Integrand size = 23, antiderivative size = 123 \[ \int \frac {1}{\sqrt [3]{2-3 x^2} \left (-6 d+d x^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{2-3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{2-3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d} \] Output:
-1/8*arctan(2^(1/6)*(2^(1/3)-(-3*x^2+2)^(1/3))/x)*2^(1/6)/d-1/24*arctanh(1 /6*x*6^(1/2))*2^(1/6)*3^(1/2)/d+1/24*arctanh(1/18*(2^(1/3)-(-3*x^2+2)^(1/3 ))^2*2^(5/6)*3^(1/2)/x)*2^(1/6)*3^(1/2)/d
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 4.52 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt [3]{2-3 x^2} \left (-6 d+d x^2\right )} \, dx=\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )}{d \sqrt [3]{2-3 x^2} \left (-6+x^2\right ) \left (9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )+x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )\right )\right )} \] Input:
Integrate[1/((2 - 3*x^2)^(1/3)*(-6*d + d*x^2)),x]
Output:
(9*x*AppellF1[1/2, 1/3, 1, 3/2, (3*x^2)/2, x^2/6])/(d*(2 - 3*x^2)^(1/3)*(- 6 + x^2)*(9*AppellF1[1/2, 1/3, 1, 3/2, (3*x^2)/2, x^2/6] + x^2*(AppellF1[3 /2, 1/3, 2, 5/2, (3*x^2)/2, x^2/6] + 3*AppellF1[3/2, 4/3, 1, 5/2, (3*x^2)/ 2, x^2/6])))
Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {307}
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.
\(\displaystyle \int \frac {1}{\sqrt [3]{2-3 x^2} \left (d x^2-6 d\right )} \, dx\) |
\(\Big \downarrow \) 307 |
\(\displaystyle -\frac {\arctan \left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{2-3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{2-3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}\) |
Input:
Int[1/((2 - 3*x^2)^(1/3)*(-6*d + d*x^2)),x]
Output:
-1/4*ArcTan[(2^(1/6)*(2^(1/3) - (2 - 3*x^2)^(1/3)))/x]/(2^(5/6)*d) - ArcTa nh[x/Sqrt[6]]/(4*2^(5/6)*Sqrt[3]*d) + ArcTanh[(2^(1/3) - (2 - 3*x^2)^(1/3) )^2/(3*2^(1/6)*Sqrt[3]*x)]/(4*2^(5/6)*Sqrt[3]*d)
Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[(-q)*(ArcTanh[q*(x/3)]/(12*Rt[a, 3]*d)), x] + (Si mp[q*(ArcTanh[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)]/(12*Rt[a , 3]*d)), x] - Simp[q*(ArcTan[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[ a, 3]*q*x)]/(4*Sqrt[3]*Rt[a, 3]*d)), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[ b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && NegQ[b/a]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 216.23 (sec) , antiderivative size = 724, normalized size of antiderivative = 5.89
Input:
int(1/(-3*x^2+2)^(1/3)/(d*x^2-6*d),x,method=_RETURNVERBOSE)
Output:
-1/24*(ln(-(4*RootOf(_Z^6-54)^7*x+192*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootO f(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^6*x+6*RootOf(_Z^6-54)^5*(-3*x^2+2)^(1 /3)*x+288*RootOf(_Z^6-54)^4*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54) +576*_Z^2)*(-3*x^2+2)^(1/3)*x+9*x^2*RootOf(_Z^6-54)^4+18*RootOf(_Z^6-54)^4 -108*(-3*x^2+2)^(1/3)*RootOf(_Z^6-54)^2-324*(-3*x^2+2)^(2/3))/(x^2-6))*Roo tOf(_Z^6-54)+RootOf(_Z^6-54)*ln((4*RootOf(_Z^6-54)^7*x+288*RootOf(RootOf(_ Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^6*x+4608*RootOf( RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)^2*RootOf(_Z^6-54)^5*x-14 4*RootOf(_Z^6-54)^4*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^ 2)*(-3*x^2+2)^(1/3)*x-6912*(-3*x^2+2)^(1/3)*RootOf(_Z^6-54)^3*RootOf(RootO f(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)^2*x+9*x^2*RootOf(_Z^6-54)^4+2 16*RootOf(_Z^6-54)^3*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z ^2)*x^2+18*RootOf(_Z^6-54)^4+432*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^ 6-54)+576*_Z^2)*RootOf(_Z^6-54)^3+2592*(-3*x^2+2)^(1/3)*RootOf(_Z^6-54)*Ro otOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)+324*(-3*x^2+2)^(2/3 ))/(x^2-6))+24*ln(-(4*RootOf(_Z^6-54)^7*x+192*RootOf(RootOf(_Z^6-54)^2+24* _Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^6*x+6*RootOf(_Z^6-54)^5*(-3*x ^2+2)^(1/3)*x+288*RootOf(_Z^6-54)^4*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf( _Z^6-54)+576*_Z^2)*(-3*x^2+2)^(1/3)*x+9*x^2*RootOf(_Z^6-54)^4+18*RootOf(_Z ^6-54)^4-108*(-3*x^2+2)^(1/3)*RootOf(_Z^6-54)^2-324*(-3*x^2+2)^(2/3))/(...
Timed out. \[ \int \frac {1}{\sqrt [3]{2-3 x^2} \left (-6 d+d x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(-3*x^2+2)^(1/3)/(d*x^2-6*d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\sqrt [3]{2-3 x^2} \left (-6 d+d x^2\right )} \, dx=\frac {\int \frac {1}{x^{2} \sqrt [3]{2 - 3 x^{2}} - 6 \sqrt [3]{2 - 3 x^{2}}}\, dx}{d} \] Input:
integrate(1/(-3*x**2+2)**(1/3)/(d*x**2-6*d),x)
Output:
Integral(1/(x**2*(2 - 3*x**2)**(1/3) - 6*(2 - 3*x**2)**(1/3)), x)/d
\[ \int \frac {1}{\sqrt [3]{2-3 x^2} \left (-6 d+d x^2\right )} \, dx=\int { \frac {1}{{\left (d x^{2} - 6 \, d\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(-3*x^2+2)^(1/3)/(d*x^2-6*d),x, algorithm="maxima")
Output:
integrate(1/((d*x^2 - 6*d)*(-3*x^2 + 2)^(1/3)), x)
\[ \int \frac {1}{\sqrt [3]{2-3 x^2} \left (-6 d+d x^2\right )} \, dx=\int { \frac {1}{{\left (d x^{2} - 6 \, d\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(-3*x^2+2)^(1/3)/(d*x^2-6*d),x, algorithm="giac")
Output:
integrate(1/((d*x^2 - 6*d)*(-3*x^2 + 2)^(1/3)), x)
Timed out. \[ \int \frac {1}{\sqrt [3]{2-3 x^2} \left (-6 d+d x^2\right )} \, dx=-\int \frac {1}{{\left (2-3\,x^2\right )}^{1/3}\,\left (6\,d-d\,x^2\right )} \,d x \] Input:
int(-1/((2 - 3*x^2)^(1/3)*(6*d - d*x^2)),x)
Output:
-int(1/((2 - 3*x^2)^(1/3)*(6*d - d*x^2)), x)
\[ \int \frac {1}{\sqrt [3]{2-3 x^2} \left (-6 d+d x^2\right )} \, dx=\frac {\int \frac {1}{\left (-3 x^{2}+2\right )^{\frac {1}{3}} x^{2}-6 \left (-3 x^{2}+2\right )^{\frac {1}{3}}}d x}{d} \] Input:
int(1/(-3*x^2+2)^(1/3)/(d*x^2-6*d),x)
Output:
int(1/(( - 3*x**2 + 2)**(1/3)*x**2 - 6*( - 3*x**2 + 2)**(1/3)),x)/d