3.24.0 ∫ (−1+x4) 4 √ _____ x2 + x6 1−x4+x8 dx [2362]

Integrand size = 29, antiderivative size = 187 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1-x^4+x^8} \, dx=\frac {\arctan \left (\frac {\sqrt [8]{3} x}{\sqrt [4]{x^2+x^6}}\right )}{2\ 3^{3/8}}-\frac {\arctan \left (\frac {\sqrt {2} 3^{7/8} x \sqrt [4]{x^2+x^6}}{-3 x^2+3^{3/4} \sqrt {x^2+x^6}}\right )}{2 \sqrt {2} 3^{3/8}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{3} x}{\sqrt [4]{x^2+x^6}}\right )}{2\ 3^{3/8}}+\frac {\text {arctanh}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2}}+\frac {\sqrt {x^2+x^6}}{\sqrt {2} \sqrt [8]{3}}}{x \sqrt [4]{x^2+x^6}}\right )}{2 \sqrt {2} 3^{3/8}} \]

1/6*arctan(3^(1/8)*x/(x^6+x^2)^(1/4))*3^(5/8)-1/12*arctan(2^(1/2)*3^(7/8)*x*(x^6+x^2)^(1/4)/(-3*x^2+3^(3/4)*(x

^6+x^2)^(1/2)))*2^(1/2)*3^(5/8)-1/6*arctanh(3^(1/8)*x/(x^6+x^2)^(1/4))*3^(5/8)+1/12*arctanh((1/2*3^(1/8)*x^2*2

^(1/2)+1/6*(x^6+x^2)^(1/2)*2^(1/2)*3^(7/8))/x/(x^6+x^2)^(1/4))*2^(1/2)*3^(5/8)

Rubi [C] (veriﬁed)

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

Time = 0.33 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2081, 6860, 477, 524} \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1-x^4+x^8} \, dx=-\frac {2 \left (\sqrt {3}+3 i\right ) x \sqrt [4]{x^6+x^2} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^4,\frac {2 x^4}{1-i \sqrt {3}}\right )}{9 \left (\sqrt {3}+i\right ) \sqrt [4]{x^4+1}}-\frac {2 \left (-\sqrt {3}+3 i\right ) x \sqrt [4]{x^6+x^2} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^4,\frac {2 x^4}{1+i \sqrt {3}}\right )}{9 \left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+1}} \]

(-2*(3*I + Sqrt[3])*x*(x^2 + x^6)^(1/4)*AppellF1[3/8, -1/4, 1, 11/8, -x^4, (2*x^4)/(1 - I*Sqrt[3])])/(9*(I + S

qrt[3])*(1 + x^4)^(1/4)) - (2*(3*I - Sqrt[3])*x*(x^2 + x^6)^(1/4)*AppellF1[3/8, -1/4, 1, 11/8, -x^4, (2*x^4)/(

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

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

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

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +

b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{x^2+x^6} \int \frac {\sqrt {x} \left (-1+x^4\right ) \sqrt [4]{1+x^4}}{1-x^4+x^8} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}} \\ & = \frac {\sqrt [4]{x^2+x^6} \int \left (\frac {\left (1+\frac {i}{\sqrt {3}}\right ) \sqrt {x} \sqrt [4]{1+x^4}}{-1-i \sqrt {3}+2 x^4}+\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \sqrt {x} \sqrt [4]{1+x^4}}{-1+i \sqrt {3}+2 x^4}\right ) \, dx}{\sqrt {x} \sqrt [4]{1+x^4}} \\ & = \frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x} \sqrt [4]{1+x^4}}{-1+i \sqrt {3}+2 x^4} \, dx}{3 \sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x} \sqrt [4]{1+x^4}}{-1-i \sqrt {3}+2 x^4} \, dx}{3 \sqrt {x} \sqrt [4]{1+x^4}} \\ & = \frac {\left (2 \left (3-i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{-1+i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (2 \left (3+i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{-1-i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt [4]{1+x^4}} \\ & = -\frac {2 \left (3 i+\sqrt {3}\right ) x \sqrt [4]{x^2+x^6} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^4,\frac {2 x^4}{1-i \sqrt {3}}\right )}{9 \left (i+\sqrt {3}\right ) \sqrt [4]{1+x^4}}-\frac {2 \left (3 i-\sqrt {3}\right ) x \sqrt [4]{x^2+x^6} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^4,\frac {2 x^4}{1+i \sqrt {3}}\right )}{9 \left (i-\sqrt {3}\right ) \sqrt [4]{1+x^4}} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 1.05 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.01 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1-x^4+x^8} \, dx=\frac {\sqrt [4]{x^2+x^6} \left (2 \arctan \left (\frac {\sqrt [8]{3} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} 3^{7/8} \sqrt {x} \sqrt [4]{1+x^4}}{3 x-3^{3/4} \sqrt {1+x^4}}\right )-2 \text {arctanh}\left (\frac {\sqrt [8]{3} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} 3^{7/8} \sqrt {x} \sqrt [4]{1+x^4}}{3 x+3^{3/4} \sqrt {1+x^4}}\right )\right )}{4\ 3^{3/8} \sqrt {x} \sqrt [4]{1+x^4}} \]

((x^2 + x^6)^(1/4)*(2*ArcTan[(3^(1/8)*Sqrt[x])/(1 + x^4)^(1/4)] + Sqrt[2]*ArcTan[(Sqrt[2]*3^(7/8)*Sqrt[x]*(1 +

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

Sqrt[2]*3^(7/8)*Sqrt[x]*(1 + x^4)^(1/4))/(3*x + 3^(3/4)*Sqrt[1 + x^4])]))/(4*3^(3/8)*Sqrt[x]*(1 + x^4)^(1/4))

Maple [C] (veriﬁed)

Time = 53.92 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.20


method	result	size

pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3}}\right )}{4}\)	\(37\)
trager	\(\text {Expression too large to display}\)	\(1494\)

Fricas [C] (veriﬁcation not implemented)

Time = 11.07 (sec) , antiderivative size = 1185, normalized size of antiderivative = 6.34 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1-x^4+x^8} \, dx=\text {Too large to display} \]

(1/432*I - 1/432)*27^(7/8)*sqrt(2)*log(-(2*27^(3/4)*(I*x^7 + I*x^3) - (x^6 + x^2)^(3/4)*((9*I + 9)*27^(1/8)*sq

rt(2)*x^2 + 27^(5/8)*sqrt(2)*(-(I + 1)*x^4 - I - 1)) - 18*sqrt(x^6 + x^2)*(x^5 - sqrt(3)*x^3 + x) + 3*27^(1/4)

*(-I*x^9 - 5*I*x^5 - I*x) - (x^6 + x^2)^(1/4)*(-(I - 1)*27^(7/8)*sqrt(2)*x^4 - 3*27^(3/8)*sqrt(2)*(-(I - 1)*x^

6 - (I - 1)*x^2)))/(x^9 - x^5 + x)) - (1/432*I - 1/432)*27^(7/8)*sqrt(2)*log(-(2*27^(3/4)*(I*x^7 + I*x^3) - (x

^6 + x^2)^(3/4)*(-(9*I + 9)*27^(1/8)*sqrt(2)*x^2 + 27^(5/8)*sqrt(2)*((I + 1)*x^4 + I + 1)) - 18*sqrt(x^6 + x^2

)*(x^5 - sqrt(3)*x^3 + x) + 3*27^(1/4)*(-I*x^9 - 5*I*x^5 - I*x) - (x^6 + x^2)^(1/4)*((I - 1)*27^(7/8)*sqrt(2)*

x^4 - 3*27^(3/8)*sqrt(2)*((I - 1)*x^6 + (I - 1)*x^2)))/(x^9 - x^5 + x)) - (1/432*I + 1/432)*27^(7/8)*sqrt(2)*l

og(-(2*27^(3/4)*(-I*x^7 - I*x^3) - (x^6 + x^2)^(3/4)*(-(9*I - 9)*27^(1/8)*sqrt(2)*x^2 + 27^(5/8)*sqrt(2)*((I -

1)*x^4 + I - 1)) - 18*sqrt(x^6 + x^2)*(x^5 - sqrt(3)*x^3 + x) + 3*27^(1/4)*(I*x^9 + 5*I*x^5 + I*x) - (x^6 + x

^2)^(1/4)*((I + 1)*27^(7/8)*sqrt(2)*x^4 - 3*27^(3/8)*sqrt(2)*((I + 1)*x^6 + (I + 1)*x^2)))/(x^9 - x^5 + x)) +

(1/432*I + 1/432)*27^(7/8)*sqrt(2)*log(-(2*27^(3/4)*(-I*x^7 - I*x^3) - (x^6 + x^2)^(3/4)*((9*I - 9)*27^(1/8)*s

qrt(2)*x^2 + 27^(5/8)*sqrt(2)*(-(I - 1)*x^4 - I + 1)) - 18*sqrt(x^6 + x^2)*(x^5 - sqrt(3)*x^3 + x) + 3*27^(1/4

)*(I*x^9 + 5*I*x^5 + I*x) - (x^6 + x^2)^(1/4)*(-(I + 1)*27^(7/8)*sqrt(2)*x^4 - 3*27^(3/8)*sqrt(2)*(-(I + 1)*x^

6 - (I + 1)*x^2)))/(x^9 - x^5 + x)) - 1/216*27^(7/8)*log((2*27^(3/4)*(x^7 + x^3) + 2*(x^6 + x^2)^(3/4)*(9*27^(

1/8)*x^2 + 27^(5/8)*(x^4 + 1)) + 18*sqrt(x^6 + x^2)*(x^5 + sqrt(3)*x^3 + x) + 3*27^(1/4)*(x^9 + 5*x^5 + x) + 2

*(x^6 + x^2)^(1/4)*(27^(7/8)*x^4 + 3*27^(3/8)*(x^6 + x^2)))/(x^9 - x^5 + x)) + 1/216*27^(7/8)*log((2*27^(3/4)*

(x^7 + x^3) - 2*(x^6 + x^2)^(3/4)*(9*27^(1/8)*x^2 + 27^(5/8)*(x^4 + 1)) + 18*sqrt(x^6 + x^2)*(x^5 + sqrt(3)*x^

3 + x) + 3*27^(1/4)*(x^9 + 5*x^5 + x) - 2*(x^6 + x^2)^(1/4)*(27^(7/8)*x^4 + 3*27^(3/8)*(x^6 + x^2)))/(x^9 - x^

5 + x)) - 1/216*I*27^(7/8)*log(-(2*27^(3/4)*(x^7 + x^3) + 2*(x^6 + x^2)^(3/4)*(9*I*27^(1/8)*x^2 + 27^(5/8)*(I*

x^4 + I)) - 18*sqrt(x^6 + x^2)*(x^5 + sqrt(3)*x^3 + x) + 3*27^(1/4)*(x^9 + 5*x^5 + x) + 2*(x^6 + x^2)^(1/4)*(-

I*27^(7/8)*x^4 + 3*27^(3/8)*(-I*x^6 - I*x^2)))/(x^9 - x^5 + x)) + 1/216*I*27^(7/8)*log(-(2*27^(3/4)*(x^7 + x^3

) + 2*(x^6 + x^2)^(3/4)*(-9*I*27^(1/8)*x^2 + 27^(5/8)*(-I*x^4 - I)) - 18*sqrt(x^6 + x^2)*(x^5 + sqrt(3)*x^3 +

x) + 3*27^(1/4)*(x^9 + 5*x^5 + x) + 2*(x^6 + x^2)^(1/4)*(I*27^(7/8)*x^4 + 3*27^(3/8)*(I*x^6 + I*x^2)))/(x^9 -

Sympy [F]

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

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

Maxima [F]

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

Giac [F]

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

Mupad [F(-1)]

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

\(\int \frac {(-1+x^4) \sqrt [4]{x^2+x^6}}{1-x^4+x^8} \, dx\) [2362]

Optimal result