Integrand size = 27, antiderivative size = 80 \[ \int \frac {1}{\sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}} \, dx=-\frac {2 (2 c+3 d x) \sqrt {1-\frac {3 d x}{c}} \text {arctanh}\left (\frac {\sqrt {1-\frac {3 d x}{c}}}{\sqrt {3}}\right )}{3 \sqrt {3} d \sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}} \] Output:
-2/9*(3*d*x+2*c)*(1-3*d*x/c)^(1/2)*arctanh(1/3*(1-3*d*x/c)^(1/2)*3^(1/2))* 3^(1/2)/d/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(1/2)
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}} \, dx=-\frac {2 \sqrt {c-3 d x} (2 c+3 d x) \text {arctanh}\left (\frac {\sqrt {c-3 d x}}{\sqrt {3} \sqrt {c}}\right )}{3 \sqrt {3} \sqrt {c} d \sqrt {(c-3 d x) (2 c+3 d x)^2}} \] Input:
Integrate[1/Sqrt[4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3],x]
Output:
(-2*Sqrt[c - 3*d*x]*(2*c + 3*d*x)*ArcTanh[Sqrt[c - 3*d*x]/(Sqrt[3]*Sqrt[c] )])/(3*Sqrt[3]*Sqrt[c]*d*Sqrt[(c - 3*d*x)*(2*c + 3*d*x)^2])
Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.36, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2480, 27, 73, 221}
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.
\(\displaystyle \int \frac {1}{\sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}} \, dx\) |
\(\Big \downarrow \) 2480 |
\(\displaystyle \frac {39366 c^2 d^3 (2 c+3 d x) \sqrt {c^3 d^6-3 c^2 d^7 x} \int \frac {1}{39366 c^2 d^3 (2 c+3 d x) \sqrt {c^3 d^6-3 c^2 d^7 x}}dx}{\sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(2 c+3 d x) \sqrt {c^3 d^6-3 c^2 d^7 x} \int \frac {1}{(2 c+3 d x) \sqrt {c^3 d^6-3 c^2 d^7 x}}dx}{\sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {2 (2 c+3 d x) \sqrt {c^3 d^6-3 c^2 d^7 x} \int \frac {1}{3 c-\frac {c^3 d^6-3 c^2 d^7 x}{c^2 d^6}}d\sqrt {c^3 d^6-3 c^2 d^7 x}}{3 c^2 d^7 \sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 (2 c+3 d x) \sqrt {c^3 d^6-3 c^2 d^7 x} \text {arctanh}\left (\frac {\sqrt {c^3 d^6-3 c^2 d^7 x}}{\sqrt {3} c^{3/2} d^3}\right )}{3 \sqrt {3} c^{3/2} d^4 \sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}}\) |
Input:
Int[1/Sqrt[4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3],x]
Output:
(-2*(2*c + 3*d*x)*Sqrt[c^3*d^6 - 3*c^2*d^7*x]*ArcTanh[Sqrt[c^3*d^6 - 3*c^2 *d^7*x]/(Sqrt[3]*c^(3/2)*d^3)])/(3*Sqrt[3]*c^(3/2)*d^4*Sqrt[4*c^3 - 27*c*d ^2*x^2 - 27*d^3*x^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] :> 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]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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]
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {2 \left (3 d x +2 c \right ) \sqrt {-3 d x +c}\, \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 d x +c}\, \sqrt {3}}{3 \sqrt {c}}\right )}{9 \sqrt {-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}}\, d \sqrt {c}}\) | \(70\) |
Input:
int(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/9/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(1/2)*(3*d*x+2*c)*(-3*d*x+c)^(1/2)/d *3^(1/2)/c^(1/2)*arctanh(1/3*(-3*d*x+c)^(1/2)*3^(1/2)/c^(1/2))
Time = 0.11 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.99 \[ \int \frac {1}{\sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}} \, dx=\left [\frac {\sqrt {3} \log \left (\frac {9 \, d^{2} x^{2} - 6 \, c d x - 8 \, c^{2} + 2 \, \sqrt {3} \sqrt {-27 \, d^{3} x^{3} - 27 \, c d^{2} x^{2} + 4 \, c^{3}} \sqrt {c}}{9 \, d^{2} x^{2} + 12 \, c d x + 4 \, c^{2}}\right )}{9 \, \sqrt {c} d}, -\frac {2 \, \sqrt {3} \sqrt {-c} \arctan \left (\frac {\sqrt {3} \sqrt {-27 \, d^{3} x^{3} - 27 \, c d^{2} x^{2} + 4 \, c^{3}} \sqrt {-c}}{9 \, d^{2} x^{2} + 3 \, c d x - 2 \, c^{2}}\right )}{9 \, c d}\right ] \] Input:
integrate(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(1/2),x, algorithm="fricas")
Output:
[1/9*sqrt(3)*log((9*d^2*x^2 - 6*c*d*x - 8*c^2 + 2*sqrt(3)*sqrt(-27*d^3*x^3 - 27*c*d^2*x^2 + 4*c^3)*sqrt(c))/(9*d^2*x^2 + 12*c*d*x + 4*c^2))/(sqrt(c) *d), -2/9*sqrt(3)*sqrt(-c)*arctan(sqrt(3)*sqrt(-27*d^3*x^3 - 27*c*d^2*x^2 + 4*c^3)*sqrt(-c)/(9*d^2*x^2 + 3*c*d*x - 2*c^2))/(c*d)]
\[ \int \frac {1}{\sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}} \, dx=\int \frac {1}{\sqrt {4 c^{3} - 27 c d^{2} x^{2} - 27 d^{3} x^{3}}}\, dx \] Input:
integrate(1/(-27*d**3*x**3-27*c*d**2*x**2+4*c**3)**(1/2),x)
Output:
Integral(1/sqrt(4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3), x)
\[ \int \frac {1}{\sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}} \, dx=\int { \frac {1}{\sqrt {-27 \, d^{3} x^{3} - 27 \, c d^{2} x^{2} + 4 \, c^{3}}} \,d x } \] Input:
integrate(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(-27*d^3*x^3 - 27*c*d^2*x^2 + 4*c^3), x)
Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.54 \[ \int \frac {1}{\sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}} \, dx=-\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3 \, d x + c}}{3 \, \sqrt {-c}}\right )}{9 \, \sqrt {-c} d \mathrm {sgn}\left (-3 \, d x - 2 \, c\right )} \] Input:
integrate(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(1/2),x, algorithm="giac")
Output:
-2/9*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(-3*d*x + c)/sqrt(-c))/(sqrt(-c)*d*sgn (-3*d*x - 2*c))
Timed out. \[ \int \frac {1}{\sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}} \, dx=\int \frac {1}{\sqrt {4\,c^3-27\,c\,d^2\,x^2-27\,d^3\,x^3}} \,d x \] Input:
int(1/(4*c^3 - 27*d^3*x^3 - 27*c*d^2*x^2)^(1/2),x)
Output:
int(1/(4*c^3 - 27*d^3*x^3 - 27*c*d^2*x^2)^(1/2), x)
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\sqrt {4 c^3-27 c d^2 x^2-27 d^3 x^3}} \, dx=\frac {\sqrt {c}\, \sqrt {3}\, \left (\mathrm {log}\left (\sqrt {-3 d x +c}-\sqrt {c}\, \sqrt {3}\right )-\mathrm {log}\left (\sqrt {-3 d x +c}+\sqrt {c}\, \sqrt {3}\right )\right )}{9 c d} \] Input:
int(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(1/2),x)
Output:
(sqrt(c)*sqrt(3)*(log(sqrt(c - 3*d*x) - sqrt(c)*sqrt(3)) - log(sqrt(c - 3* d*x) + sqrt(c)*sqrt(3))))/(9*c*d)