\(\int \frac {1}{\sqrt [3]{-1+x^2} (3+x^2)} \, dx\) [2626]

Optimal result

Integrand size = 17, antiderivative size = 231 \[ \int \frac {1}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx=\frac {\arctan \left (\frac {2^{2/3} x}{2^{2/3} \sqrt {3}+2 \sqrt {3} \sqrt [3]{-1+x^2}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {3\ 2^{2/3} x \sqrt [3]{-1+x^2}}{-3 \sqrt [3]{2}-\sqrt [3]{2} x^2+3\ 2^{2/3} \sqrt [3]{-1+x^2}-6 \left (-1+x^2\right )^{2/3}}\right )}{6\ 2^{2/3}}-\frac {i \text {arctanh}\left (\frac {2 i \sqrt [3]{2} \sqrt {3} x-i 2^{2/3} \sqrt {3} x \sqrt [3]{-1+x^2}}{-3 \sqrt [3]{2}+\sqrt [3]{2} x^2+3\ 2^{2/3} \sqrt [3]{-1+x^2}-6 \left (-1+x^2\right )^{2/3}}\right )}{6\ 2^{2/3} \sqrt {3}} \]

[Out]

1/18*arctan(2^(2/3)*x/(2^(2/3)*3^(1/2)+2*(x^2-1)^(1/3)*3^(1/2)))*2^(1/3)*3^(1/2)-1/12*arctanh(3*2^(2/3)*x*(x^2
-1)^(1/3)/(-3*2^(1/3)-2^(1/3)*x^2+3*2^(2/3)*(x^2-1)^(1/3)-6*(x^2-1)^(2/3)))*2^(1/3)-1/36*I*arctanh((2*I*2^(1/3
)*3^(1/2)*x-I*2^(2/3)*3^(1/2)*x*(x^2-1)^(1/3))/(-3*2^(1/3)+2^(1/3)*x^2+3*2^(2/3)*(x^2-1)^(1/3)-6*(x^2-1)^(2/3)
))*2^(1/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.59, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {402} \[ \int \frac {1}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx=-\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3} \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{x^2-1}+1\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{2} \sqrt [3]{x^2-1}+\sqrt [3]{-1}}\right )+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}(x) \]

[In]

Int[1/((-1 + x^2)^(1/3)*(3 + x^2)),x]

[Out]

-1/2*((-1)^(2/3)*ArcTan[Sqrt[3]/x])/(2^(2/3)*Sqrt[3]) - ((-1)^(2/3)*ArcTan[(Sqrt[3]*(1 + (-1)^(2/3)*2^(1/3)*(-
1 + x^2)^(1/3)))/x])/(2*2^(2/3)*Sqrt[3]) + ((-1/2)^(2/3)*ArcTanh[x])/6 - ((-1/2)^(2/3)*ArcTanh[((-1)^(1/3)*x)/
((-1)^(1/3) + 2^(1/3)*(-1 + x^2)^(1/3))])/2

Rule 402

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[q*(ArcTan
[Sqrt[3]/(q*x)]/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d)), x] + (Simp[q*(ArcTanh[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a + b*
x^2)^(1/3))]/(2*2^(2/3)*a^(1/3)*d)), x] - Simp[q*(ArcTanh[q*x]/(6*2^(2/3)*a^(1/3)*d)), x] + Simp[q*(ArcTan[Sqr
t[3]*((a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3))/(a^(1/3)*q*x))]/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d)), x])] /; FreeQ[{a,
b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && NegQ[b/a]

Rubi steps \begin{align*} \text {integral}& = -\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3} \left (1+(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{-1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}(x)-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{-1}+\sqrt [3]{2} \sqrt [3]{-1+x^2}}\right ) \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 4.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx=-\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )}{\sqrt [3]{-1+x^2} \left (3+x^2\right ) \left (-9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},x^2,-\frac {x^2}{3}\right )-\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )\right )\right )} \]

[In]

Integrate[1/((-1 + x^2)^(1/3)*(3 + x^2)),x]

[Out]

(-9*x*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/((-1 + x^2)^(1/3)*(3 + x^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, x^
2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2])))

Maple [C] (verified)

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

Time = 10.62 (sec) , antiderivative size = 874, normalized size of antiderivative = 3.78

method	result	size
trager	\(\text {Expression too large to display}\)	\(874\)

[In]

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

[Out]

1/432*ln((RootOf(_Z^6+108)^4*x^6+72*x^5*RootOf(_Z^6+108)^4-36*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^5+225*RootOf(
_Z^6+108)^4*x^4-648*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^4-72*RootOf(_Z^6+108)^4*x^3+648*(x^2-1)^(2/3)*x^4-864*R
ootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^3-189*RootOf(_Z^6+108)^4*x^2+4536*x^3*(x^2-1)^(2/3)+648*RootOf(_Z^6+108)^2*(
x^2-1)^(1/3)*x^2+1944*(x^2-1)^(2/3)*x^2+324*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x+27*RootOf(_Z^6+108)^4-1944*x*(x
^2-1)^(2/3))/(x^2+3)^3)*RootOf(_Z^6+108)^4+1/72*ln((RootOf(_Z^6+108)^4*x^6+72*x^5*RootOf(_Z^6+108)^4-36*RootOf
(_Z^6+108)^2*(x^2-1)^(1/3)*x^5+225*RootOf(_Z^6+108)^4*x^4-648*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^4-72*RootOf(_
Z^6+108)^4*x^3+648*(x^2-1)^(2/3)*x^4-864*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^3-189*RootOf(_Z^6+108)^4*x^2+4536*
x^3*(x^2-1)^(2/3)+648*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^2+1944*(x^2-1)^(2/3)*x^2+324*RootOf(_Z^6+108)^2*(x^2-
1)^(1/3)*x+27*RootOf(_Z^6+108)^4-1944*x*(x^2-1)^(2/3))/(x^2+3)^3)*RootOf(_Z^6+108)+1/36*RootOf(_Z^6+108)*ln((1
8*RootOf(_Z^6+108)*x^6+4050*RootOf(_Z^6+108)*x^4-RootOf(_Z^6+108)^4*x^6-225*RootOf(_Z^6+108)^4*x^4-72*x^5*Root
Of(_Z^6+108)^4+1296*x^5*RootOf(_Z^6+108)+189*RootOf(_Z^6+108)^4*x^2-3402*RootOf(_Z^6+108)*x^2+72*RootOf(_Z^6+1
08)^4*x^3-1296*RootOf(_Z^6+108)*x^3+9072*x^3*(x^2-1)^(2/3)-3888*x*(x^2-1)^(2/3)+1296*(x^2-1)^(2/3)*x^4+3888*(x
^2-1)^(2/3)*x^2-27*RootOf(_Z^6+108)^4+486*RootOf(_Z^6+108)+36*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^5+648*RootOf(
_Z^6+108)^2*(x^2-1)^(1/3)*x^4+864*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^3-648*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^
2-324*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x-6*(x^2-1)^(1/3)*RootOf(_Z^6+108)^5*x^5-108*(x^2-1)^(1/3)*RootOf(_Z^6+
108)^5*x^4-144*(x^2-1)^(1/3)*RootOf(_Z^6+108)^5*x^3+108*(x^2-1)^(1/3)*RootOf(_Z^6+108)^5*x^2+54*(x^2-1)^(1/3)*
RootOf(_Z^6+108)^5*x)/(x^2+3)^3)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1208 vs. \(2 (164) = 328\).

Time = 0.72 (sec) , antiderivative size = 1208, normalized size of antiderivative = 5.23 \[ \int \frac {1}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx=\text {Too large to display} \]

[In]

integrate(1/(x^2-1)^(1/3)/(x^2+3),x, algorithm="fricas")

[Out]

-1/10368*432^(5/6)*(-1)^(1/6)*(sqrt(-3) + 1)*log((432^(5/6)*(-1)^(1/6)*(x^6 - 69*x^4 + 63*x^2 + sqrt(-3)*(x^6
- 69*x^4 + 63*x^2 - 27) - 27) + 432*2^(1/3)*(-1)^(2/3)*(5*x^5 - 30*x^3 + sqrt(-3)*(5*x^5 - 30*x^3 + 9*x) + 9*x
) + 1728*(9*x^3 - sqrt(3)*(I*x^4 - 9*I*x^2) - 9*x)*(x^2 - 1)^(2/3) + 432*(2^(2/3)*(-1)^(1/3)*(x^5 - 18*x^3 - s
qrt(-3)*(x^5 - 18*x^3 + 9*x) + 9*x) + 4*432^(1/6)*(-1)^(5/6)*(x^4 - 3*x^2 - sqrt(-3)*(x^4 - 3*x^2)))*(x^2 - 1)
^(1/3))/(x^6 + 9*x^4 + 27*x^2 + 27)) + 1/10368*432^(5/6)*(-1)^(1/6)*(sqrt(-3) + 1)*log(-(432^(5/6)*(-1)^(1/6)*
(x^6 - 69*x^4 + 63*x^2 + sqrt(-3)*(x^6 - 69*x^4 + 63*x^2 - 27) - 27) - 432*2^(1/3)*(-1)^(2/3)*(5*x^5 - 30*x^3
+ sqrt(-3)*(5*x^5 - 30*x^3 + 9*x) + 9*x) - 1728*(9*x^3 - sqrt(3)*(-I*x^4 + 9*I*x^2) - 9*x)*(x^2 - 1)^(2/3) - 4
32*(2^(2/3)*(-1)^(1/3)*(x^5 - 18*x^3 - sqrt(-3)*(x^5 - 18*x^3 + 9*x) + 9*x) - 4*432^(1/6)*(-1)^(5/6)*(x^4 - 3*
x^2 - sqrt(-3)*(x^4 - 3*x^2)))*(x^2 - 1)^(1/3))/(x^6 + 9*x^4 + 27*x^2 + 27)) + 1/10368*432^(5/6)*(-1)^(1/6)*(s
qrt(-3) - 1)*log((432^(5/6)*(-1)^(1/6)*(x^6 - 69*x^4 + 63*x^2 - sqrt(-3)*(x^6 - 69*x^4 + 63*x^2 - 27) - 27) +
432*2^(1/3)*(-1)^(2/3)*(5*x^5 - 30*x^3 - sqrt(-3)*(5*x^5 - 30*x^3 + 9*x) + 9*x) + 1728*(9*x^3 - sqrt(3)*(I*x^4
- 9*I*x^2) - 9*x)*(x^2 - 1)^(2/3) + 432*(2^(2/3)*(-1)^(1/3)*(x^5 - 18*x^3 + sqrt(-3)*(x^5 - 18*x^3 + 9*x) + 9
*x) + 4*432^(1/6)*(-1)^(5/6)*(x^4 - 3*x^2 + sqrt(-3)*(x^4 - 3*x^2)))*(x^2 - 1)^(1/3))/(x^6 + 9*x^4 + 27*x^2 +
27)) - 1/10368*432^(5/6)*(-1)^(1/6)*(sqrt(-3) - 1)*log(-(432^(5/6)*(-1)^(1/6)*(x^6 - 69*x^4 + 63*x^2 - sqrt(-3
)*(x^6 - 69*x^4 + 63*x^2 - 27) - 27) - 432*2^(1/3)*(-1)^(2/3)*(5*x^5 - 30*x^3 - sqrt(-3)*(5*x^5 - 30*x^3 + 9*x
) + 9*x) - 1728*(9*x^3 - sqrt(3)*(-I*x^4 + 9*I*x^2) - 9*x)*(x^2 - 1)^(2/3) - 432*(2^(2/3)*(-1)^(1/3)*(x^5 - 18
*x^3 + sqrt(-3)*(x^5 - 18*x^3 + 9*x) + 9*x) - 4*432^(1/6)*(-1)^(5/6)*(x^4 - 3*x^2 + sqrt(-3)*(x^4 - 3*x^2)))*(
x^2 - 1)^(1/3))/(x^6 + 9*x^4 + 27*x^2 + 27)) + 1/5184*432^(5/6)*(-1)^(1/6)*log(-(432^(5/6)*(-1)^(1/6)*(x^6 - 6
9*x^4 + 63*x^2 - 27) + 432*2^(1/3)*(-1)^(2/3)*(5*x^5 - 30*x^3 + 9*x) - 864*(9*x^3 - sqrt(3)*(I*x^4 - 9*I*x^2)
- 9*x)*(x^2 - 1)^(2/3) + 432*(2^(2/3)*(-1)^(1/3)*(x^5 - 18*x^3 + 9*x) + 4*432^(1/6)*(-1)^(5/6)*(x^4 - 3*x^2))*
(x^2 - 1)^(1/3))/(x^6 + 9*x^4 + 27*x^2 + 27)) - 1/5184*432^(5/6)*(-1)^(1/6)*log((432^(5/6)*(-1)^(1/6)*(x^6 - 6
9*x^4 + 63*x^2 - 27) - 432*2^(1/3)*(-1)^(2/3)*(5*x^5 - 30*x^3 + 9*x) + 864*(9*x^3 - sqrt(3)*(-I*x^4 + 9*I*x^2)
- 9*x)*(x^2 - 1)^(2/3) - 432*(2^(2/3)*(-1)^(1/3)*(x^5 - 18*x^3 + 9*x) - 4*432^(1/6)*(-1)^(5/6)*(x^4 - 3*x^2))
*(x^2 - 1)^(1/3))/(x^6 + 9*x^4 + 27*x^2 + 27))

Sympy [F]

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

[In]

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

[Out]

Integral(1/(((x - 1)*(x + 1))**(1/3)*(x**2 + 3)), x)

Maxima [F]

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

[In]

integrate(1/(x^2-1)^(1/3)/(x^2+3),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(1/(x^2-1)^(1/3)/(x^2+3),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/((x^2 - 1)^(1/3)*(x^2 + 3)),x)

[Out]

int(1/((x^2 - 1)^(1/3)*(x^2 + 3)), x)