3.2.56 \(\int \frac {1}{\sqrt [3]{2+3 x^2} (6 d+d x^2)} \, dx\) [156]
Optimal. Leaf size=123 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\tan ^{-1}\left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{2+3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{2+3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d} \]
[Out]
-1/8*arctanh(2^(1/6)*(2^(1/3)-(3*x^2+2)^(1/3))/x)*2^(1/6)/d+1/24*arctan(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*arctan(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 = {403}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\text {ArcTan}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )}{x}\right )}{4\ 2^{5/6} d} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((2 + 3*x^2)^(1/3)*(6*d + d*x^2)),x]
[Out]
ArcTan[x/Sqrt[6]]/(4*2^(5/6)*Sqrt[3]*d) + ArcTan[(2^(1/3) - (2 + 3*x^2)^(1/3))^2/(3*2^(1/6)*Sqrt[3]*x)]/(4*2^(
5/6)*Sqrt[3]*d) - ArcTanh[(2^(1/6)*(2^(1/3) - (2 + 3*x^2)^(1/3)))/x]/(4*2^(5/6)*d)
Rule 403
Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b/a, 2]}, Simp[q*(ArcTan[
q*(x/3)]/(12*Rt[a, 3]*d)), x] + (Simp[q*(ArcTan[(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*(ArcTanh[(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] && PosQ[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 {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\tan ^{-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 {\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{2+3 x^2}\right )}{x}\right )}{4\ 2^{5/6} 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 = 5.29, 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 \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, -1/6*x^2])/(d*(6 + x^2)*(2 + 3*x^2)^(1/3)*(-9*AppellF1[1/2, 1/3,
1, 3/2, (-3*x^2)/2, -1/6*x^2] + x^2*(AppellF1[3/2, 1/3, 2, 5/2, (-3*x^2)/2, -1/6*x^2] + 3*AppellF1[3/2, 4/3, 1
, 5/2, (-3*x^2)/2, -1/6*x^2])))
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 76.53, size = 548, normalized size = 4.46
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(548\) |
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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*(RootOf(_Z^6+54)*ln(-(16*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*RootOf(_Z^6+54)^6*x-76
8*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)^2*RootOf(_Z^6+54)^5*x-RootOf(_Z^6+54)^5*(3*x^2+2)^(
1/3)*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*Ro
otOf(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(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+18*RootOf(_Z^6+54)^2*(3*x^2+2)^(1/3)-432*(3*x^2+2)^(1/3)*RootOf(RootOf(_Z^6+5
4)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*RootOf(_Z^6+54)+54*(3*x^2+2)^(2/3))/(x^2+6))+24*RootOf(RootOf(_Z^6+54)^2-
24*_Z*RootOf(_Z^6+54)+576*_Z^2)*ln((4*RootOf(_Z^6+54)^7*x-192*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+5
76*_Z^2)*RootOf(_Z^6+54)^6*x-6*RootOf(_Z^6+54)^5*(3*x^2+2)^(1/3)*x+288*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-9*x^2*RootOf(_Z^6+54)^4+18*RootOf(_Z^6+54)^4+108*RootOf(
_Z^6+54)^2*(3*x^2+2)^(1/3)-324*(3*x^2+2)^(2/3))/(x^2+6)))/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. 2961 vs.
\(2 (90) = 180\).
time = 51.46, size = 2961, normalized size = 24.07 \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]
1/96*sqrt(3)*(1/864)^(1/6)*(d^(-6))^(1/6)*log(1/24*(5184*sqrt(3)*(1/864)^(5/6)*(3*d^5*x^5 + 20*d^5*x^3 + 12*d^
5*x)*(d^(-6))^(5/6) + 12*(32*sqrt(3)*(1/864)^(1/6)*d*(d^(-6))^(1/6)*x^3 + 3*(1/4)^(2/3)*(d^4*x^4 + 28*d^4*x^2
+ 4*d^4)*(d^(-6))^(2/3))*(3*x^2 + 2)^(2/3) + (1/4)^(1/3)*(d^2*x^6 + 1098*d^2*x^4 + 396*d^2*x^2 - 72*d^2)*(d^(-
6))^(1/3) + 6*(27*x^4 + sqrt(3)*sqrt(1/6)*(d^3*x^5 + 140*d^3*x^3 + 36*d^3*x)*sqrt(d^(-6)) + 36*x^2 + 12)*(3*x^
2 + 2)^(1/3))/(x^6 + 18*x^4 + 108*x^2 + 216)) + 1/96*sqrt(3)*(1/864)^(1/6)*(d^(-6))^(1/6)*log(1/96*(5184*sqrt(
3)*(1/864)^(5/6)*(3*d^5*x^5 + 20*d^5*x^3 + 12*d^5*x)*(d^(-6))^(5/6) + 12*(32*sqrt(3)*(1/864)^(1/6)*d*(d^(-6))^
(1/6)*x^3 + 3*(1/4)^(2/3)*(d^4*x^4 + 28*d^4*x^2 + 4*d^4)*(d^(-6))^(2/3))*(3*x^2 + 2)^(2/3) + (1/4)^(1/3)*(d^2*
x^6 + 1098*d^2*x^4 + 396*d^2*x^2 - 72*d^2)*(d^(-6))^(1/3) + 6*(27*x^4 + sqrt(3)*sqrt(1/6)*(d^3*x^5 + 140*d^3*x
^3 + 36*d^3*x)*sqrt(d^(-6)) + 36*x^2 + 12)*(3*x^2 + 2)^(1/3))/(x^6 + 18*x^4 + 108*x^2 + 216)) - 1/96*sqrt(3)*(
1/864)^(1/6)*(d^(-6))^(1/6)*log(-1/96*(5184*sqrt(3)*(1/864)^(5/6)*(3*d^5*x^5 + 20*d^5*x^3 + 12*d^5*x)*(d^(-6))
^(5/6) + 12*(32*sqrt(3)*(1/864)^(1/6)*d*(d^(-6))^(1/6)*x^3 - 3*(1/4)^(2/3)*(d^4*x^4 + 28*d^4*x^2 + 4*d^4)*(d^(
-6))^(2/3))*(3*x^2 + 2)^(2/3) - (1/4)^(1/3)*(d^2*x^6 + 1098*d^2*x^4 + 396*d^2*x^2 - 72*d^2)*(d^(-6))^(1/3) - 6
*(27*x^4 - sqrt(3)*sqrt(1/6)*(d^3*x^5 + 140*d^3*x^3 + 36*d^3*x)*sqrt(d^(-6)) + 36*x^2 + 12)*(3*x^2 + 2)^(1/3))
/(x^6 + 18*x^4 + 108*x^2 + 216)) - 1/96*sqrt(3)*(1/864)^(1/6)*(d^(-6))^(1/6)*log(-1/24*(5184*sqrt(3)*(1/864)^(
5/6)*(3*d^5*x^5 + 20*d^5*x^3 + 12*d^5*x)*(d^(-6))^(5/6) + 12*(32*sqrt(3)*(1/864)^(1/6)*d*(d^(-6))^(1/6)*x^3 -
3*(1/4)^(2/3)*(d^4*x^4 + 28*d^4*x^2 + 4*d^4)*(d^(-6))^(2/3))*(3*x^2 + 2)^(2/3) - (1/4)^(1/3)*(d^2*x^6 + 1098*d
^2*x^4 + 396*d^2*x^2 - 72*d^2)*(d^(-6))^(1/3) - 6*(27*x^4 - sqrt(3)*sqrt(1/6)*(d^3*x^5 + 140*d^3*x^3 + 36*d^3*
x)*sqrt(d^(-6)) + 36*x^2 + 12)*(3*x^2 + 2)^(1/3))/(x^6 + 18*x^4 + 108*x^2 + 216)) - 1/6*(1/864)^(1/6)*(d^(-6))
^(1/6)*arctan(12*(288*(1/864)^(5/6)*(23*d^5*x^3 + 18*d^5*x)*(3*x^2 + 2)^(2/3)*(d^(-6))^(5/6) + 6*sqrt(1/6)*sqr
t((1/4)^(1/3)*d^2*(d^(-6))^(1/3))*(12*(1/864)^(5/6)*(d^5*x^6 + 642*d^5*x^4 - 36*d^5*x^2 - 72*d^5)*(d^(-6))^(5/
6) + sqrt(1/6)*(13*d^3*x^4 + 12*d^3)*(3*x^2 + 2)^(1/3)*sqrt(d^(-6)) + (1/864)^(1/6)*(d*x^4 + 48*d*x^2 + 12*d)*
(3*x^2 + 2)^(2/3)*(d^(-6))^(1/6)) + sqrt(1/6)*(11*d^3*x^5 + 20*d^3*x^3 - 36*d^3*x)*sqrt(d^(-6)) + (1/864)^(1/6
)*(d*x^5 - 160*d*x^3 - 36*d*x)*(3*x^2 + 2)^(1/3)*(d^(-6))^(1/6))/(x^6 - 1314*x^4 - 756*x^2 - 216)) + 1/12*(1/8
64)^(1/6)*(d^(-6))^(1/6)*arctan(-(24*sqrt(1/6)*(72*(1/864)^(5/6)*(d^5*x^12 + 33744*d^5*x^10 - 830844*d^5*x^8 -
1118016*d^5*x^6 - 1364688*d^5*x^4 + 139968*d^5)*(d^(-6))^(5/6) - 2*sqrt(3)*(1/4)^(1/3)*(43*d^2*x^11 + 9370*d^
2*x^9 - 250632*d^2*x^7 - 387504*d^2*x^5 - 177552*d^2*x^3 + 23328*d^2*x)*(d^(-6))^(1/3) - 24*(8*sqrt(3)*(1/4)^(
2/3)*(11*d^4*x^9 - 174*d^4*x^7 - 2700*d^4*x^5 - 648*d^4*x^3)*(d^(-6))^(2/3) - (1/864)^(1/6)*(d*x^10 + 1080*d*x
^8 - 28296*d*x^6 - 12960*d*x^4 + 11664*d*x^2)*(d^(-6))^(1/6))*(3*x^2 + 2)^(2/3) + (3*x^2 + 2)^(1/3)*(6*sqrt(1/
6)*(151*d^3*x^10 + 4698*d^3*x^8 - 219816*d^3*x^6 - 138672*d^3*x^4 - 58320*d^3*x^2 + 23328*d^3)*sqrt(d^(-6)) -
sqrt(3)*(x^11 + 5990*x^9 - 155160*x^7 + 34992*x^5 + 6480*x^3 - 23328*x)))*sqrt((5184*sqrt(3)*(1/864)^(5/6)*(3*
d^5*x^5 + 20*d^5*x^3 + 12*d^5*x)*(d^(-6))^(5/6) + 12*(32*sqrt(3)*(1/864)^(1/6)*d*(d^(-6))^(1/6)*x^3 + 3*(1/4)^
(2/3)*(d^4*x^4 + 28*d^4*x^2 + 4*d^4)*(d^(-6))^(2/3))*(3*x^2 + 2)^(2/3) + (1/4)^(1/3)*(d^2*x^6 + 1098*d^2*x^4 +
396*d^2*x^2 - 72*d^2)*(d^(-6))^(1/3) + 6*(27*x^4 + sqrt(3)*sqrt(1/6)*(d^3*x^5 + 140*d^3*x^3 + 36*d^3*x)*sqrt(
d^(-6)) + 36*x^2 + 12)*(3*x^2 + 2)^(1/3))/(x^6 + 18*x^4 + 108*x^2 + 216)) + 192*sqrt(1/6)*(4*d^3*x^11 - 905*d^
3*x^9 + 11016*d^3*x^7 - 1944*d^3*x^5 - 2592*d^3*x^3 - 11664*d^3*x)*sqrt(d^(-6)) + 24*(576*(1/864)^(5/6)*(97*d^
5*x^9 - 4644*d^5*x^7 + 33696*d^5*x^5 + 1296*d^5*x^3 - 11664*d^5*x)*(d^(-6))^(5/6) - sqrt(3)*(1/4)^(1/3)*(d^2*x
^10 + 366*d^2*x^8 - 3576*d^2*x^6 + 38448*d^2*x^4 - 3888*d^2*x^2 + 7776*d^2)*(d^(-6))^(1/3))*(3*x^2 + 2)^(2/3)
- sqrt(3)*(x^12 - 156*x^10 + 101916*x^8 - 359712*x^6 + 1346544*x^4 + 793152*x^2 + 46656) - 24*(2*sqrt(3)*(1/4)
^(2/3)*(37*d^4*x^10 - 450*d^4*x^8 + 15432*d^4*x^6 - 28944*d^4*x^4 + 11664*d^4*x^2 + 7776*d^4)*(d^(-6))^(2/3) -
(1/864)^(1/6)*(d*x^11 - 4282*d*x^9 + 100008*d*x^7 - 327888*d*x^5 - 304560*d*x^3 - 23328*d*x)*(d^(-6))^(1/6))*
(3*x^2 + 2)^(1/3))/(x^12 - 32652*x^10 + 1036476*x^8 - 4865184*x^6 - 4770576*x^4 - 1259712*x^2 + 419904)) - 1/1
2*(1/864)^(1/6)*(d^(-6))^(1/6)*arctan((24*sqrt(1/6)*(72*(1/864)^(5/6)*(d^5*x^12 + 33744*d^5*x^10 - 830844*d^5*
x^8 - 1118016*d^5*x^6 - 1364688*d^5*x^4 + 139968*d^5)*(d^(-6))^(5/6) + 2*sqrt(3)*(1/4)^(1/3)*(43*d^2*x^11 + 93
70*d^2*x^9 - 250632*d^2*x^7 - 387504*d^2*x^5 - 177552*d^2*x^3 + 23328*d^2*x)*(d^(-6))^(1/3) + 24*(8*sqrt(3)*(1
/4)^(2/3)*(11*d^4*x^9 - 174*d^4*x^7 - 2700*d^4*x^5 - 648*d^4*x^3)*(d^(-6))^(2/3) + (1/864)^(1/6)*(d*x^10 + 108
0*d*x^8 - 28296*d*x^6 - 12960*d*x^4 + 11664*d*x...
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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 (d\,x^2+6\,d\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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