\(\int \frac {1+x^2}{(-1+x^2) \sqrt [3]{x^2+x^4}} \, dx\) [2237]

Optimal result

Integrand size = 24, antiderivative size = 167 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt [3]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^2+x^4}}\right )}{\sqrt [3]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [3]{2} x^2+\frac {\left (x^2+x^4\right )^{2/3}}{\sqrt [3]{2}}}{x \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt [3]{2}} \]

[Out]

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

Rubi [C] (verified)

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

Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.25, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2081, 477, 440} \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=-\frac {3 x \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},x^2,-x^2\right )}{\sqrt [3]{x^4+x^2}} \]

[In]

Int[(1 + x^2)/((-1 + x^2)*(x^2 + x^4)^(1/3)),x]

[Out]

(-3*x*(1 + x^2)^(1/3)*AppellF1[1/6, 1, -2/3, 7/6, x^2, -x^2])/(x^2 + x^4)^(1/3)

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {\left (1+x^2\right )^{2/3}}{x^{2/3} \left (-1+x^2\right )} \, dx}{\sqrt [3]{x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{-1+x^6} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = -\frac {3 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},x^2,-x^2\right )}{\sqrt [3]{x^2+x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.10 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=-\frac {x^{2/3} \sqrt [3]{1+x^2} \left (\sqrt {3} \left (\arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{-\sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}}\right )+\arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}}\right )\right )+2 \text {arctanh}\left (\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x^2}}\right )+\text {arctanh}\left (\frac {2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}{2 x^{2/3}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}}\right )\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2+x^4}} \]

[In]

Integrate[(1 + x^2)/((-1 + x^2)*(x^2 + x^4)^(1/3)),x]

[Out]

-1/2*(x^(2/3)*(1 + x^2)^(1/3)*(Sqrt[3]*(ArcTan[(Sqrt[3]*x^(1/3))/(-x^(1/3) + 2^(2/3)*(1 + x^2)^(1/3))] + ArcTa
n[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2^(2/3)*(1 + x^2)^(1/3))]) + 2*ArcTanh[(2^(1/3)*x^(1/3))/(1 + x^2)^(1/3)] + Arc
Tanh[(2^(2/3)*x^(1/3)*(1 + x^2)^(1/3))/(2*x^(2/3) + 2^(1/3)*(1 + x^2)^(2/3))]))/(2^(1/3)*(x^2 + x^4)^(1/3))

Maple [A] (verified)

Time = 18.94 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.20

method	result	size
pseudoelliptic	\(-\frac {2^{\frac {2}{3}} \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x \right )}{3 x}\right )-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x \right )}{3 x}\right )-\frac {\ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\frac {\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\ln \left (\frac {2^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x}\right )\right )}{4}\)	\(201\)
trager	\(\text {Expression too large to display}\)	\(3878\)

[In]

int((x^2+1)/(x^2-1)/(x^4+x^2)^(1/3),x,method=_RETURNVERBOSE)

[Out]

-1/4*2^(2/3)*(3^(1/2)*arctan(1/3*3^(1/2)*(-2^(2/3)*(x^2*(x^2+1))^(1/3)+x)/x)-3^(1/2)*arctan(1/3*3^(1/2)*(2^(2/
3)*(x^2*(x^2+1))^(1/3)+x)/x)-1/2*ln((2^(2/3)*x^2-2^(1/3)*(x^2*(x^2+1))^(1/3)*x+(x^2*(x^2+1))^(2/3))/x^2)+1/2*l
n((2^(2/3)*x^2+2^(1/3)*(x^2*(x^2+1))^(1/3)*x+(x^2*(x^2+1))^(2/3))/x^2)+ln((2^(1/3)*x+(x^2*(x^2+1))^(1/3))/x)-l
n((-2^(1/3)*x+(x^2*(x^2+1))^(1/3))/x))

Fricas [A] (verification not implemented)

none

Time = 0.57 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.39 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=-\frac {1}{4} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{5} + 8 \, x^{4} - 2 \, x^{3} + 8 \, x^{2} + x\right )} + 8 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 2 \, x^{2} + x\right )} + 8 \cdot 2^{\frac {1}{6}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} - 2 \, x + 1\right )}\right )}}{6 \, {\left (x^{5} - 8 \, x^{4} - 2 \, x^{3} - 8 \, x^{2} + x\right )}}\right ) + \frac {1}{4} \cdot 2^{\frac {2}{3}} \log \left (-\frac {2^{\frac {2}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} + 4 \cdot 2^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x - 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} + 2 \, x^{2} + x}\right ) - \frac {1}{8} \cdot 2^{\frac {2}{3}} \log \left (\frac {2 \cdot 2^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{3} + 2 \, x^{2} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x}{x^{3} + 2 \, x^{2} + x}\right ) \]

[In]

integrate((x^2+1)/(x^2-1)/(x^4+x^2)^(1/3),x, algorithm="fricas")

[Out]

-1/4*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(x^5 + 8*x^4 - 2*x^3 + 8*x^2 + x) + 8*sqrt(2)*(x^4 +
x^2)^(1/3)*(x^3 + 2*x^2 + x) + 8*2^(1/6)*(x^4 + x^2)^(2/3)*(x^2 - 2*x + 1))/(x^5 - 8*x^4 - 2*x^3 - 8*x^2 + x))
+ 1/4*2^(2/3)*log(-(2^(2/3)*(x^3 - 2*x^2 + x) + 4*2^(1/3)*(x^4 + x^2)^(1/3)*x - 4*(x^4 + x^2)^(2/3))/(x^3 + 2
*x^2 + x)) - 1/8*2^(2/3)*log((2*2^(2/3)*(x^4 + x^2)^(2/3) + 2^(1/3)*(x^3 + 2*x^2 + x) + 4*(x^4 + x^2)^(1/3)*x)
/(x^3 + 2*x^2 + x))

Sympy [F]

\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=\int \frac {x^{2} + 1}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right )}\, dx \]

[In]

integrate((x**2+1)/(x**2-1)/(x**4+x**2)**(1/3),x)

[Out]

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

Maxima [F]

\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}} \,d x } \]

[In]

integrate((x^2+1)/(x^2-1)/(x^4+x^2)^(1/3),x, algorithm="maxima")

[Out]

integrate((x^2 + 1)/((x^4 + x^2)^(1/3)*(x^2 - 1)), x)

Giac [F]

\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}} \,d x } \]

[In]

integrate((x^2+1)/(x^2-1)/(x^4+x^2)^(1/3),x, algorithm="giac")

[Out]

integrate((x^2 + 1)/((x^4 + x^2)^(1/3)*(x^2 - 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=\int \frac {x^2+1}{{\left (x^4+x^2\right )}^{1/3}\,\left (x^2-1\right )} \,d x \]

[In]

int((x^2 + 1)/((x^2 + x^4)^(1/3)*(x^2 - 1)),x)

[Out]

int((x^2 + 1)/((x^2 + x^4)^(1/3)*(x^2 - 1)), x)