∫ 1_________ 3√_____ −2 + 3x2 (−6d+dx2) dx [158]

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

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

[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 = 0.01, 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 = {404} \begin {gather*} \frac {\text {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 {\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} \end {gather*}

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 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*} \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] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 4.86, size = 136, normalized size = 1.14 \begin {gather*} \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 (-6+x^2\right ) \sqrt [3]{-2+3 x^2} \left (9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {3 x^2}{2},\frac {x^2}{6}\right )+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 )\right )} \end {gather*}

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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 98.38, size = 723, normalized size = 6.08


method	result	size

trager	\(\text {Expression too large to display}\)	\(723\)

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,method=_RETURNVERBOSE)

[Out]

-1/24*(ln((4*RootOf(_Z^6-54)^7*x+192*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^

6*x-6*(3*x^2-2)^(1/3)*RootOf(_Z^6-54)^5*x-288*(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-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^6-54)+RootOf(_Z^6-54)*ln(-(4*RootOf(_Z^6-54)^7*x+288*RootOf(RootOf(_

Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf(_Z^6-54)^6*x+4608*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-

54)+576*_Z^2)^2*RootOf(_Z^6-54)^5*x+144*(3*x^2-2)^(1/3)*RootOf(_Z^6-54)^4*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootO

f(_Z^6-54)+576*_Z^2)*x+6912*(3*x^2-2)^(1/3)*RootOf(_Z^6-54)^3*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+5

76*_Z^2)^2*x-9*x^2*RootOf(_Z^6-54)^4-216*RootOf(_Z^6-54)^3*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*

_Z^2)*x^2-18*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+

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))+24*ln((4*RootOf(_Z^6-54)^7*x+192*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*RootOf

(_Z^6-54)^6*x-6*(3*x^2-2)^(1/3)*RootOf(_Z^6-54)^5*x-288*(3*x^2-2)^(1/3)*RootOf(_Z^6-54)^4*RootOf(RootOf(_Z^6-5

4)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*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(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2))/d

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2681 vs. \(2 (86) = 172\).
time = 46.32, size = 2681, normalized size = 22.53 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/12*sqrt(3)*(1/864)^(1/6)*(d^(-6))^(1/6)*arctan(1/3*(144*sqrt(3)*sqrt(1/6)*(3*d^3*x^11 - 2234*d^3*x^9 + 15672

*d^3*x^7 - 14928*d^3*x^5 + 4080*d^3*x^3 - 288*d^3*x)*sqrt(d^(-6)) + 24*sqrt(1/6)*(72*sqrt(3)*(1/864)^(5/6)*(d^

5*x^12 + 4368*d^5*x^10 - 844860*d^5*x^8 + 753216*d^5*x^6 - 217296*d^5*x^4 + 13824*d^5*x^2 + 1728*d^5)*(d^(-6))

^(5/6) + 2*sqrt(3)*(1/4)^(1/3)*(29*d^2*x^11 + 586*d^2*x^9 - 10680*d^2*x^7 + 39888*d^2*x^5 - 19440*d^2*x^3 + 25

92*d^2*x)*(d^(-6))^(1/3) + 12*(16*sqrt(3)*(1/4)^(2/3)*(5*d^4*x^9 - 490*d^4*x^7 + 732*d^4*x^5 - 120*d^4*x^3)*(d

^(-6))^(2/3) + sqrt(3)*(1/864)^(1/6)*(d*x^10 - 214*d*x^8 - 29048*d*x^6 + 18576*d*x^4 - 3888*d*x^2 + 288*d)*(d^

(-6))^(1/6))*(3*x^2 - 2)^(2/3) + (6*sqrt(3)*sqrt(1/6)*(49*d^3*x^10 - 10086*d^3*x^8 + 14632*d^3*x^6 + 3024*d^3*

x^4 - 2736*d^3*x^2 + 288*d^3)*sqrt(d^(-6)) + sqrt(3)*(x^11 + 1834*x^9 - 162264*x^7 + 126288*x^5 - 32688*x^3 +

2592*x))*(3*x^2 - 2)^(1/3))*sqrt(-(3456*(1/864)^(5/6)*(d^5*x^5 + 26*d^5*x^3)*(d^(-6))^(5/6) - 12*(8*(1/864)^(1

/6)*d*(d^(-6))^(1/6)*x^3 + (1/4)^(2/3)*(d^4*x^4 + 12*d^4*x^2 - 12*d^4)*(d^(-6))^(2/3))*(3*x^2 - 2)^(2/3) - (1/

4)^(1/3)*(d^2*x^6 - 186*d^2*x^4 - 468*d^2*x^2 + 72*d^2)*(d^(-6))^(1/3) + 6*(x^4 - 36*x^2 + sqrt(1/6)*(d^3*x^5

- 20*d^3*x^3 - 108*d^3*x)*sqrt(d^(-6)) - 12)*(3*x^2 - 2)^(1/3))/(x^6 - 18*x^4 + 108*x^2 - 216)) + 24*(576*sqrt

(3)*(1/864)^(5/6)*(35*d^5*x^9 - 9720*d^5*x^7 + 8424*d^5*x^5 - 3168*d^5*x^3 + 432*d^5*x)*(d^(-6))^(5/6) + sqrt(

3)*(1/4)^(1/3)*(d^2*x^10 + 1178*d^2*x^8 + 15784*d^2*x^6 - 6192*d^2*x^4 - 432*d^2*x^2 + 288*d^2)*(d^(-6))^(1/3)

)*(3*x^2 - 2)^(2/3) + sqrt(3)*(x^12 + 6300*x^10 + 311964*x^8 + 34080*x^6 - 229392*x^4 + 91584*x^2 - 8640) + 24

*(2*sqrt(3)*(1/4)^(2/3)*(27*d^4*x^10 + 4614*d^4*x^8 + 18296*d^4*x^6 - 20304*d^4*x^4 + 4464*d^4*x^2 - 288*d^4)*

(d^(-6))^(2/3) + sqrt(3)*(1/864)^(1/6)*(d*x^11 - 3182*d*x^9 + 169704*d*x^7 - 120816*d*x^5 + 20304*d*x^3 - 864*

d*x)*(d^(-6))^(1/6))*(3*x^2 - 2)^(1/3))/(x^12 - 9972*x^10 + 1310076*x^8 - 1277280*x^6 + 413424*x^4 - 67392*x^2

+ 5184)) - 1/12*sqrt(3)*(1/864)^(1/6)*(d^(-6))^(1/6)*arctan(-1/3*(144*sqrt(3)*sqrt(1/6)*(3*d^3*x^11 - 2234*d^

3*x^9 + 15672*d^3*x^7 - 14928*d^3*x^5 + 4080*d^3*x^3 - 288*d^3*x)*sqrt(d^(-6)) + 24*sqrt(1/6)*(72*sqrt(3)*(1/8

64)^(5/6)*(d^5*x^12 + 4368*d^5*x^10 - 844860*d^5*x^8 + 753216*d^5*x^6 - 217296*d^5*x^4 + 13824*d^5*x^2 + 1728*

d^5)*(d^(-6))^(5/6) - 2*sqrt(3)*(1/4)^(1/3)*(29*d^2*x^11 + 586*d^2*x^9 - 10680*d^2*x^7 + 39888*d^2*x^5 - 19440

*d^2*x^3 + 2592*d^2*x)*(d^(-6))^(1/3) - 12*(16*sqrt(3)*(1/4)^(2/3)*(5*d^4*x^9 - 490*d^4*x^7 + 732*d^4*x^5 - 12

0*d^4*x^3)*(d^(-6))^(2/3) - sqrt(3)*(1/864)^(1/6)*(d*x^10 - 214*d*x^8 - 29048*d*x^6 + 18576*d*x^4 - 3888*d*x^2

+ 288*d)*(d^(-6))^(1/6))*(3*x^2 - 2)^(2/3) + (6*sqrt(3)*sqrt(1/6)*(49*d^3*x^10 - 10086*d^3*x^8 + 14632*d^3*x^

6 + 3024*d^3*x^4 - 2736*d^3*x^2 + 288*d^3)*sqrt(d^(-6)) - sqrt(3)*(x^11 + 1834*x^9 - 162264*x^7 + 126288*x^5 -

32688*x^3 + 2592*x))*(3*x^2 - 2)^(1/3))*sqrt((3456*(1/864)^(5/6)*(d^5*x^5 + 26*d^5*x^3)*(d^(-6))^(5/6) - 12*(

8*(1/864)^(1/6)*d*(d^(-6))^(1/6)*x^3 - (1/4)^(2/3)*(d^4*x^4 + 12*d^4*x^2 - 12*d^4)*(d^(-6))^(2/3))*(3*x^2 - 2)

^(2/3) + (1/4)^(1/3)*(d^2*x^6 - 186*d^2*x^4 - 468*d^2*x^2 + 72*d^2)*(d^(-6))^(1/3) - 6*(x^4 - 36*x^2 - sqrt(1/

6)*(d^3*x^5 - 20*d^3*x^3 - 108*d^3*x)*sqrt(d^(-6)) - 12)*(3*x^2 - 2)^(1/3))/(x^6 - 18*x^4 + 108*x^2 - 216)) +

24*(576*sqrt(3)*(1/864)^(5/6)*(35*d^5*x^9 - 9720*d^5*x^7 + 8424*d^5*x^5 - 3168*d^5*x^3 + 432*d^5*x)*(d^(-6))^(

5/6) - sqrt(3)*(1/4)^(1/3)*(d^2*x^10 + 1178*d^2*x^8 + 15784*d^2*x^6 - 6192*d^2*x^4 - 432*d^2*x^2 + 288*d^2)*(d

^(-6))^(1/3))*(3*x^2 - 2)^(2/3) - sqrt(3)*(x^12 + 6300*x^10 + 311964*x^8 + 34080*x^6 - 229392*x^4 + 91584*x^2

- 8640) - 24*(2*sqrt(3)*(1/4)^(2/3)*(27*d^4*x^10 + 4614*d^4*x^8 + 18296*d^4*x^6 - 20304*d^4*x^4 + 4464*d^4*x^2

- 288*d^4)*(d^(-6))^(2/3) - sqrt(3)*(1/864)^(1/6)*(d*x^11 - 3182*d*x^9 + 169704*d*x^7 - 120816*d*x^5 + 20304*

d*x^3 - 864*d*x)*(d^(-6))^(1/6))*(3*x^2 - 2)^(1/3))/(x^12 - 9972*x^10 + 1310076*x^8 - 1277280*x^6 + 413424*x^4

- 67392*x^2 + 5184)) + 1/48*(1/864)^(1/6)*(d^(-6))^(1/6)*log(-1/24*(3456*(1/864)^(5/6)*(d^5*x^5 + 26*d^5*x^3)

*(d^(-6))^(5/6) - 12*(8*(1/864)^(1/6)*d*(d^(-6))^(1/6)*x^3 + (1/4)^(2/3)*(d^4*x^4 + 12*d^4*x^2 - 12*d^4)*(d^(-

6))^(2/3))*(3*x^2 - 2)^(2/3) - (1/4)^(1/3)*(d^2*x^6 - 186*d^2*x^4 - 468*d^2*x^2 + 72*d^2)*(d^(-6))^(1/3) + 6*(

x^4 - 36*x^2 + sqrt(1/6)*(d^3*x^5 - 20*d^3*x^3 - 108*d^3*x)*sqrt(d^(-6)) - 12)*(3*x^2 - 2)^(1/3))/(x^6 - 18*x^

4 + 108*x^2 - 216)) - 1/48*(1/864)^(1/6)*(d^(-6))^(1/6)*log(1/24*(3456*(1/864)^(5/6)*(d^5*x^5 + 26*d^5*x^3)*(d

^(-6))^(5/6) - 12*(8*(1/864)^(1/6)*d*(d^(-6))^(1/6)*x^3 - (1/4)^(2/3)*(d^4*x^4 + 12*d^4*x^2 - 12*d^4)*(d^(-6))

^(2/3))*(3*x^2 - 2)^(2/3) + (1/4)^(1/3)*(d^2*x^6 - 186*d^2*x^4 - 468*d^2*x^2 + 72*d^2)*(d^(-6))^(1/3) - 6*(x^4

- 36*x^2 - sqrt(1/6)*(d^3*x^5 - 20*d^3*x^3 - 108*d^3*x)*sqrt(d^(-6)) - 12)*(3*x^2 - 2)^(1/3))/(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...

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

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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