\(\int \frac {x^4}{\sqrt [4]{x^2+x^6} (-1+x^8)} \, dx\) [2361]

Optimal result

Integrand size = 22, antiderivative size = 187 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\frac {\left (x^2+x^6\right )^{3/4}}{2 x \left (1+x^4\right )}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{8 \sqrt [4]{2}}+\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{8\ 2^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{8 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{8\ 2^{3/4}} \]

[Out]

1/2*(x^6+x^2)^(3/4)/x/(x^4+1)-1/16*arctan(2^(1/4)*x/(x^6+x^2)^(1/4))*2^(3/4)+1/16*arctan(2^(3/4)*x*(x^6+x^2)^(
1/4)/(2^(1/2)*x^2-(x^6+x^2)^(1/2)))*2^(1/4)-1/16*arctanh(2^(1/4)*x/(x^6+x^2)^(1/4))*2^(3/4)-1/16*arctanh((1/2*
x^2*2^(3/4)+1/2*(x^6+x^2)^(1/2)*2^(1/4))/x/(x^6+x^2)^(1/4))*2^(1/4)

Rubi [C] (verified)

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

Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.25, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2081, 1493, 477, 524} \[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=-\frac {2 x^5 \sqrt [4]{x^4+1} \operatorname {AppellF1}\left (\frac {9}{8},1,\frac {5}{4},\frac {17}{8},x^4,-x^4\right )}{9 \sqrt [4]{x^6+x^2}} \]

[In]

Int[x^4/((x^2 + x^6)^(1/4)*(-1 + x^8)),x]

[Out]

(-2*x^5*(1 + x^4)^(1/4)*AppellF1[9/8, 1, 5/4, 17/8, x^4, -x^4])/(9*(x^2 + x^6)^(1/4))

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 524

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

Rule 1493

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(f*x)^
m*(d + e*x^n)^(q + p)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, f, q, m, n, q}, x] && EqQ[n2, 2*n] && EqQ[
c*d^2 + a*e^2, 0] && IntegerQ[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 (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/2}}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx}{\sqrt [4]{x^2+x^6}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/2}}{\left (-1+x^4\right ) \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x^2+x^6}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^8}{\left (-1+x^8\right ) \left (1+x^8\right )^{5/4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}} \\ & = -\frac {2 x^5 \sqrt [4]{1+x^4} \operatorname {AppellF1}\left (\frac {9}{8},1,\frac {5}{4},\frac {17}{8},x^4,-x^4\right )}{9 \sqrt [4]{x^2+x^6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.17 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\frac {\sqrt {x} \left (8 \sqrt {x}-2^{3/4} \sqrt [4]{1+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt [4]{2} \sqrt [4]{1+x^4} \arctan \left (\frac {2^{3/4} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt {2} x-\sqrt {1+x^4}}\right )-2^{3/4} \sqrt [4]{1+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \sqrt [4]{1+x^4} \text {arctanh}\left (\frac {2 \sqrt [4]{2} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+\sqrt {2} \sqrt {1+x^4}}\right )\right )}{16 \sqrt [4]{x^2+x^6}} \]

[In]

Integrate[x^4/((x^2 + x^6)^(1/4)*(-1 + x^8)),x]

[Out]

(Sqrt[x]*(8*Sqrt[x] - 2^(3/4)*(1 + x^4)^(1/4)*ArcTan[(2^(1/4)*Sqrt[x])/(1 + x^4)^(1/4)] + 2^(1/4)*(1 + x^4)^(1
/4)*ArcTan[(2^(3/4)*Sqrt[x]*(1 + x^4)^(1/4))/(Sqrt[2]*x - Sqrt[1 + x^4])] - 2^(3/4)*(1 + x^4)^(1/4)*ArcTanh[(2
^(1/4)*Sqrt[x])/(1 + x^4)^(1/4)] - 2^(1/4)*(1 + x^4)^(1/4)*ArcTanh[(2*2^(1/4)*Sqrt[x]*(1 + x^4)^(1/4))/(2*x +
Sqrt[2]*Sqrt[1 + x^4])]))/(16*(x^2 + x^6)^(1/4))

Maple [A] (verified)

Time = 52.77 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.50

method	result	size
pseudoelliptic	\(\frac {2 \arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right ) 2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+\ln \left (\frac {-2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}{2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}\right ) 2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+x}{x}\right ) 2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-x}{x}\right ) 2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+16 x}{32 \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\)	\(281\)
risch	\(\text {Expression too large to display}\)	\(645\)
trager	\(\text {Expression too large to display}\)	\(654\)

[In]

int(x^4/(x^6+x^2)^(1/4)/(x^8-1),x,method=_RETURNVERBOSE)

[Out]

1/32*(2*arctan(1/2*(x^2*(x^4+1))^(1/4)/x*2^(3/4))*2^(3/4)*(x^2*(x^4+1))^(1/4)-ln((-2^(1/4)*x-(x^2*(x^4+1))^(1/
4))/(2^(1/4)*x-(x^2*(x^4+1))^(1/4)))*2^(3/4)*(x^2*(x^4+1))^(1/4)+ln((-2^(3/4)*(x^2*(x^4+1))^(1/4)*x+2^(1/2)*x^
2+(x^2*(x^4+1))^(1/2))/(2^(3/4)*(x^2*(x^4+1))^(1/4)*x+2^(1/2)*x^2+(x^2*(x^4+1))^(1/2)))*2^(1/4)*(x^2*(x^4+1))^
(1/4)+2*arctan((2^(1/4)*(x^2*(x^4+1))^(1/4)+x)/x)*2^(1/4)*(x^2*(x^4+1))^(1/4)+2*arctan((2^(1/4)*(x^2*(x^4+1))^
(1/4)-x)/x)*2^(1/4)*(x^2*(x^4+1))^(1/4)+16*x)/(x^2*(x^4+1))^(1/4)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.65 (sec) , antiderivative size = 718, normalized size of antiderivative = 3.84 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=-\frac {2^{\frac {3}{4}} {\left (x^{5} + x\right )} \log \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) - 2^{\frac {3}{4}} {\left (x^{5} + x\right )} \log \left (-\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} - 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) + 2^{\frac {3}{4}} {\left (-i \, x^{5} - i \, x\right )} \log \left (\frac {4 i \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2 i \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) + 2^{\frac {3}{4}} {\left (i \, x^{5} + i \, x\right )} \log \left (\frac {-4 i \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 i \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) - 2^{\frac {1}{4}} {\left (\left (i - 1\right ) \, x^{5} + \left (i - 1\right ) \, x\right )} \log \left (\frac {\left (2 i + 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (-i \, x^{5} + 2 i \, x^{3} - i \, x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x - \left (2 i - 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}\right ) - 2^{\frac {1}{4}} {\left (-\left (i + 1\right ) \, x^{5} - \left (i + 1\right ) \, x\right )} \log \left (\frac {-\left (2 i - 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (i \, x^{5} - 2 i \, x^{3} + i \, x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x + \left (2 i + 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}\right ) - 2^{\frac {1}{4}} {\left (\left (i + 1\right ) \, x^{5} + \left (i + 1\right ) \, x\right )} \log \left (\frac {\left (2 i - 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (i \, x^{5} - 2 i \, x^{3} + i \, x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x - \left (2 i + 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}\right ) - 2^{\frac {1}{4}} {\left (-\left (i - 1\right ) \, x^{5} - \left (i - 1\right ) \, x\right )} \log \left (\frac {-\left (2 i + 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (-i \, x^{5} + 2 i \, x^{3} - i \, x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x + \left (2 i - 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}\right ) - 32 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{64 \, {\left (x^{5} + x\right )}} \]

[In]

integrate(x^4/(x^6+x^2)^(1/4)/(x^8-1),x, algorithm="fricas")

[Out]

-1/64*(2^(3/4)*(x^5 + x)*log((4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 2*2^(3/4)*(x^6 + x^2)^(3/4) + sqrt(2)*(x^5 + 2
*x^3 + x) + 4*sqrt(x^6 + x^2)*x)/(x^5 - 2*x^3 + x)) - 2^(3/4)*(x^5 + x)*log(-(4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2
+ 2*2^(3/4)*(x^6 + x^2)^(3/4) - sqrt(2)*(x^5 + 2*x^3 + x) - 4*sqrt(x^6 + x^2)*x)/(x^5 - 2*x^3 + x)) + 2^(3/4)*
(-I*x^5 - I*x)*log((4*I*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 - 2*I*2^(3/4)*(x^6 + x^2)^(3/4) - sqrt(2)*(x^5 + 2*x^3 +
x) + 4*sqrt(x^6 + x^2)*x)/(x^5 - 2*x^3 + x)) + 2^(3/4)*(I*x^5 + I*x)*log((-4*I*2^(1/4)*(x^6 + x^2)^(1/4)*x^2
+ 2*I*2^(3/4)*(x^6 + x^2)^(3/4) - sqrt(2)*(x^5 + 2*x^3 + x) + 4*sqrt(x^6 + x^2)*x)/(x^5 - 2*x^3 + x)) - 2^(1/4
)*((I - 1)*x^5 + (I - 1)*x)*log(((2*I + 2)*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(-I*x^5 + 2*I*x^3 - I*x) +
4*sqrt(x^6 + x^2)*x - (2*I - 2)*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) - 2^(1/4)*(-(I + 1)*x^5 - (I + 1
)*x)*log((-(2*I - 2)*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(I*x^5 - 2*I*x^3 + I*x) + 4*sqrt(x^6 + x^2)*x + (
2*I + 2)*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) - 2^(1/4)*((I + 1)*x^5 + (I + 1)*x)*log(((2*I - 2)*2^(3
/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(I*x^5 - 2*I*x^3 + I*x) + 4*sqrt(x^6 + x^2)*x - (2*I + 2)*2^(1/4)*(x^6 + x
^2)^(3/4))/(x^5 + 2*x^3 + x)) - 2^(1/4)*(-(I - 1)*x^5 - (I - 1)*x)*log((-(2*I + 2)*2^(3/4)*(x^6 + x^2)^(1/4)*x
^2 + sqrt(2)*(-I*x^5 + 2*I*x^3 - I*x) + 4*sqrt(x^6 + x^2)*x + (2*I - 2)*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^
3 + x)) - 32*(x^6 + x^2)^(3/4))/(x^5 + x)

Sympy [F]

\[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\int \frac {x^{4}}{\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \]

[In]

integrate(x**4/(x**6+x**2)**(1/4)/(x**8-1),x)

[Out]

Integral(x**4/((x**2*(x**4 + 1))**(1/4)*(x - 1)*(x + 1)*(x**2 + 1)*(x**4 + 1)), x)

Maxima [F]

\[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\int { \frac {x^{4}}{{\left (x^{8} - 1\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x^4/(x^6+x^2)^(1/4)/(x^8-1),x, algorithm="maxima")

[Out]

integrate(x^4/((x^8 - 1)*(x^6 + x^2)^(1/4)), x)

Giac [F]

\[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\int { \frac {x^{4}}{{\left (x^{8} - 1\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x^4/(x^6+x^2)^(1/4)/(x^8-1),x, algorithm="giac")

[Out]

integrate(x^4/((x^8 - 1)*(x^6 + x^2)^(1/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\int \frac {x^4}{{\left (x^6+x^2\right )}^{1/4}\,\left (x^8-1\right )} \,d x \]

[In]

int(x^4/((x^2 + x^6)^(1/4)*(x^8 - 1)),x)

[Out]

int(x^4/((x^2 + x^6)^(1/4)*(x^8 - 1)), x)