∫ 1_________ 3√____ 2 − 3x2 (−6d+dx2) dx [157]

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

Optimal. Leaf size=123 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{2-3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\tanh ^{-1}\left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{2-3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d} \]

[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 = 123, 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]{2}-\sqrt [3]{2-3 x^2}\right )}{x}\right )}{4\ 2^{5/6} 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}-\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]

-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*} \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 = 1.11 \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 \sqrt [3]{2-3 x^2} \left (-6+x^2\right ) \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*(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])))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 72.68, size = 1064, normalized size = 8.65


method	result	size

trager	\(\text {Expression too large to display}\)	\(1064\)

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((16*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+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-(-3*x^2+2)^(1/3)*RootOf(_Z^6-54)^5*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(_Z^6-54)^3*RootOf(Roo

tOf(_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)^3-18*(-3*x^2+2)^(1/3)*RootOf(_Z^6-54)^2-432*(-3*x^2+2)^(1/3)*RootOf(RootOf(_Z^6-54)^2+24*_Z*R

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

4)^2+24*_Z*RootOf(_Z^6-54)+576*_Z^2)*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*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-691

2*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-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)*Roo

tOf(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((16

*RootOf(RootOf(_Z^6-54)^2+24*_Z*RootOf(_Z^6-54)+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-(-3*x^2+2)^(1/3)*RootOf(_Z^6-54)^5*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(_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)^3-18*(-3*x^2+2)^(1/3)*RootOf(_Z^6-54)^2-432*(-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)+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))/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. 2831 vs. \(2 (90) = 180\).
time = 49.52, size = 2831, normalized size = 23.02 \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*(13824*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)*(-3*x^2 + 2)^(2/3)*(d^(-6))^(5/6) + 48*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)*(-3*x^2 + 2)^(1/3)*(d^(-6))^(2/3

) - 24*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)*(-

3*x^2 + 2)^(2/3)*(d^(-6))^(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(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)*(-3*x^2 + 2)^(1/3)*(d^(-6))^(1/6) - 24*sqrt(1/6)*(192*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)*(-3*x^2 + 2)^(2/3)*(d^(-6))^(2/3) - 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) + 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)*(-3*x^2 + 2)^(1/3)*sqrt(d^(-6)) - 12*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)*(-3*x^2 + 2)^(2/3)*(d^(-6))^(1/6) + 2*sqrt(3)*(1/4)^(1/3)*(29*d^2*x^11 + 58

6*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) - 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(-(96*(1/864)^(1/6)*(-3*x^2 + 2)^(

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

3456*(1/864)^(5/6)*(d^5*x^5 + 26*d^5*x^3)*(d^(-6))^(5/6) + 6*sqrt(1/6)*(d^3*x^5 - 20*d^3*x^3 - 108*d^3*x)*(-3

*x^2 + 2)^(1/3)*sqrt(d^(-6)) - (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 - 12)*(-3*x^2 + 2)^(1/3))/(x^6 - 18*x^4 + 108*x^2 - 216)) - sqrt(3)*(x^12 + 6300*x^10 + 311964*x^

8 + 34080*x^6 - 229392*x^4 + 91584*x^2 - 8640))/(x^12 - 9972*x^10 + 1310076*x^8 - 1277280*x^6 + 413424*x^4 - 6

7392*x^2 + 5184)) - 1/12*sqrt(3)*(1/864)^(1/6)*(d^(-6))^(1/6)*arctan(1/3*(13824*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)*(-3*x^2 + 2)^(2/3)*(d^(-6))^(5/6) - 48*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)*(-3*x^2 + 2)^

(1/3)*(d^(-6))^(2/3) + 24*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)*(-3*x^2 + 2)^(2/3)*(d^(-6))^(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(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)*(-3*x^2 + 2)^(1/3)*(d^(-6))^(1/6) + 24*sqrt(1/6)*(19

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

+ 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 + 1382

4*d^5*x^2 + 1728*d^5)*(d^(-6))^(5/6) - 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)*(-3*x^2 + 2)^(1/3)*sqrt(d^(-6)) + 12*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)*(-3*x^2 + 2)^(2/3)*(d^(-6))^(1/6) + 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) - sq

rt(3)*(x^11 + 1834*x^9 - 162264*x^7 + 126288*x^5 - 32688*x^3 + 2592*x)*(-3*x^2 + 2)^(1/3))*sqrt((96*(1/864)^(1

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

)*(d^(-6))^(2/3) - 3456*(1/864)^(5/6)*(d^5*x^5 + 26*d^5*x^3)*(d^(-6))^(5/6) + 6*sqrt(1/6)*(d^3*x^5 - 20*d^3*x^

3 - 108*d^3*x)*(-3*x^2 + 2)^(1/3)*sqrt(d^(-6)) + (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 - 12)*(-3*x^2 + 2)^(1/3))/(x^6 - 18*x^4 + 108*x^2 - 216)) + sqrt(3)*(x^12 + 630

0*x^10 + 311964*x^8 + 34080*x^6 - 229392*x^4 + 91584*x^2 - 8640))/(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*(96*(1/864)^(1/6)*(-3*x^2 + 2

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

) - 3456*(1/864)^(5/6)*(d^5*x^5 + 26*d^5*x^3)*(d^(-6))^(5/6) + 6*sqrt(1/6)*(d^3*x^5 - 20*d^3*x^3 - 108*d^3*x)*

(-3*x^2 + 2)^(1/3)*sqrt(d^(-6)) + (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 - 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*(96*(1/864)^(1/6)*(-3*x^2 + 2)^(2/3)*d*(d^(-6))^(1/6)*x^3 - 12*(1/4)^(2/3)*(d^4*x^4 + 12*d^4*x^2 - 1

2*d^4)*(-3*x^2 + 2)^(2/3)*(d^(-6))^(2/3) - 3456*(1/864)^(5/6)*(d^5*x^5 + 26*d^5*x^3)*(d^(-6))^(5/6) + 6*sqrt(1

/6)*(d^3*x^5 - 20*d^3*x^3 - 108*d^3*x)*(-3*x^2 ...

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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]{2 - 3 x^{2}} - 6 \sqrt [3]{2 - 3 x^{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*(2 - 3*x**2)**(1/3) - 6*(2 - 3*x**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 (2-3\,x^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/((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)

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