\(\int \frac {1}{\sqrt [3]{-2+3 x^2} (-6 d+d x^2)} \, dx\) [158]

Optimal result

Integrand size = 23, antiderivative size = 119 \[ \int \frac {1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}+\sqrt [3]{-2+3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{2}+\sqrt [3]{-2+3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d} \]

[Out]

1/8*arctan(2^(1/6)*(2^(1/3)+(3*x^2-2)^(1/3))/x)*2^(1/6)/d-1/24*arctanh(1/18*(2^(1/3)+(3*x^2-2)^(1/3))^2*2^(5/6
)/x*3^(1/2))*2^(1/6)/d*3^(1/2)+1/24*arctanh(1/6*x*6^(1/2))*2^(1/6)/d*3^(1/2)

Rubi [A] (verified)

Time = 0.01 (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 = {404} \[ \int \frac {1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [6]{2} \left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )}{x}\right )}{4\ 2^{5/6} d}-\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d} \]

[In]

Int[1/((-2 + 3*x^2)^(1/3)*(-6*d + d*x^2)),x]

[Out]

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)

Rule 404

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[(-q)*(Arc
Tanh[q*(x/3)]/(12*Rt[a, 3]*d)), x] + (Simp[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}+\sqrt [3]{-2+3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\tanh ^{-1}\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} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 4.50 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx=\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )}{d \left (-6+x^2\right ) \sqrt [3]{-2+3 x^2} \left (9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )+x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )\right )\right )} \]

[In]

Integrate[1/((-2 + 3*x^2)^(1/3)*(-6*d + d*x^2)),x]

[Out]

(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])))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 74.92 (sec) , antiderivative size = 1063, normalized size of antiderivative = 8.93

method	result	size
trager	\(\text {Expression too large to display}\)	\(1063\)

[In]

int(1/(3*x^2-2)^(1/3)/(d*x^2-6*d),x,method=_RETURNVERBOSE)

[Out]

-1/24*(24*ln(-(768*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)^2*RootOf(_Z^6-54)^5*x+16*RootOf(Ro
otOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^6*x+1152*(3*x^2-2)^(1/3)*RootOf(RootOf(_Z^6-54
)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)^2*RootOf(_Z^6-54)^3*x+72*(3*x^2-2)^(1/3)*RootOf(RootOf(_Z^6-54)^2+24*_Z*Ro
otOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^4*x+(3*x^2-2)^(1/3)*RootOf(_Z^6-54)^5*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(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2
)*RootOf(_Z^6-54)^3-432*(3*x^2-2)^(1/3)*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-5
4)-18*RootOf(_Z^6-54)^2*(3*x^2-2)^(1/3)-54*(3*x^2-2)^(2/3))/(x^2-6))*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^
6-54)+576*_Z^2)+24*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*ln((4608*RootOf(RootOf(_Z^6-54)^2+
24*_Z*RootOf(_Z^6-54)+576*_Z^2)^2*RootOf(_Z^6-54)^5*x+288*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_
Z^2)*RootOf(_Z^6-54)^6*x+4*RootOf(_Z^6-54)^7*x+6912*(3*x^2-2)^(1/3)*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6
-54)+576*_Z^2)^2*RootOf(_Z^6-54)^3*x+144*(3*x^2-2)^(1/3)*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z
^2)*RootOf(_Z^6-54)^4*x+216*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^3*x^2+9*x
^2*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+18*RootOf(
_Z^6-54)^4-2592*(3*x^2-2)^(1/3)*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)+324*(
3*x^2-2)^(2/3))/(x^2-6))+ln(-(768*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)^2*RootOf(_Z^6-54)^5
*x+16*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^6*x+1152*(3*x^2-2)^(1/3)*RootOf
(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)^2*RootOf(_Z^6-54)^3*x+72*(3*x^2-2)^(1/3)*RootOf(RootOf(_Z^6
-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^4*x+(3*x^2-2)^(1/3)*RootOf(_Z^6-54)^5*x+36*RootOf(RootO
f(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^3*x^2+72*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z
^6-54)+576*_Z^2)*RootOf(_Z^6-54)^3-432*(3*x^2-2)^(1/3)*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2
)*RootOf(_Z^6-54)-18*RootOf(_Z^6-54)^2*(3*x^2-2)^(1/3)-54*(3*x^2-2)^(2/3))/(x^2-6))*RootOf(_Z^6-54))/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1771 vs. \(2 (86) = 172\).

Time = 190.80 (sec) , antiderivative size = 1771, normalized size of antiderivative = 14.88 \[ \int \frac {1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx=\text {Too large to display} \]

[In]

integrate(1/(3*x^2-2)^(1/3)/(d*x^2-6*d),x, algorithm="fricas")

[Out]

1/48*(1/864)^(1/6)*(sqrt(-3) + 1)*(d^(-6))^(1/6)*log(-1/4*(4*(1/4)^(2/3)*(7*d^4*x^5 + 92*d^4*x^3 - 36*d^4*x +
sqrt(-3)*(7*d^4*x^5 + 92*d^4*x^3 - 36*d^4*x))*(d^(-6))^(2/3) - 4*(10*x^3 + 3*sqrt(1/6)*(d^3*x^4 + 24*d^3*x^2 +
12*d^3)*sqrt(d^(-6)) + 36*x)*(3*x^2 - 2)^(2/3) + 2*(432*(1/864)^(5/6)*(5*d^5*x^4 + 36*d^5*x^2 - 12*d^5 - sqrt
(-3)*(5*d^5*x^4 + 36*d^5*x^2 - 12*d^5))*(d^(-6))^(5/6) + (1/4)^(1/3)*(d^2*x^5 + 52*d^2*x^3 + 36*d^2*x - sqrt(-
3)*(d^2*x^5 + 52*d^2*x^3 + 36*d^2*x))*(d^(-6))^(1/3))*(3*x^2 - 2)^(1/3) + (1/864)^(1/6)*(d*x^6 + 210*d*x^4 + 2
52*d*x^2 + sqrt(-3)*(d*x^6 + 210*d*x^4 + 252*d*x^2 - 72*d) - 72*d)*(d^(-6))^(1/6))/(x^6 - 18*x^4 + 108*x^2 - 2
16)) - 1/48*(1/864)^(1/6)*(sqrt(-3) + 1)*(d^(-6))^(1/6)*log(-1/4*(4*(1/4)^(2/3)*(7*d^4*x^5 + 92*d^4*x^3 - 36*d
^4*x + sqrt(-3)*(7*d^4*x^5 + 92*d^4*x^3 - 36*d^4*x))*(d^(-6))^(2/3) - 4*(10*x^3 - 3*sqrt(1/6)*(d^3*x^4 + 24*d^
3*x^2 + 12*d^3)*sqrt(d^(-6)) + 36*x)*(3*x^2 - 2)^(2/3) - 2*(432*(1/864)^(5/6)*(5*d^5*x^4 + 36*d^5*x^2 - 12*d^5
- sqrt(-3)*(5*d^5*x^4 + 36*d^5*x^2 - 12*d^5))*(d^(-6))^(5/6) - (1/4)^(1/3)*(d^2*x^5 + 52*d^2*x^3 + 36*d^2*x -
sqrt(-3)*(d^2*x^5 + 52*d^2*x^3 + 36*d^2*x))*(d^(-6))^(1/3))*(3*x^2 - 2)^(1/3) - (1/864)^(1/6)*(d*x^6 + 210*d*
x^4 + 252*d*x^2 + sqrt(-3)*(d*x^6 + 210*d*x^4 + 252*d*x^2 - 72*d) - 72*d)*(d^(-6))^(1/6))/(x^6 - 18*x^4 + 108*
x^2 - 216)) - 1/48*(1/864)^(1/6)*(sqrt(-3) - 1)*(d^(-6))^(1/6)*log(-1/4*(4*(1/4)^(2/3)*(7*d^4*x^5 + 92*d^4*x^3
- 36*d^4*x - sqrt(-3)*(7*d^4*x^5 + 92*d^4*x^3 - 36*d^4*x))*(d^(-6))^(2/3) - 4*(10*x^3 + 3*sqrt(1/6)*(d^3*x^4
+ 24*d^3*x^2 + 12*d^3)*sqrt(d^(-6)) + 36*x)*(3*x^2 - 2)^(2/3) + 2*(432*(1/864)^(5/6)*(5*d^5*x^4 + 36*d^5*x^2 -
12*d^5 + sqrt(-3)*(5*d^5*x^4 + 36*d^5*x^2 - 12*d^5))*(d^(-6))^(5/6) + (1/4)^(1/3)*(d^2*x^5 + 52*d^2*x^3 + 36*
d^2*x + sqrt(-3)*(d^2*x^5 + 52*d^2*x^3 + 36*d^2*x))*(d^(-6))^(1/3))*(3*x^2 - 2)^(1/3) + (1/864)^(1/6)*(d*x^6 +
210*d*x^4 + 252*d*x^2 - sqrt(-3)*(d*x^6 + 210*d*x^4 + 252*d*x^2 - 72*d) - 72*d)*(d^(-6))^(1/6))/(x^6 - 18*x^4
+ 108*x^2 - 216)) + 1/48*(1/864)^(1/6)*(sqrt(-3) - 1)*(d^(-6))^(1/6)*log(-1/4*(4*(1/4)^(2/3)*(7*d^4*x^5 + 92*
d^4*x^3 - 36*d^4*x - sqrt(-3)*(7*d^4*x^5 + 92*d^4*x^3 - 36*d^4*x))*(d^(-6))^(2/3) - 4*(10*x^3 - 3*sqrt(1/6)*(d
^3*x^4 + 24*d^3*x^2 + 12*d^3)*sqrt(d^(-6)) + 36*x)*(3*x^2 - 2)^(2/3) - 2*(432*(1/864)^(5/6)*(5*d^5*x^4 + 36*d^
5*x^2 - 12*d^5 + sqrt(-3)*(5*d^5*x^4 + 36*d^5*x^2 - 12*d^5))*(d^(-6))^(5/6) - (1/4)^(1/3)*(d^2*x^5 + 52*d^2*x^
3 + 36*d^2*x + sqrt(-3)*(d^2*x^5 + 52*d^2*x^3 + 36*d^2*x))*(d^(-6))^(1/3))*(3*x^2 - 2)^(1/3) - (1/864)^(1/6)*(
d*x^6 + 210*d*x^4 + 252*d*x^2 - sqrt(-3)*(d*x^6 + 210*d*x^4 + 252*d*x^2 - 72*d) - 72*d)*(d^(-6))^(1/6))/(x^6 -
18*x^4 + 108*x^2 - 216)) - 1/24*(1/864)^(1/6)*(d^(-6))^(1/6)*log(1/4*(4*(1/4)^(2/3)*(7*d^4*x^5 + 92*d^4*x^3 -
36*d^4*x)*(d^(-6))^(2/3) + 2*(10*x^3 + 3*sqrt(1/6)*(d^3*x^4 + 24*d^3*x^2 + 12*d^3)*sqrt(d^(-6)) + 36*x)*(3*x^
2 - 2)^(2/3) + 2*(432*(1/864)^(5/6)*(5*d^5*x^4 + 36*d^5*x^2 - 12*d^5)*(d^(-6))^(5/6) + (1/4)^(1/3)*(d^2*x^5 +
52*d^2*x^3 + 36*d^2*x)*(d^(-6))^(1/3))*(3*x^2 - 2)^(1/3) + (1/864)^(1/6)*(d*x^6 + 210*d*x^4 + 252*d*x^2 - 72*d
)*(d^(-6))^(1/6))/(x^6 - 18*x^4 + 108*x^2 - 216)) + 1/24*(1/864)^(1/6)*(d^(-6))^(1/6)*log(1/4*(4*(1/4)^(2/3)*(
7*d^4*x^5 + 92*d^4*x^3 - 36*d^4*x)*(d^(-6))^(2/3) + 2*(10*x^3 - 3*sqrt(1/6)*(d^3*x^4 + 24*d^3*x^2 + 12*d^3)*sq
rt(d^(-6)) + 36*x)*(3*x^2 - 2)^(2/3) - 2*(432*(1/864)^(5/6)*(5*d^5*x^4 + 36*d^5*x^2 - 12*d^5)*(d^(-6))^(5/6) -
(1/4)^(1/3)*(d^2*x^5 + 52*d^2*x^3 + 36*d^2*x)*(d^(-6))^(1/3))*(3*x^2 - 2)^(1/3) - (1/864)^(1/6)*(d*x^6 + 210*
d*x^4 + 252*d*x^2 - 72*d)*(d^(-6))^(1/6))/(x^6 - 18*x^4 + 108*x^2 - 216))

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx=\frac {\int \frac {1}{x^{2} \sqrt [3]{3 x^{2} - 2} - 6 \sqrt [3]{3 x^{2} - 2}}\, dx}{d} \]

[In]

integrate(1/(3*x**2-2)**(1/3)/(d*x**2-6*d),x)

[Out]

Integral(1/(x**2*(3*x**2 - 2)**(1/3) - 6*(3*x**2 - 2)**(1/3)), x)/d

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx=\int { \frac {1}{{\left (d x^{2} - 6 \, d\right )} {\left (3 \, x^{2} - 2\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(1/(3*x^2-2)^(1/3)/(d*x^2-6*d),x, algorithm="maxima")

[Out]

integrate(1/((d*x^2 - 6*d)*(3*x^2 - 2)^(1/3)), x)

Giac [F]

\[ \int \frac {1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx=\int { \frac {1}{{\left (d x^{2} - 6 \, d\right )} {\left (3 \, x^{2} - 2\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(1/(3*x^2-2)^(1/3)/(d*x^2-6*d),x, algorithm="giac")

[Out]

integrate(1/((d*x^2 - 6*d)*(3*x^2 - 2)^(1/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx=-\int \frac {1}{{\left (3\,x^2-2\right )}^{1/3}\,\left (6\,d-d\,x^2\right )} \,d x \]

[In]

int(-1/((3*x^2 - 2)^(1/3)*(6*d - d*x^2)),x)

[Out]

-int(1/((3*x^2 - 2)^(1/3)*(6*d - d*x^2)), x)