} x^3+24 \sqrt {3} x^5}{-4+\left (-20+60 \sqrt {3}\right ) x^2+\left (-100+10 \sqrt {3}\right ) x^4+\left (12+60 \sqrt {3}\right ) x^6-36 x^8} \, dx=-\sqrt [4]{3} \arctan \left (\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {3}}}-\frac {1}{2} 3^{3/4} x^2\right )-\sqrt [4]{3} \text {arctanh}\left (\sqrt {\frac {1}{2}+\frac {1}{\sqrt {3}}}-\frac {x^2}{\sqrt [4]{3}}\right ) \] Output:

3^(1/4)*arctan(-1/6*(-18+12*3^(1/2))^(1/2)+1/2*3^(3/4)*x^2)+3^(1/4)*arctan 
h(-1/6*(18+12*3^(1/2))^(1/2)+1/3*3^(3/4)*x^2)
 

Mathematica [C] (verified)

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

Time = 0.09 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.60 \[ \int \frac {8 \left (15-2 \sqrt {3}\right ) x+96 \sqrt {3} x^3+24 \sqrt {3} x^5}{-4+\left (-20+60 \sqrt {3}\right ) x^2+\left (-100+10 \sqrt {3}\right ) x^4+\left (12+60 \sqrt {3}\right ) x^6-36 x^8} \, dx=\text {RootSum}\left [2+10 \text {$\#$1}^2-30 \sqrt {3} \text {$\#$1}^2+50 \text {$\#$1}^4-5 \sqrt {3} \text {$\#$1}^4-6 \text {$\#$1}^6-30 \sqrt {3} \text {$\#$1}^6+18 \text {$\#$1}^8\&,\frac {15 \log (x-\text {$\#$1})-2 \sqrt {3} \log (x-\text {$\#$1})+12 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^2+3 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^4}{-5+15 \sqrt {3}-50 \text {$\#$1}^2+5 \sqrt {3} \text {$\#$1}^2+9 \text {$\#$1}^4+45 \sqrt {3} \text {$\#$1}^4-36 \text {$\#$1}^6}\&\right ] \] Input:

Integrate[(8*(15 - 2*Sqrt[3])*x + 96*Sqrt[3]*x^3 + 24*Sqrt[3]*x^5)/(-4 + ( 
-20 + 60*Sqrt[3])*x^2 + (-100 + 10*Sqrt[3])*x^4 + (12 + 60*Sqrt[3])*x^6 - 
36*x^8),x]
 

Output:

RootSum[2 + 10*#1^2 - 30*Sqrt[3]*#1^2 + 50*#1^4 - 5*Sqrt[3]*#1^4 - 6*#1^6 
- 30*Sqrt[3]*#1^6 + 18*#1^8 & , (15*Log[x - #1] - 2*Sqrt[3]*Log[x - #1] + 
12*Sqrt[3]*Log[x - #1]*#1^2 + 3*Sqrt[3]*Log[x - #1]*#1^4)/(-5 + 15*Sqrt[3] 
 - 50*#1^2 + 5*Sqrt[3]*#1^2 + 9*#1^4 + 45*Sqrt[3]*#1^4 - 36*#1^6) & ]
 

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.

\(\displaystyle \int \frac {24 \sqrt {3} x^5+96 \sqrt {3} x^3+8 \left (15-2 \sqrt {3}\right ) x}{-36 x^8+\left (12+60 \sqrt {3}\right ) x^6+\left (10 \sqrt {3}-100\right ) x^4+\left (60 \sqrt {3}-20\right ) x^2-4} \, dx\)

\(\Big \downarrow \) 2028

\(\displaystyle \int \frac {x \left (24 \sqrt {3} x^4+96 \sqrt {3} x^2+8 \left (15-2 \sqrt {3}\right )\right )}{-36 x^8+\left (12+60 \sqrt {3}\right ) x^6+\left (10 \sqrt {3}-100\right ) x^4+\left (60 \sqrt {3}-20\right ) x^2-4}dx\)

\(\Big \downarrow \) 7266

\(\displaystyle \frac {1}{2} \int -\frac {4 \left (3 \sqrt {3} x^4+12 \sqrt {3} x^2-2 \sqrt {3}+15\right )}{18 x^8-6 \left (1+5 \sqrt {3}\right ) x^6+5 \left (10-\sqrt {3}\right ) x^4+10 \left (1-3 \sqrt {3}\right ) x^2+2}dx^2\)

\(\Big \downarrow \) 27

\(\displaystyle -2 \int \frac {3 \sqrt {3} x^4+12 \sqrt {3} x^2-2 \sqrt {3}+15}{18 x^8-6 \left (1+5 \sqrt {3}\right ) x^6+5 \left (10-\sqrt {3}\right ) x^4+10 \left (1-3 \sqrt {3}\right ) x^2+2}dx^2\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {3 \sqrt {3} x^4}{18 x^8-6 \left (1+5 \sqrt {3}\right ) x^6+5 \left (10-\sqrt {3}\right ) x^4+10 \left (1-3 \sqrt {3}\right ) x^2+2}+\frac {12 \sqrt {3} x^2}{18 x^8-6 \left (1+5 \sqrt {3}\right ) x^6+5 \left (10-\sqrt {3}\right ) x^4+10 \left (1-3 \sqrt {3}\right ) x^2+2}+\frac {2 \sqrt {3} \left (1-\frac {5 \sqrt {3}}{2}\right )}{-18 x^8+6 \left (1+5 \sqrt {3}\right ) x^6-5 \left (10-\sqrt {3}\right ) x^4-10 \left (1-3 \sqrt {3}\right ) x^2-2}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (-\left (\left (15-2 \sqrt {3}\right ) \int \frac {1}{-18 x^8+6 \left (1+5 \sqrt {3}\right ) x^6-5 \left (10-\sqrt {3}\right ) x^4-10 \left (1-3 \sqrt {3}\right ) x^2-2}dx^2\right )+12 \sqrt {3} \int \frac {x^2}{18 x^8-6 \left (1+5 \sqrt {3}\right ) x^6+5 \left (10-\sqrt {3}\right ) x^4+10 \left (1-3 \sqrt {3}\right ) x^2+2}dx^2+3 \sqrt {3} \int \frac {x^4}{18 x^8-6 \left (1+5 \sqrt {3}\right ) x^6+5 \left (10-\sqrt {3}\right ) x^4+10 \left (1-3 \sqrt {3}\right ) x^2+2}dx^2\right )\)

Input:

Int[(8*(15 - 2*Sqrt[3])*x + 96*Sqrt[3]*x^3 + 24*Sqrt[3]*x^5)/(-4 + (-20 + 
60*Sqrt[3])*x^2 + (-100 + 10*Sqrt[3])*x^4 + (12 + 60*Sqrt[3])*x^6 - 36*x^8 
),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.69

method	result	size
default	\(3^{\frac {1}{4}} \operatorname {arctanh}\left (\frac {\left (4 x^{2}-2 \sqrt {3}-2\right ) 3^{\frac {3}{4}}}{12}\right )+3^{\frac {1}{4}} \arctan \left (\frac {\left (18 x^{2}-6 \sqrt {3}+6\right ) 3^{\frac {3}{4}}}{36}\right )\)	\(46\)

Input:

int((8*(15-2*3^(1/2))*x+96*3^(1/2)*x^3+24*3^(1/2)*x^5)/(-4+(-20+60*3^(1/2) 
)*x^2+(-100+10*3^(1/2))*x^4+(12+60*3^(1/2))*x^6-36*x^8),x,method=_RETURNVE 
RBOSE)
 

Output:

3^(1/4)*arctanh(1/12*(4*x^2-2*3^(1/2)-2)*3^(3/4))+3^(1/4)*arctan(1/36*(18* 
x^2-6*3^(1/2)+6)*3^(3/4))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (51) = 102\).

Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.55 \[ \int \frac {8 \left (15-2 \sqrt {3}\right ) x+96 \sqrt {3} x^3+24 \sqrt {3} x^5}{-4+\left (-20+60 \sqrt {3}\right ) x^2+\left (-100+10 \sqrt {3}\right ) x^4+\left (12+60 \sqrt {3}\right ) x^6-36 x^8} \, dx=-3^{\frac {1}{4}} \arctan \left (-\frac {1}{12} \cdot 3^{\frac {3}{4}} {\left (5 \, x^{2} - 2 \, \sqrt {3}\right )} - \frac {1}{12} \cdot 3^{\frac {3}{4}} {\left (x^{2} + 2\right )}\right ) + \frac {1}{2} \cdot 3^{\frac {1}{4}} \log \left (36 \, x^{2} - \sqrt {3} {\left (5 \, \sqrt {3} + 3\right )} - 15 \, \sqrt {3} + 36 \cdot 3^{\frac {1}{4}} - 3\right ) - \frac {1}{2} \cdot 3^{\frac {1}{4}} \log \left (36 \, x^{2} - \sqrt {3} {\left (5 \, \sqrt {3} + 3\right )} - 15 \, \sqrt {3} - 36 \cdot 3^{\frac {1}{4}} - 3\right ) \] Input:

integrate((8*(15-2*3^(1/2))*x+96*3^(1/2)*x^3+24*3^(1/2)*x^5)/(-4+(-20+60*3 
^(1/2))*x^2+(-100+10*3^(1/2))*x^4+(12+60*3^(1/2))*x^6-36*x^8),x, algorithm 
="fricas")
 

Output:

-3^(1/4)*arctan(-1/12*3^(3/4)*(5*x^2 - 2*sqrt(3)) - 1/12*3^(3/4)*(x^2 + 2) 
) + 1/2*3^(1/4)*log(36*x^2 - sqrt(3)*(5*sqrt(3) + 3) - 15*sqrt(3) + 36*3^( 
1/4) - 3) - 1/2*3^(1/4)*log(36*x^2 - sqrt(3)*(5*sqrt(3) + 3) - 15*sqrt(3) 
- 36*3^(1/4) - 3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {8 \left (15-2 \sqrt {3}\right ) x+96 \sqrt {3} x^3+24 \sqrt {3} x^5}{-4+\left (-20+60 \sqrt {3}\right ) x^2+\left (-100+10 \sqrt {3}\right ) x^4+\left (12+60 \sqrt {3}\right ) x^6-36 x^8} \, dx=\text {Timed out} \] Input:

integrate((8*(15-2*3**(1/2))*x+96*3**(1/2)*x**3+24*3**(1/2)*x**5)/(-4+(-20 
+60*3**(1/2))*x**2+(-100+10*3**(1/2))*x**4+(12+60*3**(1/2))*x**6-36*x**8), 
x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {8 \left (15-2 \sqrt {3}\right ) x+96 \sqrt {3} x^3+24 \sqrt {3} x^5}{-4+\left (-20+60 \sqrt {3}\right ) x^2+\left (-100+10 \sqrt {3}\right ) x^4+\left (12+60 \sqrt {3}\right ) x^6-36 x^8} \, dx=\int { -\frac {4 \, {\left (3 \, \sqrt {3} x^{5} + 12 \, \sqrt {3} x^{3} - x {\left (2 \, \sqrt {3} - 15\right )}\right )}}{18 \, x^{8} - 6 \, x^{6} {\left (5 \, \sqrt {3} + 1\right )} - 5 \, x^{4} {\left (\sqrt {3} - 10\right )} - 10 \, x^{2} {\left (3 \, \sqrt {3} - 1\right )} + 2} \,d x } \] Input:

integrate((8*(15-2*3^(1/2))*x+96*3^(1/2)*x^3+24*3^(1/2)*x^5)/(-4+(-20+60*3 
^(1/2))*x^2+(-100+10*3^(1/2))*x^4+(12+60*3^(1/2))*x^6-36*x^8),x, algorithm 
="maxima")
 

Output:

-4*integrate((3*sqrt(3)*x^5 + 12*sqrt(3)*x^3 - x*(2*sqrt(3) - 15))/(18*x^8 
 - 6*x^6*(5*sqrt(3) + 1) - 5*x^4*(sqrt(3) - 10) - 10*x^2*(3*sqrt(3) - 1) + 
 2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {8 \left (15-2 \sqrt {3}\right ) x+96 \sqrt {3} x^3+24 \sqrt {3} x^5}{-4+\left (-20+60 \sqrt {3}\right ) x^2+\left (-100+10 \sqrt {3}\right ) x^4+\left (12+60 \sqrt {3}\right ) x^6-36 x^8} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((8*(15-2*3^(1/2))*x+96*3^(1/2)*x^3+24*3^(1/2)*x^5)/(-4+(-20+60*3 
^(1/2))*x^2+(-100+10*3^(1/2))*x^4+(12+60*3^(1/2))*x^6-36*x^8),x, algorithm 
="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to find common minimal polyn 
omial Error: Bad Argument ValueUnable to find common minimal polynomial Er 
ror: Bad
 

Mupad [B] (verification not implemented)

Time = 11.06 (sec) , antiderivative size = 761, normalized size of antiderivative = 11.36 \[ \int \frac {8 \left (15-2 \sqrt {3}\right ) x+96 \sqrt {3} x^3+24 \sqrt {3} x^5}{-4+\left (-20+60 \sqrt {3}\right ) x^2+\left (-100+10 \sqrt {3}\right ) x^4+\left (12+60 \sqrt {3}\right ) x^6-36 x^8} \, dx=\text {Too large to display} \] Input:

int((96*3^(1/2)*x^3 + 24*3^(1/2)*x^5 - 8*x*(2*3^(1/2) - 15))/(x^6*(60*3^(1 
/2) + 12) + x^2*(60*3^(1/2) - 20) + x^4*(10*3^(1/2) - 100) - 36*x^8 - 4),x 
)
 

Output:

symsum(log(456889680*root(26866560*3^(1/2)*z^4 - 194630448*z^4 - 5037480*3 
^(1/2) + 36493209, z, k) + 1247412816*3^(1/2)*root(26866560*3^(1/2)*z^4 - 
194630448*z^4 - 5037480*3^(1/2) + 36493209, z, k) + 116925120*3^(1/2) + 98 
7505680*3^(1/2)*root(26866560*3^(1/2)*z^4 - 194630448*z^4 - 5037480*3^(1/2 
) + 36493209, z, k)^2 + 12462219648*3^(1/2)*root(26866560*3^(1/2)*z^4 - 19 
4630448*z^4 - 5037480*3^(1/2) + 36493209, z, k)^3 + 6711293680*3^(1/2)*roo 
t(26866560*3^(1/2)*z^4 - 194630448*z^4 - 5037480*3^(1/2) + 36493209, z, k) 
^4 + 25395954432*3^(1/2)*root(26866560*3^(1/2)*z^4 - 194630448*z^4 - 50374 
80*3^(1/2) + 36493209, z, k)^5 + 7845886720*3^(1/2)*root(26866560*3^(1/2)* 
z^4 - 194630448*z^4 - 5037480*3^(1/2) + 36493209, z, k)^6 - 86845934655*ro 
ot(26866560*3^(1/2)*z^4 - 194630448*z^4 - 5037480*3^(1/2) + 36493209, z, k 
)*x^2 - 9785804940*3^(1/2)*x^2 + 8919043926*root(26866560*3^(1/2)*z^4 - 19 
4630448*z^4 - 5037480*3^(1/2) + 36493209, z, k)^2 + 5990014080*root(268665 
60*3^(1/2)*z^4 - 194630448*z^4 - 5037480*3^(1/2) + 36493209, z, k)^3 + 371 
22771840*root(26866560*3^(1/2)*z^4 - 194630448*z^4 - 5037480*3^(1/2) + 364 
93209, z, k)^4 + 22080787200*root(26866560*3^(1/2)*z^4 - 194630448*z^4 - 5 
037480*3^(1/2) + 36493209, z, k)^5 + 28883711264*root(26866560*3^(1/2)*z^4 
 - 194630448*z^4 - 5037480*3^(1/2) + 36493209, z, k)^6 - 5212268892*x^2 - 
29104473855*root(26866560*3^(1/2)*z^4 - 194630448*z^4 - 5037480*3^(1/2) + 
36493209, z, k)^2*x^2 - 540228895440*root(26866560*3^(1/2)*z^4 - 194630...
 

Reduce [F]

\[ \int \frac {8 \left (15-2 \sqrt {3}\right ) x+96 \sqrt {3} x^3+24 \sqrt {3} x^5}{-4+\left (-20+60 \sqrt {3}\right ) x^2+\left (-100+10 \sqrt {3}\right ) x^4+\left (12+60 \sqrt {3}\right ) x^6-36 x^8} \, dx=-216 \sqrt {3}\, \left (\int \frac {x^{13}}{324 x^{16}-216 x^{14}-864 x^{12}-1140 x^{10}-3023 x^{8}+76 x^{6}-2400 x^{4}+40 x^{2}+4}d x \right )-792 \sqrt {3}\, \left (\int \frac {x^{11}}{324 x^{16}-216 x^{14}-864 x^{12}-1140 x^{10}-3023 x^{8}+76 x^{6}-2400 x^{4}+40 x^{2}+4}d x \right )-168 \sqrt {3}\, \left (\int \frac {x^{9}}{324 x^{16}-216 x^{14}-864 x^{12}-1140 x^{10}-3023 x^{8}+76 x^{6}-2400 x^{4}+40 x^{2}+4}d x \right )-4368 \sqrt {3}\, \left (\int \frac {x^{7}}{324 x^{16}-216 x^{14}-864 x^{12}-1140 x^{10}-3023 x^{8}+76 x^{6}-2400 x^{4}+40 x^{2}+4}d x \right )-404 \sqrt {3}\, \left (\int \frac {x^{5}}{324 x^{16}-216 x^{14}-864 x^{12}-1140 x^{10}-3023 x^{8}+76 x^{6}-2400 x^{4}+40 x^{2}+4}d x \right )-1816 \sqrt {3}\, \left (\int \frac {x^{3}}{324 x^{16}-216 x^{14}-864 x^{12}-1140 x^{10}-3023 x^{8}+76 x^{6}-2400 x^{4}+40 x^{2}+4}d x \right )+16 \sqrt {3}\, \left (\int \frac {x}{324 x^{16}-216 x^{14}-864 x^{12}-1140 x^{10}-3023 x^{8}+76 x^{6}-2400 x^{4}+40 x^{2}+4}d x \right )-1080 \left (\int \frac {x^{11}}{324 x^{16}-216 x^{14}-864 x^{12}-1140 x^{10}-3023 x^{8}+76 x^{6}-2400 x^{4}+40 x^{2}+4}d x \right )-5580 \left (\int \frac {x^{9}}{324 x^{16}-216 x^{14}-864 x^{12}-1140 x^{10}-3023 x^{8}+76 x^{6}-2400 x^{4}+40 x^{2}+4}d x \right )-720 \left (\int \frac {x^{7}}{324 x^{16}-216 x^{14}-864 x^{12}-1140 x^{10}-3023 x^{8}+76 x^{6}-2400 x^{4}+40 x^{2}+4}d x \right )-7200 \left (\int \frac {x^{5}}{324 x^{16}-216 x^{14}-864 x^{12}-1140 x^{10}-3023 x^{8}+76 x^{6}-2400 x^{4}+40 x^{2}+4}d x \right )+120 \left (\int \frac {x^{3}}{324 x^{16}-216 x^{14}-864 x^{12}-1140 x^{10}-3023 x^{8}+76 x^{6}-2400 x^{4}+40 x^{2}+4}d x \right )-120 \left (\int \frac {x}{324 x^{16}-216 x^{14}-864 x^{12}-1140 x^{10}-3023 x^{8}+76 x^{6}-2400 x^{4}+40 x^{2}+4}d x \right ) \] Input:

int((8*(15-2*3^(1/2))*x+96*3^(1/2)*x^3+24*3^(1/2)*x^5)/(-4+(-20+60*3^(1/2) 
)*x^2+(-100+10*3^(1/2))*x^4+(12+60*3^(1/2))*x^6-36*x^8),x)
 

Output:

4*( - 54*sqrt(3)*int(x**13/(324*x**16 - 216*x**14 - 864*x**12 - 1140*x**10 
 - 3023*x**8 + 76*x**6 - 2400*x**4 + 40*x**2 + 4),x) - 198*sqrt(3)*int(x** 
11/(324*x**16 - 216*x**14 - 864*x**12 - 1140*x**10 - 3023*x**8 + 76*x**6 - 
 2400*x**4 + 40*x**2 + 4),x) - 42*sqrt(3)*int(x**9/(324*x**16 - 216*x**14 
- 864*x**12 - 1140*x**10 - 3023*x**8 + 76*x**6 - 2400*x**4 + 40*x**2 + 4), 
x) - 1092*sqrt(3)*int(x**7/(324*x**16 - 216*x**14 - 864*x**12 - 1140*x**10 
 - 3023*x**8 + 76*x**6 - 2400*x**4 + 40*x**2 + 4),x) - 101*sqrt(3)*int(x** 
5/(324*x**16 - 216*x**14 - 864*x**12 - 1140*x**10 - 3023*x**8 + 76*x**6 - 
2400*x**4 + 40*x**2 + 4),x) - 454*sqrt(3)*int(x**3/(324*x**16 - 216*x**14 
- 864*x**12 - 1140*x**10 - 3023*x**8 + 76*x**6 - 2400*x**4 + 40*x**2 + 4), 
x) + 4*sqrt(3)*int(x/(324*x**16 - 216*x**14 - 864*x**12 - 1140*x**10 - 302 
3*x**8 + 76*x**6 - 2400*x**4 + 40*x**2 + 4),x) - 270*int(x**11/(324*x**16 
- 216*x**14 - 864*x**12 - 1140*x**10 - 3023*x**8 + 76*x**6 - 2400*x**4 + 4 
0*x**2 + 4),x) - 1395*int(x**9/(324*x**16 - 216*x**14 - 864*x**12 - 1140*x 
**10 - 3023*x**8 + 76*x**6 - 2400*x**4 + 40*x**2 + 4),x) - 180*int(x**7/(3 
24*x**16 - 216*x**14 - 864*x**12 - 1140*x**10 - 3023*x**8 + 76*x**6 - 2400 
*x**4 + 40*x**2 + 4),x) - 1800*int(x**5/(324*x**16 - 216*x**14 - 864*x**12 
 - 1140*x**10 - 3023*x**8 + 76*x**6 - 2400*x**4 + 40*x**2 + 4),x) + 30*int 
(x**3/(324*x**16 - 216*x**14 - 864*x**12 - 1140*x**10 - 3023*x**8 + 76*x** 
6 - 2400*x**4 + 40*x**2 + 4),x) - 30*int(x/(324*x**16 - 216*x**14 - 864...