Integrand size = 32, antiderivative size = 88 \[ \int \frac {1}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=-\frac {1}{9 \sqrt {1-6 b} \left (1-\sqrt {1-6 b}-6 x\right )}-\frac {\log \left (1-\sqrt {1-6 b}-6 x\right )}{27 (1-6 b)}+\frac {\log \left (1+2 \sqrt {1-6 b}-6 x\right )}{27 (1-6 b)} \] Output:
-1/9/(1-6*b)^(1/2)/(1-(1-6*b)^(1/2)-6*x)-ln(1-(1-6*b)^(1/2)-6*x)/(27-162*b )+ln(1+2*(1-6*b)^(1/2)-6*x)/(27-162*b)
Time = 0.39 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.90 \[ \int \frac {1}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\frac {1}{54} \left (\frac {1+\sqrt {1-6 b}-6 b-6 \sqrt {1-6 b} x}{(-1+6 b) (b+2 x (-1+3 x))}-\frac {2 \sqrt {1-6 b} \arctan \left (\frac {-1+6 x}{2 \sqrt {-1+6 b}}\right )}{(-1+6 b)^{3/2}}-\frac {2 \sqrt {1-6 b} \arctan \left (\frac {-1+6 x}{\sqrt {-1+6 b}}\right )}{(-1+6 b)^{3/2}}+\frac {\log \left (1-8 b+4 x-12 x^2\right )}{1-6 b}+\frac {\log \left (b-2 x+6 x^2\right )}{-1+6 b}\right ) \] Input:
Integrate[(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3)^(-1),x]
Output:
((1 + Sqrt[1 - 6*b] - 6*b - 6*Sqrt[1 - 6*b]*x)/((-1 + 6*b)*(b + 2*x*(-1 + 3*x))) - (2*Sqrt[1 - 6*b]*ArcTan[(-1 + 6*x)/(2*Sqrt[-1 + 6*b])])/(-1 + 6*b )^(3/2) - (2*Sqrt[1 - 6*b]*ArcTan[(-1 + 6*x)/Sqrt[-1 + 6*b]])/(-1 + 6*b)^( 3/2) + Log[1 - 8*b + 4*x - 12*x^2]/(1 - 6*b) + Log[b - 2*x + 6*x^2]/(-1 + 6*b))/54
Time = 0.52 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2479, 27, 27, 54, 2009}
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.
\(\displaystyle \int \frac {1}{54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \, dx\) |
\(\Big \downarrow \) 2479 |
\(\displaystyle 99179645184 (1-6 b)^3 \int -\frac {1}{49589822592 \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \left (\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)-6 (1-6 b) x\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -2 (1-6 b)^3 \int \frac {1}{(1-6 b) \left (-6 x+2 \sqrt {1-6 b}+1\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -2 (1-6 b)^2 \int \frac {1}{\left (-6 x+2 \sqrt {1-6 b}+1\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2}dx\) |
\(\Big \downarrow \) 54 |
\(\displaystyle -2 (1-6 b)^2 \int \left (-\frac {1}{9 (6 b-1)^3 \left (-6 x+2 \sqrt {1-6 b}+1\right )}-\frac {1}{9 (6 b-1)^3 \left (6 x+\sqrt {1-6 b}-1\right )}+\frac {1}{3 (6 b-1)^2 \left (6 x+\sqrt {1-6 b}-1\right )^2 \sqrt {1-6 b}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 (1-6 b)^2 \left (\frac {1}{18 (1-6 b)^{5/2} \left (-\sqrt {1-6 b}-6 x+1\right )}+\frac {\log \left (-\sqrt {1-6 b}-6 x+1\right )}{54 (1-6 b)^3}-\frac {\log \left (2 \sqrt {1-6 b}-6 x+1\right )}{54 (1-6 b)^3}\right )\) |
Input:
Int[(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3)^(-1),x]
Output:
-2*(1 - 6*b)^2*(1/(18*(1 - 6*b)^(5/2)*(1 - Sqrt[1 - 6*b] - 6*x)) + Log[1 - Sqrt[1 - 6*b] - 6*x]/(54*(1 - 6*b)^3) - Log[1 + 2*Sqrt[1 - 6*b] - 6*x]/(5 4*(1 - 6*b)^3))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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[1/(4^p*(c^2 - 3*b*d)^(3*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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.42 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64
method | result | size |
default | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-\left (1-6 b \right )^{\frac {3}{2}}+108 \textit {\_Z}^{3}+54 b \textit {\_Z} -54 \textit {\_Z}^{2}-9 b +1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{-6 \textit {\_R}^{2}+2 \textit {\_R} -b}\right )}{54}\) | \(56\) |
parallelrisch | \(\frac {\sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right )+6 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x -\sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right )-6 \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x -\ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right )+\ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right )-3 \sqrt {1-6 b}}{27 \left (-1+6 b \right ) \left (-1+6 x +\sqrt {1-6 b}\right )}\) | \(137\) |
Input:
int(1/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x,method=_RETURNVERBOSE)
Output:
-1/54*sum(1/(-6*_R^2+2*_R-b)*ln(x-_R),_R=RootOf(-(1-6*b)^(3/2)+108*_Z^3+54 *b*_Z-54*_Z^2-9*b+1))
Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.16 \[ \int \frac {1}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\frac {2 \, {\left (6 \, x^{2} + b - 2 \, x\right )} \log \left (6 \, x + \sqrt {-6 \, b + 1} - 1\right ) - 2 \, {\left (6 \, x^{2} + b - 2 \, x\right )} \log \left (6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1\right ) - \sqrt {-6 \, b + 1} {\left (6 \, x - 1\right )} - 6 \, b + 1}{54 \, {\left (6 \, {\left (6 \, b - 1\right )} x^{2} + 6 \, b^{2} - 2 \, {\left (6 \, b - 1\right )} x - b\right )}} \] Input:
integrate(1/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, algorithm="fric as")
Output:
1/54*(2*(6*x^2 + b - 2*x)*log(6*x + sqrt(-6*b + 1) - 1) - 2*(6*x^2 + b - 2 *x)*log(6*x - 2*sqrt(-6*b + 1) - 1) - sqrt(-6*b + 1)*(6*x - 1) - 6*b + 1)/ (6*(6*b - 1)*x^2 + 6*b^2 - 2*(6*b - 1)*x - b)
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (66) = 132\).
Time = 0.60 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.09 \[ \int \frac {1}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\frac {6}{- 324 b + 324 x \sqrt {1 - 6 b} - 54 \sqrt {1 - 6 b} + 54} - \frac {\log {\left (\frac {b}{2 \sqrt {1 - 6 b}} + x - \frac {1}{6} - \frac {- \frac {243 b^{2}}{\sqrt {1 - 6 b}} + \frac {81 b}{\sqrt {1 - 6 b}} - \frac {27}{4 \sqrt {1 - 6 b}}}{27 \cdot \left (6 b - 1\right )} - \frac {1}{12 \sqrt {1 - 6 b}} \right )}}{27 \cdot \left (6 b - 1\right )} + \frac {\log {\left (\frac {b}{2 \sqrt {1 - 6 b}} + x - \frac {1}{6} + \frac {- \frac {243 b^{2}}{\sqrt {1 - 6 b}} + \frac {81 b}{\sqrt {1 - 6 b}} - \frac {27}{4 \sqrt {1 - 6 b}}}{27 \cdot \left (6 b - 1\right )} - \frac {1}{12 \sqrt {1 - 6 b}} \right )}}{27 \cdot \left (6 b - 1\right )} \] Input:
integrate(1/(1-(1-6*b)**(3/2)-9*b+54*b*x-54*x**2+108*x**3),x)
Output:
6/(-324*b + 324*x*sqrt(1 - 6*b) - 54*sqrt(1 - 6*b) + 54) - log(b/(2*sqrt(1 - 6*b)) + x - 1/6 - (-243*b**2/sqrt(1 - 6*b) + 81*b/sqrt(1 - 6*b) - 27/(4 *sqrt(1 - 6*b)))/(27*(6*b - 1)) - 1/(12*sqrt(1 - 6*b)))/(27*(6*b - 1)) + l og(b/(2*sqrt(1 - 6*b)) + x - 1/6 + (-243*b**2/sqrt(1 - 6*b) + 81*b/sqrt(1 - 6*b) - 27/(4*sqrt(1 - 6*b)))/(27*(6*b - 1)) - 1/(12*sqrt(1 - 6*b)))/(27* (6*b - 1))
\[ \int \frac {1}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\int { \frac {1}{108 \, x^{3} + 54 \, b x - 54 \, x^{2} - {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 9 \, b + 1} \,d x } \] Input:
integrate(1/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, algorithm="maxi ma")
Output:
integrate(1/(108*x^3 + 54*b*x - 54*x^2 - (-6*b + 1)^(3/2) - 9*b + 1), x)
Exception generated. \[ \int \frac {1}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, algorithm="giac ")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{-972,[1]%%%}+%%%{162,[0]%%%},[2]%%%}+%%%{%%{[%%%{-324, [1]%%%}+%
Time = 0.16 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.50 \[ \int \frac {1}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=-\frac {{\left (1-6\,b\right )}^{3/2}}{1944\,\left (\frac {{\left (1-6\,b\right )}^{3/2}}{36\,\left (b-\frac {1}{6}\right )}-x+\frac {1}{6}\right )\,\left (b^2-\frac {b}{3}+\frac {1}{36}\right )}-\frac {\mathrm {atan}\left (\frac {-b\,12{}\mathrm {i}-x\,12{}\mathrm {i}+b\,x\,72{}\mathrm {i}+{\left (1-6\,b\right )}^{3/2}\,1{}\mathrm {i}+2{}\mathrm {i}}{\sqrt {864\,b^2-{\left (6\,b-1\right )}^3-144\,b-1728\,b^3+8}}\right )\,\sqrt {1-6\,b}\,2{}\mathrm {i}}{9\,\sqrt {864\,b^2-{\left (6\,b-1\right )}^3-144\,b-1728\,b^3+8}} \] Input:
int(-1/(9*b - 54*b*x + (1 - 6*b)^(3/2) + 54*x^2 - 108*x^3 - 1),x)
Output:
- (1 - 6*b)^(3/2)/(1944*((1 - 6*b)^(3/2)/(36*(b - 1/6)) - x + 1/6)*(b^2 - b/3 + 1/36)) - (atan((b*x*72i - x*12i - b*12i + (1 - 6*b)^(3/2)*1i + 2i)/( 864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2))*(1 - 6*b)^(1/2)*2i)/( 9*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2))
Time = 0.15 (sec) , antiderivative size = 498, normalized size of antiderivative = 5.66 \[ \int \frac {1}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\frac {-1+12 b -108 \sqrt {-6 b +1}\, b \,x^{2}+\mathrm {log}\left (12 x^{2}+8 b -4 x -1\right ) b -\mathrm {log}\left (6 x^{2}+b -2 x \right ) b -6 \,\mathrm {log}\left (12 x^{2}+8 b -4 x -1\right ) b^{2}+6 \,\mathrm {log}\left (12 x^{2}+8 b -4 x -1\right ) x^{2}-2 \,\mathrm {log}\left (12 x^{2}+8 b -4 x -1\right ) x +6 \,\mathrm {log}\left (6 x^{2}+b -2 x \right ) b^{2}-6 \,\mathrm {log}\left (6 x^{2}+b -2 x \right ) x^{2}+2 \,\mathrm {log}\left (6 x^{2}+b -2 x \right ) x -18 \sqrt {-6 b +1}\, b^{2}+9 \sqrt {-6 b +1}\, b +18 \sqrt {-6 b +1}\, x^{2}-36 \,\mathrm {log}\left (12 x^{2}+8 b -4 x -1\right ) b \,x^{2}+12 \,\mathrm {log}\left (12 x^{2}+8 b -4 x -1\right ) b x +36 \,\mathrm {log}\left (6 x^{2}+b -2 x \right ) b \,x^{2}-12 \,\mathrm {log}\left (6 x^{2}+b -2 x \right ) b x -\sqrt {-6 b +1}-2 \sqrt {6 b -1}\, \sqrt {-6 b +1}\, \mathit {atan} \left (\frac {6 x -1}{\sqrt {6 b -1}}\right ) b -12 \sqrt {6 b -1}\, \sqrt {-6 b +1}\, \mathit {atan} \left (\frac {6 x -1}{\sqrt {6 b -1}}\right ) x^{2}+4 \sqrt {6 b -1}\, \sqrt {-6 b +1}\, \mathit {atan} \left (\frac {6 x -1}{\sqrt {6 b -1}}\right ) x -2 \sqrt {6 b -1}\, \sqrt {-6 b +1}\, \mathit {atan} \left (\frac {6 x -1}{2 \sqrt {6 b -1}}\right ) b -12 \sqrt {6 b -1}\, \sqrt {-6 b +1}\, \mathit {atan} \left (\frac {6 x -1}{2 \sqrt {6 b -1}}\right ) x^{2}+4 \sqrt {6 b -1}\, \sqrt {-6 b +1}\, \mathit {atan} \left (\frac {6 x -1}{2 \sqrt {6 b -1}}\right ) x -36 b^{2}}{11664 b^{2} x^{2}+1944 b^{3}-3888 b^{2} x -3888 b \,x^{2}-648 b^{2}+1296 b x +324 x^{2}+54 b -108 x} \] Input:
int(1/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x)
Output:
( - 2*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b - 12* sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*x**2 + 4*sqrt (6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*x - 2*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/(2*sqrt(6*b - 1)))*b - 12*sqrt(6*b - 1) *sqrt( - 6*b + 1)*atan((6*x - 1)/(2*sqrt(6*b - 1)))*x**2 + 4*sqrt(6*b - 1) *sqrt( - 6*b + 1)*atan((6*x - 1)/(2*sqrt(6*b - 1)))*x - 18*sqrt( - 6*b + 1 )*b**2 - 108*sqrt( - 6*b + 1)*b*x**2 + 9*sqrt( - 6*b + 1)*b + 18*sqrt( - 6 *b + 1)*x**2 - sqrt( - 6*b + 1) - 6*log(8*b + 12*x**2 - 4*x - 1)*b**2 - 36 *log(8*b + 12*x**2 - 4*x - 1)*b*x**2 + 12*log(8*b + 12*x**2 - 4*x - 1)*b*x + log(8*b + 12*x**2 - 4*x - 1)*b + 6*log(8*b + 12*x**2 - 4*x - 1)*x**2 - 2*log(8*b + 12*x**2 - 4*x - 1)*x + 6*log(b + 6*x**2 - 2*x)*b**2 + 36*log(b + 6*x**2 - 2*x)*b*x**2 - 12*log(b + 6*x**2 - 2*x)*b*x - log(b + 6*x**2 - 2*x)*b - 6*log(b + 6*x**2 - 2*x)*x**2 + 2*log(b + 6*x**2 - 2*x)*x - 36*b** 2 + 12*b - 1)/(54*(36*b**3 + 216*b**2*x**2 - 72*b**2*x - 12*b**2 - 72*b*x* *2 + 24*b*x + b + 6*x**2 - 2*x))