Integrand size = 23, antiderivative size = 119 \[ \int \frac {1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}+\sqrt [3]{-2+3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{2}+\sqrt [3]{-2+3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d} \] 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 = 4.62 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx=\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )}{d \left (-6+x^2\right ) \sqrt [3]{-2+3 x^2} \left (9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )+x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )\right )\right )} \] 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*(-6 + x^2)*(-2 + 3*x ^2)^(1/3)*(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 = 0.18 (sec) , antiderivative size = 119, 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.
\(\displaystyle \int \frac {1}{\sqrt [3]{3 x^2-2} \left (d x^2-6 d\right )} \, dx\) |
\(\Big \downarrow \) 307 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt [6]{2} \left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )}{x}\right )}{4\ 2^{5/6} d}-\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}\) |
Input:
Int[1/((-2 + 3*x^2)^(1/3)*(-6*d + d*x^2)),x]
Output:
ArcTan[(2^(1/6)*(2^(1/3) + (-2 + 3*x^2)^(1/3)))/x]/(4*2^(5/6)*d) + ArcTanh [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 = 153.56 (sec) , antiderivative size = 1063, normalized size of antiderivative = 8.93
Input:
int(1/(3*x^2-2)^(1/3)/(d*x^2-6*d),x,method=_RETURNVERBOSE)
Output:
-1/24*(ln((-16*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*Ro otOf(_Z^6-54)^6*x-768*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_ Z^2)^2*RootOf(_Z^6-54)^5*x-RootOf(_Z^6-54)^5*(3*x^2-2)^(1/3)*x-72*RootOf(R ootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^4*(3*x^2- 2)^(1/3)*x-1152*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)^2 *RootOf(_Z^6-54)^3*(3*x^2-2)^(1/3)*x-36*RootOf(_Z^6-54)^3*RootOf(RootOf(_Z ^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*x^2-72*RootOf(RootOf(_Z^6-54)^2+2 4*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^3+18*RootOf(_Z^6-54)^2*(3*x ^2-2)^(1/3)+432*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*R ootOf(_Z^6-54)*(3*x^2-2)^(1/3)+54*(3*x^2-2)^(2/3))/(x^2-6))*RootOf(_Z^6-54 )+24*ln((-16*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*Root Of(_Z^6-54)^6*x-768*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^ 2)^2*RootOf(_Z^6-54)^5*x-RootOf(_Z^6-54)^5*(3*x^2-2)^(1/3)*x-72*RootOf(Roo tOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^4*(3*x^2-2) ^(1/3)*x-1152*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)^2*R ootOf(_Z^6-54)^3*(3*x^2-2)^(1/3)*x-36*RootOf(_Z^6-54)^3*RootOf(RootOf(_Z^6 -54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*x^2-72*RootOf(RootOf(_Z^6-54)^2+24* _Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^3+18*RootOf(_Z^6-54)^2*(3*x^2 -2)^(1/3)+432*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*Roo tOf(_Z^6-54)*(3*x^2-2)^(1/3)+54*(3*x^2-2)^(2/3))/(x^2-6))*RootOf(RootOf...
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
\[ \int \frac {1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx=\frac {\int \frac {1}{x^{2} \sqrt [3]{3 x^{2} - 2} - 6 \sqrt [3]{3 x^{2} - 2}}\, dx}{d} \] Input:
integrate(1/(3*x**2-2)**(1/3)/(d*x**2-6*d),x)
Output:
Integral(1/(x**2*(3*x**2 - 2)**(1/3) - 6*(3*x**2 - 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. \[ \int \frac {1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx=-\int \frac {1}{{\left (3\,x^2-2\right )}^{1/3}\,\left (6\,d-d\,x^2\right )} \,d x \] Input:
int(-1/((3*x^2 - 2)^(1/3)*(6*d - d*x^2)),x)
Output:
-int(1/((3*x^2 - 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