\(\int \frac {1}{\sqrt [3]{2-3 x^2} (-6 d+d x^2)} \, dx\) [157]

Optimal result

Integrand size = 23, antiderivative size = 123 \[ \int \frac {1}{\sqrt [3]{2-3 x^2} \left (-6 d+d x^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{2-3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{2-3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d} \]

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

[In]

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

[Out]

-1/4*ArcTan[(2^(1/6)*(2^(1/3) - (2 - 3*x^2)^(1/3)))/x]/(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 = 4.15 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt [3]{2-3 x^2} \left (-6 d+d x^2\right )} \, dx=\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )}{d \sqrt [3]{2-3 x^2} \left (-6+x^2\right ) \left (9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )+x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},\frac {3 x^2}{2},\frac {x^2}{6}\right )\right )\right )} \]

[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*(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])))

Maple [C] (warning: unable to verify)

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

Time = 81.70 (sec) , antiderivative size = 547, normalized size of antiderivative = 4.45

method	result	size
trager	\(\text {Expression too large to display}\)	\(547\)

[In]

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

[Out]

-1/24*(24*RootOf(RootOf(_Z^6-54)^2-24*_Z*RootOf(_Z^6-54)+576*_Z^2)*ln((-192*RootOf(RootOf(_Z^6-54)^2-24*_Z*Roo
tOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^6*x+4*RootOf(_Z^6-54)^7*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+6*RootOf(_Z^6-54)^5*(-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))+RootOf(_Z^6-5
4)*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*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+72*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-RootOf(_Z^6-54)^5*(-3*x^2+2)^(1/3)*x+36*RootOf(RootOf(_Z^6-54)^2-24*_Z*R
ootOf(_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)*Roo
tOf(_Z^6-54)^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)-1
8*(-3*x^2+2)^(1/3)*RootOf(_Z^6-54)^2+54*(-3*x^2+2)^(2/3))/(x^2-6)))/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1867 vs. \(2 (90) = 180\).

Time = 193.66 (sec) , antiderivative size = 1867, normalized size of antiderivative = 15.18 \[ \int \frac {1}{\sqrt [3]{2-3 x^2} \left (-6 d+d x^2\right )} \, dx=\text {Too large to display} \]

[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*(864*(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))*(-3*x^2 + 2)^(1/3)*(d^(-6))^(5/6) + 12*sqrt(1/6)*(d^3*x^4 + 24*
d^3*x^2 + 12*d^3)*(-3*x^2 + 2)^(2/3)*sqrt(d^(-6)) + 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) - 2*(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))*(-3*x^2 + 2)^(1/3)*(d^(-6))^(1/3) - 8*(5*x^3 + 18*x)*(-3*x^2 + 2)^(2/3)
- (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*(864*(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))*(-3*x^2 + 2)^(1/3)*
(d^(-6))^(5/6) + 12*sqrt(1/6)*(d^3*x^4 + 24*d^3*x^2 + 12*d^3)*(-3*x^2 + 2)^(2/3)*sqrt(d^(-6)) - 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) + 2*(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))*(-3*x^2 + 2)^(1/3)*(d^(-6))
^(1/3) + 8*(5*x^3 + 18*x)*(-3*x^2 + 2)^(2/3) - (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*(864*(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))*(-3*x^2 + 2)^(1/3)*(d^(-6))^(5/6) + 12*sqrt(1/6)*(d^3*x^4 + 24*d^3*x^2 + 12*d^3)*(-3
*x^2 + 2)^(2/3)*sqrt(d^(-6)) + 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) - 2*(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))*(-3*x^2 + 2)^(1/3)*(d^(-6))^(1/3) - 8*(5*x^3 + 18*x)*(-3*x^2 + 2)^(2/3) - (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*(864*(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))*(-3*x^2 + 2)^(1/3)*(d^(-6))^(5/6) + 12*s
qrt(1/6)*(d^3*x^4 + 24*d^3*x^2 + 12*d^3)*(-3*x^2 + 2)^(2/3)*sqrt(d^(-6)) - 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) + 2*(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))*(-3*x^2 + 2)^(1/3)*(d^(-6))^(1/3) + 8*(5*x^3 + 1
8*x)*(-3*x^2 + 2)^(2/3) - (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
*(864*(1/864)^(5/6)*(5*d^5*x^4 + 36*d^5*x^2 - 12*d^5)*(-3*x^2 + 2)^(1/3)*(d^(-6))^(5/6) - 6*sqrt(1/6)*(d^3*x^4
+ 24*d^3*x^2 + 12*d^3)*(-3*x^2 + 2)^(2/3)*sqrt(d^(-6)) + 4*(1/4)^(2/3)*(7*d^4*x^5 + 92*d^4*x^3 - 36*d^4*x)*(d
^(-6))^(2/3) - 2*(1/4)^(1/3)*(d^2*x^5 + 52*d^2*x^3 + 36*d^2*x)*(-3*x^2 + 2)^(1/3)*(d^(-6))^(1/3) + 4*(5*x^3 +
18*x)*(-3*x^2 + 2)^(2/3) - (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*(864*(1/864)^(5/6)*(5*d^5*x^4 + 36*d^5*x^2 - 12
*d^5)*(-3*x^2 + 2)^(1/3)*(d^(-6))^(5/6) - 6*sqrt(1/6)*(d^3*x^4 + 24*d^3*x^2 + 12*d^3)*(-3*x^2 + 2)^(2/3)*sqrt(
d^(-6)) - 4*(1/4)^(2/3)*(7*d^4*x^5 + 92*d^4*x^3 - 36*d^4*x)*(d^(-6))^(2/3) + 2*(1/4)^(1/3)*(d^2*x^5 + 52*d^2*x
^3 + 36*d^2*x)*(-3*x^2 + 2)^(1/3)*(d^(-6))^(1/3) - 4*(5*x^3 + 18*x)*(-3*x^2 + 2)^(2/3) - (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]

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

[In]

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

[Out]

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

[In]

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

[Out]

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