\(\int \frac {1}{(3-x^2) \sqrt [3]{1+x^2}} \, dx\) [147]

Optimal result

Integrand size = 19, antiderivative size = 109 \[ \int \frac {1}{\left (3-x^2\right ) \sqrt [3]{1+x^2}} \, dx=-\frac {\arctan (x)}{6\ 2^{2/3}}+\frac {\arctan \left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}} \]

[Out]

-1/12*arctan(x)*2^(1/3)+1/4*arctan(x/(1+2^(1/3)*(x^2+1)^(1/3)))*2^(1/3)-1/12*arctanh(3^(1/2)/x)*2^(1/3)*3^(1/2
)-1/12*arctanh((1-2^(1/3)*(x^2+1)^(1/3))*3^(1/2)/x)*2^(1/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {401} \[ \int \frac {1}{\left (3-x^2\right ) \sqrt [3]{1+x^2}} \, dx=\frac {\arctan \left (\frac {x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac {\arctan (x)}{6\ 2^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}} \]

[In]

Int[1/((3 - x^2)*(1 + x^2)^(1/3)),x]

[Out]

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

Rule 401

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b/a, 2]}, Simp[q*(ArcTanh
[Sqrt[3]/(q*x)]/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d)), x] + (-Simp[q*(ArcTan[(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*(ArcTan[q*x]/(6*2^(2/3)*a^(1/3)*d)), x] + Simp[q*(ArcTanh[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] && PosQ[b/a]

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

Mathematica [C] (warning: unable to verify)

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

Time = 0.02 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (3-x^2\right ) \sqrt [3]{1+x^2}} \, dx=-\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-x^2,\frac {x^2}{3}\right )}{\left (-3+x^2\right ) \sqrt [3]{1+x^2} \left (9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-x^2,\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},-x^2,\frac {x^2}{3}\right )-\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},-x^2,\frac {x^2}{3}\right )\right )\right )} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1103 vs. \(2 (77) = 154\).

Time = 0.70 (sec) , antiderivative size = 1103, normalized size of antiderivative = 10.12 \[ \int \frac {1}{\left (3-x^2\right ) \sqrt [3]{1+x^2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/10368*432^(5/6)*(sqrt(-3) + 1)*log((432^(5/6)*(x^6 + 69*x^4 + 63*x^2 + sqrt(-3)*(x^6 + 69*x^4 + 63*x^2 + 27
) + 27) - 1728*(9*x^3 + sqrt(3)*(x^4 + 9*x^2) + 9*x)*(x^2 + 1)^(2/3) + 432*2^(1/3)*(5*x^5 + 30*x^3 + sqrt(-3)*
(5*x^5 + 30*x^3 + 9*x) + 9*x) + 432*(x^2 + 1)^(1/3)*(2^(2/3)*(x^5 + 18*x^3 - sqrt(-3)*(x^5 + 18*x^3 + 9*x) + 9
*x) + 4*432^(1/6)*(x^4 + 3*x^2 - sqrt(-3)*(x^4 + 3*x^2))))/(x^6 - 9*x^4 + 27*x^2 - 27)) + 1/10368*432^(5/6)*(s
qrt(-3) + 1)*log(-(432^(5/6)*(x^6 + 69*x^4 + 63*x^2 + sqrt(-3)*(x^6 + 69*x^4 + 63*x^2 + 27) + 27) + 1728*(9*x^
3 - sqrt(3)*(x^4 + 9*x^2) + 9*x)*(x^2 + 1)^(2/3) - 432*2^(1/3)*(5*x^5 + 30*x^3 + sqrt(-3)*(5*x^5 + 30*x^3 + 9*
x) + 9*x) - 432*(x^2 + 1)^(1/3)*(2^(2/3)*(x^5 + 18*x^3 - sqrt(-3)*(x^5 + 18*x^3 + 9*x) + 9*x) - 4*432^(1/6)*(x
^4 + 3*x^2 - sqrt(-3)*(x^4 + 3*x^2))))/(x^6 - 9*x^4 + 27*x^2 - 27)) + 1/10368*432^(5/6)*(sqrt(-3) - 1)*log((43
2^(5/6)*(x^6 + 69*x^4 + 63*x^2 - sqrt(-3)*(x^6 + 69*x^4 + 63*x^2 + 27) + 27) - 1728*(9*x^3 + sqrt(3)*(x^4 + 9*
x^2) + 9*x)*(x^2 + 1)^(2/3) + 432*2^(1/3)*(5*x^5 + 30*x^3 - sqrt(-3)*(5*x^5 + 30*x^3 + 9*x) + 9*x) + 432*(x^2
+ 1)^(1/3)*(2^(2/3)*(x^5 + 18*x^3 + sqrt(-3)*(x^5 + 18*x^3 + 9*x) + 9*x) + 4*432^(1/6)*(x^4 + 3*x^2 + sqrt(-3)
*(x^4 + 3*x^2))))/(x^6 - 9*x^4 + 27*x^2 - 27)) - 1/10368*432^(5/6)*(sqrt(-3) - 1)*log(-(432^(5/6)*(x^6 + 69*x^
4 + 63*x^2 - sqrt(-3)*(x^6 + 69*x^4 + 63*x^2 + 27) + 27) + 1728*(9*x^3 - sqrt(3)*(x^4 + 9*x^2) + 9*x)*(x^2 + 1
)^(2/3) - 432*2^(1/3)*(5*x^5 + 30*x^3 - sqrt(-3)*(5*x^5 + 30*x^3 + 9*x) + 9*x) - 432*(x^2 + 1)^(1/3)*(2^(2/3)*
(x^5 + 18*x^3 + sqrt(-3)*(x^5 + 18*x^3 + 9*x) + 9*x) - 4*432^(1/6)*(x^4 + 3*x^2 + sqrt(-3)*(x^4 + 3*x^2))))/(x
^6 - 9*x^4 + 27*x^2 - 27)) + 1/5184*432^(5/6)*log(-(432^(5/6)*(x^6 + 69*x^4 + 63*x^2 + 27) + 864*(9*x^3 + sqrt
(3)*(x^4 + 9*x^2) + 9*x)*(x^2 + 1)^(2/3) + 432*2^(1/3)*(5*x^5 + 30*x^3 + 9*x) + 432*(x^2 + 1)^(1/3)*(2^(2/3)*(
x^5 + 18*x^3 + 9*x) + 4*432^(1/6)*(x^4 + 3*x^2)))/(x^6 - 9*x^4 + 27*x^2 - 27)) - 1/5184*432^(5/6)*log((432^(5/
6)*(x^6 + 69*x^4 + 63*x^2 + 27) - 864*(9*x^3 - sqrt(3)*(x^4 + 9*x^2) + 9*x)*(x^2 + 1)^(2/3) - 432*2^(1/3)*(5*x
^5 + 30*x^3 + 9*x) - 432*(x^2 + 1)^(1/3)*(2^(2/3)*(x^5 + 18*x^3 + 9*x) - 4*432^(1/6)*(x^4 + 3*x^2)))/(x^6 - 9*
x^4 + 27*x^2 - 27))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(-1/((x^2 + 1)^(1/3)*(x^2 - 3)),x)

[Out]

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