Integrand size = 23, antiderivative size = 119 \[ \int \frac {1}{\sqrt [3]{-2-3 x^2} \left (6 d+d x^2\right )} \, dx=-\frac {\arctan \left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\arctan \left (\frac {\left (\sqrt [3]{2}+\sqrt [3]{-2-3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}+\sqrt [3]{-2-3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d} \] Output:
-1/24*arctan(1/6*x*6^(1/2))*2^(1/6)*3^(1/2)/d-1/24*arctan(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+1/8*arctanh(2^(1/6) *(2^(1/3)+(-3*x^2-2)^(1/3))/x)*2^(1/6)/d
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 5.17 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt [3]{-2-3 x^2} \left (6 d+d x^2\right )} \, dx=-\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {3 x^2}{2},-\frac {x^2}{6}\right )}{d \sqrt [3]{-2-3 x^2} \left (6+x^2\right ) \left (-9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {3 x^2}{2},-\frac {x^2}{6}\right )+x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},-\frac {3 x^2}{2},-\frac {x^2}{6}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},-\frac {3 x^2}{2},-\frac {x^2}{6}\right )\right )\right )} \] 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, -1/6*x^2])/(d*(-2 - 3*x^2)^(1 /3)*(6 + x^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, (-3*x^2)/2, -1/6*x^2] + x^2*( AppellF1[3/2, 1/3, 2, 5/2, (-3*x^2)/2, -1/6*x^2] + 3*AppellF1[3/2, 4/3, 1, 5/2, (-3*x^2)/2, -1/6*x^2])))
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 = {306}
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.
\(\displaystyle \int \frac {1}{\sqrt [3]{-3 x^2-2} \left (d x^2+6 d\right )} \, dx\) |
\(\Big \downarrow \) 306 |
\(\displaystyle -\frac {\arctan \left (\frac {\left (\sqrt [3]{-3 x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\arctan \left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{-3 x^2-2}+\sqrt [3]{2}\right )}{x}\right )}{4\ 2^{5/6} d}\) |
Input:
Int[1/((-2 - 3*x^2)^(1/3)*(6*d + d*x^2)),x]
Output:
-1/4*ArcTan[x/Sqrt[6]]/(2^(5/6)*Sqrt[3]*d) - ArcTan[(2^(1/3) + (-2 - 3*x^2 )^(1/3))^2/(3*2^(1/6)*Sqrt[3]*x)]/(4*2^(5/6)*Sqrt[3]*d) + ArcTanh[(2^(1/6) *(2^(1/3) + (-2 - 3*x^2)^(1/3)))/x]/(4*2^(5/6)*d)
Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit h[{q = Rt[b/a, 2]}, Simp[q*(ArcTan[q*(x/3)]/(12*Rt[a, 3]*d)), x] + (Simp[q* (ArcTan[(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*(ArcTanh[(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] && PosQ[b/a]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 157.74 (sec) , antiderivative size = 548, normalized size of antiderivative = 4.61
Input:
int(1/(-3*x^2-2)^(1/3)/(d*x^2+6*d),x,method=_RETURNVERBOSE)
Output:
1/24*(RootOf(_Z^6+54)*ln(-(16*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+5 4)+576*_Z^2)*RootOf(_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(RootOf(_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(RootOf(_Z ^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*RootOf(_Z^6+54)^3*x^2+72*RootOf(R ootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*RootOf(_Z^6+54)^3-18*Root Of(_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)*RootOf(_Z^6+54)*(-3*x^2-2)^(1/3)+54*(-3*x^2-2)^(2/3))/(x ^2+6))+24*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*ln(-(-4 *RootOf(_Z^6+54)^7*x+192*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+57 6*_Z^2)*RootOf(_Z^6+54)^6*x-6*RootOf(_Z^6+54)^5*(-3*x^2-2)^(1/3)*x+288*Roo tOf(RootOf(_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+9*x^2*RootOf(_Z^6+54)^4-18*RootOf(_Z^6+54)^4+108*RootOf(_ Z^6+54)^2*(-3*x^2-2)^(1/3)+324*(-3*x^2-2)^(2/3))/(x^2+6)))/d
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. \[ \int \frac {1}{\sqrt [3]{-2-3 x^2} \left (6 d+d x^2\right )} \, dx=\int \frac {1}{{\left (-3\,x^2-2\right )}^{1/3}\,\left (d\,x^2+6\,d\right )} \,d x \] 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