\(\int \frac {1}{\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx\) [157]

Optimal result

Integrand size = 34, antiderivative size = 124 \[ \int \frac {1}{\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx=-\frac {\sqrt {\frac {2}{3}} \sqrt {1-6 b} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right ) \sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}} \text {arctanh}\left (\frac {\sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}}}{\sqrt {3}}\right )}{3 \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}} \] Output:

-1/9*6^(1/2)*(1-6*b)^(1/2)*(1-(1-6*x)/(1-6*b)^(1/2))*(2+(1-6*x)/(1-6*b)^(1 
/2))^(1/2)*arctanh(1/3*(2+(1-6*x)/(1-6*b)^(1/2))^(1/2)*3^(1/2))/(-2*(1-6*b 
)^(3/2)+3*(1-6*b)*(1-6*x)-(1-6*x)^3)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

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

Time = 0.42 (sec) , antiderivative size = 934, normalized size of antiderivative = 7.53 \[ \int \frac {1}{\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx =\text {Too large to display} \] Input:

Integrate[1/Sqrt[1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3],x]
 

Output:

(2*EllipticF[ArcSin[Sqrt[(-x + Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6 
*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 3])/(-Root[1 - Sqrt[1 - 6*b] - 9* 
b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 2] + Root[1 - Sqr 
t[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 3] 
)]], (Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 
 + 108*#1^3 & , 2] - Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54 
*b*#1 - 54*#1^2 + 108*#1^3 & , 3])/(Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[ 
1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 1] - Root[1 - Sqrt[1 - 6*b] 
- 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 3])]*(x - Roo 
t[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1 
^3 & , 3])*Sqrt[(-x + Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 5 
4*b*#1 - 54*#1^2 + 108*#1^3 & , 1])/(Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt 
[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 1] - Root[1 - Sqrt[1 - 6*b] 
 - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 3])]*Sqrt[(- 
x + Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 
 108*#1^3 & , 2])/(Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b 
*#1 - 54*#1^2 + 108*#1^3 & , 2] - Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 
- 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 3])])/(Sqrt[1 - Sqrt[1 - 6*b] 
- 54*x^2 + 108*x^3 + b*(-9 + 6*Sqrt[1 - 6*b] + 54*x)]*Sqrt[(x - Root[1 - S 
qrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 &...
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.31, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2480, 27, 73, 217}

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.

\(\displaystyle \int \frac {1}{\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1}} \, dx\)

\(\Big \downarrow \) 2480

\(\displaystyle -\frac {157464 \sqrt {2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \int -\frac {1}{157464 \sqrt {2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}dx}{\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \int \frac {1}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}dx}{\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \int \frac {1}{-3 (1-6 b)^{3/2}+\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)-6 x (1-6 b)}d\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{3 (1-6 b) \sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1}}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \arctan \left (\frac {\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{\sqrt {3} (1-6 b)^{3/4}}\right )}{3 \sqrt {3} (1-6 b)^{7/4} \sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1}}\)

Input:

Int[1/Sqrt[1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3],x]
 

Output:

-1/3*(((1 - Sqrt[1 - 6*b])*(1 - 6*b) - 6*(1 - 6*b)*x)*Sqrt[-((1 + 2*Sqrt[1 
 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x]*ArcTan[Sqrt[-((1 + 2*Sqrt[1 - 6*b])*( 
1 - 6*b)) + 6*(1 - 6*b)*x]/(Sqrt[3]*(1 - 6*b)^(3/4))])/(Sqrt[3]*(1 - 6*b)^ 
(7/4)*Sqrt[1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 2480

Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] 
, c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Simp[Px^p/((c^3 - 4*b*c*d + 9* 
a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3*b*d)*x)^(2*p))   Int 
[(c^3 - 4*b*c*d + 9*a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3* 
b*d)*x)^(2*p), x], x] /; EqQ[b^2*c^2 - 4*a*c^3 - 4*b^3*d + 18*a*b*c*d - 27* 
a^2*d^2, 0] && NeQ[c^2 - 3*b*d, 0]] /; FreeQ[p, x] && PolyQ[Px, x, 3] &&  ! 
IntegerQ[p]
 

Maple [F]

\[\int \frac {1}{\sqrt {1-\left (1-6 b \right )^{\frac {3}{2}}-9 b +54 b x -54 x^{2}+108 x^{3}}}d x\]

Input:

int(1/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x)
 

Output:

int(1/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.97 \[ \int \frac {1}{\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx=\left [\frac {1}{6} \, \sqrt {\frac {2}{3}} \sqrt {\frac {\sqrt {-6 \, b + 1}}{6 \, b - 1}} \log \left (-\frac {108 \, x^{4} - 6 \, {\left (12 \, b - 5\right )} x^{2} - 72 \, x^{3} + \sqrt {\frac {2}{3}} \sqrt {108 \, x^{3} + 54 \, b x - 54 \, x^{2} + {\left (6 \, b - 1\right )} \sqrt {-6 \, b + 1} - 9 \, b + 1} {\left (6 \, {\left (6 \, b - 1\right )} x + {\left (18 \, x^{2} - 3 \, b - 6 \, x + 1\right )} \sqrt {-6 \, b + 1} - 6 \, b + 1\right )} \sqrt {\frac {\sqrt {-6 \, b + 1}}{6 \, b - 1}} - 15 \, b^{2} + 6 \, {\left (4 \, b - 1\right )} x - 3 \, {\left (36 \, x^{3} + 2 \, {\left (3 \, b + 1\right )} x - 18 \, x^{2} - b\right )} \sqrt {-6 \, b + 1} + 3 \, b}{36 \, x^{4} + 4 \, {\left (3 \, b + 1\right )} x^{2} - 24 \, x^{3} + b^{2} - 4 \, b x}\right ), -\frac {1}{3} \, \sqrt {\frac {2}{3}} \sqrt {-\frac {\sqrt {-6 \, b + 1}}{6 \, b - 1}} \arctan \left (-\frac {\sqrt {\frac {2}{3}} \sqrt {108 \, x^{3} + 54 \, b x - 54 \, x^{2} + {\left (6 \, b - 1\right )} \sqrt {-6 \, b + 1} - 9 \, b + 1} {\left (6 \, {\left (6 \, b - 1\right )} x - {\left (36 \, x^{2} + 12 \, b - 12 \, x - 1\right )} \sqrt {-6 \, b + 1} - 6 \, b + 1\right )} \sqrt {-\frac {\sqrt {-6 \, b + 1}}{6 \, b - 1}}}{6 \, {\left (72 \, x^{4} + 2 \, {\left (30 \, b + 1\right )} x^{2} - 48 \, x^{3} + 8 \, b^{2} - 2 \, {\left (10 \, b - 1\right )} x - b\right )}}\right )\right ] \] Input:

integrate(1/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x, algorithm 
="fricas")
 

Output:

[1/6*sqrt(2/3)*sqrt(sqrt(-6*b + 1)/(6*b - 1))*log(-(108*x^4 - 6*(12*b - 5) 
*x^2 - 72*x^3 + sqrt(2/3)*sqrt(108*x^3 + 54*b*x - 54*x^2 + (6*b - 1)*sqrt( 
-6*b + 1) - 9*b + 1)*(6*(6*b - 1)*x + (18*x^2 - 3*b - 6*x + 1)*sqrt(-6*b + 
 1) - 6*b + 1)*sqrt(sqrt(-6*b + 1)/(6*b - 1)) - 15*b^2 + 6*(4*b - 1)*x - 3 
*(36*x^3 + 2*(3*b + 1)*x - 18*x^2 - b)*sqrt(-6*b + 1) + 3*b)/(36*x^4 + 4*( 
3*b + 1)*x^2 - 24*x^3 + b^2 - 4*b*x)), -1/3*sqrt(2/3)*sqrt(-sqrt(-6*b + 1) 
/(6*b - 1))*arctan(-1/6*sqrt(2/3)*sqrt(108*x^3 + 54*b*x - 54*x^2 + (6*b - 
1)*sqrt(-6*b + 1) - 9*b + 1)*(6*(6*b - 1)*x - (36*x^2 + 12*b - 12*x - 1)*s 
qrt(-6*b + 1) - 6*b + 1)*sqrt(-sqrt(-6*b + 1)/(6*b - 1))/(72*x^4 + 2*(30*b 
 + 1)*x^2 - 48*x^3 + 8*b^2 - 2*(10*b - 1)*x - b))]
 

Sympy [F]

\[ \int \frac {1}{\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx=\int \frac {1}{\sqrt {54 b x - 9 b + 108 x^{3} - 54 x^{2} - \left (1 - 6 b\right )^{\frac {3}{2}} + 1}}\, dx \] Input:

integrate(1/(1-(1-6*b)**(3/2)-9*b+54*b*x-54*x**2+108*x**3)**(1/2),x)
 

Output:

Integral(1/sqrt(54*b*x - 9*b + 108*x**3 - 54*x**2 - (1 - 6*b)**(3/2) + 1), 
 x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx=\int { \frac {1}{\sqrt {108 \, x^{3} + 54 \, b x - 54 \, x^{2} - {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 9 \, b + 1}} \,d x } \] Input:

integrate(1/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x, algorithm 
="maxima")
 

Output:

integrate(1/sqrt(108*x^3 + 54*b*x - 54*x^2 - (-6*b + 1)^(3/2) - 9*b + 1), 
x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.48 \[ \int \frac {1}{\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx=\frac {\sqrt {3} \sqrt {2} \arctan \left (\frac {\sqrt {3} \sqrt {6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1}}{3 \, {\left (-6 \, b + 1\right )}^{\frac {1}{4}}}\right )}{9 \, {\left (-6 \, b + 1\right )}^{\frac {1}{4}} \mathrm {sgn}\left (6 \, x + \sqrt {-6 \, b + 1} - 1\right )} \] Input:

integrate(1/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x, algorithm 
="giac")
 

Output:

1/9*sqrt(3)*sqrt(2)*arctan(1/3*sqrt(3)*sqrt(6*x - 2*sqrt(-6*b + 1) - 1)/(- 
6*b + 1)^(1/4))/((-6*b + 1)^(1/4)*sgn(6*x + sqrt(-6*b + 1) - 1))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx=\int \frac {1}{\sqrt {54\,b\,x-9\,b-{\left (1-6\,b\right )}^{3/2}-54\,x^2+108\,x^3+1}} \,d x \] Input:

int(1/(54*b*x - 9*b - (1 - 6*b)^(3/2) - 54*x^2 + 108*x^3 + 1)^(1/2),x)
 

Output:

int(1/(54*b*x - 9*b - (1 - 6*b)^(3/2) - 54*x^2 + 108*x^3 + 1)^(1/2), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx=\text {too large to display} \] Input:

int(1/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x)
 

Output:

(282175488*sqrt( - 6*b + 1)*int(sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 
1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)/(23079168*sqrt( - 6*b + 1)*b** 
8 + 311568768*sqrt( - 6*b + 1)*b**7*x**2 - 103856256*sqrt( - 6*b + 1)*b**7 
*x - 4588992*sqrt( - 6*b + 1)*b**7 + 1246275072*sqrt( - 6*b + 1)*b**6*x**4 
 - 830850048*sqrt( - 6*b + 1)*b**6*x**3 + 80850960*sqrt( - 6*b + 1)*b**6*x 
**2 + 19208016*sqrt( - 6*b + 1)*b**6*x - 2027100*sqrt( - 6*b + 1)*b**6 + 1 
246275072*sqrt( - 6*b + 1)*b**5*x**6 - 1246275072*sqrt( - 6*b + 1)*b**5*x* 
*5 + 219547584*sqrt( - 6*b + 1)*b**5*x**4 + 84426624*sqrt( - 6*b + 1)*b**5 
*x**3 - 49449528*sqrt( - 6*b + 1)*b**5*x**2 + 9228456*sqrt( - 6*b + 1)*b** 
5*x + 814462*sqrt( - 6*b + 1)*b**5 - 92021184*sqrt( - 6*b + 1)*b**4*x**6 + 
 92021184*sqrt( - 6*b + 1)*b**4*x**5 - 143971344*sqrt( - 6*b + 1)*b**4*x** 
4 + 78939936*sqrt( - 6*b + 1)*b**4*x**3 - 2013408*sqrt( - 6*b + 1)*b**4*x* 
*2 - 3525072*sqrt( - 6*b + 1)*b**4*x - 111254*sqrt( - 6*b + 1)*b**4 - 1209 
66048*sqrt( - 6*b + 1)*b**3*x**6 + 120966048*sqrt( - 6*b + 1)*b**3*x**5 - 
1381320*sqrt( - 6*b + 1)*b**3*x**4 - 21480240*sqrt( - 6*b + 1)*b**3*x**3 + 
 2925024*sqrt( - 6*b + 1)*b**3*x**2 + 467240*sqrt( - 6*b + 1)*b**3*x + 683 
6*sqrt( - 6*b + 1)*b**3 + 28860192*sqrt( - 6*b + 1)*b**2*x**6 - 28860192*s 
qrt( - 6*b + 1)*b**2*x**5 + 4814856*sqrt( - 6*b + 1)*b**2*x**4 + 2134576*s 
qrt( - 6*b + 1)*b**2*x**3 - 449960*sqrt( - 6*b + 1)*b**2*x**2 - 27984*sqrt 
( - 6*b + 1)*b**2*x - 160*sqrt( - 6*b + 1)*b**2 - 2400192*sqrt( - 6*b +...