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

Optimal. Leaf size=119 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )}{x}\right )}{4\ 2^{5/6} d}-\frac {\tanh ^{-1}\left (\frac {\left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d} \]

[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)

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Rubi [A]  time = 0.02, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {395} \[ \frac {\tan ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )}{x}\right )}{4\ 2^{5/6} d}-\frac {\tanh ^{-1}\left (\frac {\left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d} \]

Antiderivative was successfully verified.

[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 395

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

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Mathematica [C]  time = 0.10, size = 136, normalized size = 1.14 \[ \frac {9 x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {3 x^2}{2},\frac {x^2}{6}\right )}{d \left (x^2-6\right ) \sqrt [3]{3 x^2-2} \left (x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};\frac {3 x^2}{2},\frac {x^2}{6}\right )+3 F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};\frac {3 x^2}{2},\frac {x^2}{6}\right )\right )+9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {3 x^2}{2},\frac {x^2}{6}\right )\right )} \]

Warning: Unable to verify antiderivative.

[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])))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d x^{2} - 6 \, d\right )} {\left (3 \, x^{2} - 2\right )}^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 58.82, size = 1063, normalized size = 8.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(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*Ro
otOf(_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)*Roo
tOf(_Z^6-54)^6*x+4*RootOf(_Z^6-54)^7*x+6912*(3*x^2-2)^(1/3)*RootOf(_Z^6-54)^3*RootOf(RootOf(_Z^6-54)^2-24*_Z*R
ootOf(_Z^6-54)+576*_Z^2)^2*x-144*(3*x^2-2)^(1/3)*RootOf(_Z^6-54)^4*RootOf(RootOf(_Z^6-54)^2-24*_Z*RootOf(_Z^6-
54)+576*_Z^2)*x+216*RootOf(_Z^6-54)^3*RootOf(RootOf(_Z^6-54)^2-24*_Z*RootOf(_Z^6-54)+576*_Z^2)*x^2-9*x^2*RootO
f(_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(_Z^6-54)*RootOf(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((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(_Z^6-
54)^3*RootOf(RootOf(_Z^6-54)^2-24*_Z*RootOf(_Z^6-54)+576*_Z^2)^2*x-72*(3*x^2-2)^(1/3)*RootOf(_Z^6-54)^4*RootOf
(RootOf(_Z^6-54)^2-24*_Z*RootOf(_Z^6-54)+576*_Z^2)*x+RootOf(_Z^6-54)^5*(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)^3-432*(3*x^2-2)^(1/3)*RootOf(_Z^6-54)*RootOf(RootOf(_Z^6-54)^2-24*_Z*RootOf(_Z^6-54
)+576*_Z^2)+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*R
ootOf(_Z^6-54)+576*_Z^2)-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(
_Z^6-54)^3*RootOf(RootOf(_Z^6-54)^2-24*_Z*RootOf(_Z^6-54)+576*_Z^2)^2*x-72*(3*x^2-2)^(1/3)*RootOf(_Z^6-54)^4*R
ootOf(RootOf(_Z^6-54)^2-24*_Z*RootOf(_Z^6-54)+576*_Z^2)*x+RootOf(_Z^6-54)^5*(3*x^2-2)^(1/3)*x+36*RootOf(_Z^6-5
4)^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)^3-432*(3*x^2-2)^(1/3)*RootOf(_Z^6-54)*RootOf(RootOf(_Z^6-54)^2-24*_Z*RootOf(_Z
^6-54)+576*_Z^2)+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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d x^{2} - 6 \, d\right )} {\left (3 \, x^{2} - 2\right )}^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {1}{{\left (3\,x^2-2\right )}^{1/3}\,\left (6\,d-d\,x^2\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{x^{2} \sqrt [3]{3 x^{2} - 2} - 6 \sqrt [3]{3 x^{2} - 2}}\, dx}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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