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

Optimal. Leaf size=162 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{2 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^6+x^2}}{\sqrt {2} x^2-\sqrt {x^6+x^2}}\right )}{2\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^6+x^2}}{2^{3/4}}}{x \sqrt [4]{x^6+x^2}}\right )}{2\ 2^{3/4}} \]

________________________________________________________________________________________

Rubi [C]  time = 0.10, antiderivative size = 42, normalized size of antiderivative = 0.26, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2056, 466, 429} \begin {gather*} -\frac {2 x \sqrt [4]{x^4+1} F_1\left (\frac {1}{8};1,-\frac {3}{4};\frac {9}{8};x^4,-x^4\right )}{\sqrt [4]{x^6+x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 429

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 466

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 2056

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*} \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {\left (1+x^4\right )^{3/4}}{\sqrt {x} \left (-1+x^4\right )} \, dx}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^8\right )^{3/4}}{-1+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=-\frac {2 x \sqrt [4]{1+x^4} F_1\left (\frac {1}{8};1,-\frac {3}{4};\frac {9}{8};x^4,-x^4\right )}{\sqrt [4]{x^2+x^6}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 42, normalized size = 0.26 \begin {gather*} -\frac {2 x \sqrt [4]{x^4+1} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-x^4,x^4\right )}{\sqrt [4]{x^6+x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.00, size = 162, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{2\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\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 )}{2\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 52.50, size = 1002, normalized size = 6.19

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*2^(3/4)*arctan(1/2*(4*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + 2^(3/4)*(2*2^(3/4)*sqrt(x^6 + x^2)*x + 2^(1/4)*(x^5
+ 2*x^3 + x)) + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 - 2*x^3 + x)) - 1/16*2^(3/4)*log(-(4*sqrt(2)*(x^6 + x^2)^(1
/4)*x^2 + 2^(3/4)*(x^5 + 2*x^3 + x) + 4*2^(1/4)*sqrt(x^6 + x^2)*x + 4*(x^6 + x^2)^(3/4))/(x^5 - 2*x^3 + x)) +
1/16*2^(3/4)*log(-(4*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 - 2^(3/4)*(x^5 + 2*x^3 + x) - 4*2^(1/4)*sqrt(x^6 + x^2)*x +
4*(x^6 + x^2)^(3/4))/(x^5 - 2*x^3 + x)) + 1/4*2^(1/4)*arctan(-1/2*(2*x^9 + 8*x^7 + 12*x^5 + 8*x^3 + 4*2^(3/4)
*(x^6 + x^2)^(3/4)*(x^4 - 6*x^2 + 1) + 8*sqrt(2)*sqrt(x^6 + x^2)*(x^5 + 2*x^3 + x) - sqrt(2)*(32*sqrt(2)*(x^6
+ x^2)^(3/4)*x^2 + 2^(3/4)*(x^9 - 16*x^7 - 2*x^5 - 16*x^3 + x) + 4*2^(1/4)*sqrt(x^6 + x^2)*(x^5 - 6*x^3 + x) +
8*(x^6 + 2*x^4 + x^2)*(x^6 + x^2)^(1/4))*sqrt((4*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(x^5 + 2*x^3 + x) +
8*sqrt(x^6 + x^2)*x + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) + 8*2^(1/4)*(3*x^6 - 2*x^4 + 3*x^2)*(x^6
+ x^2)^(1/4) + 2*x)/(x^9 - 28*x^7 + 6*x^5 - 28*x^3 + x)) - 1/4*2^(1/4)*arctan(-1/2*(2*x^9 + 8*x^7 + 12*x^5 +
8*x^3 - 4*2^(3/4)*(x^6 + x^2)^(3/4)*(x^4 - 6*x^2 + 1) + 8*sqrt(2)*sqrt(x^6 + x^2)*(x^5 + 2*x^3 + x) - sqrt(2)*
(32*sqrt(2)*(x^6 + x^2)^(3/4)*x^2 - 2^(3/4)*(x^9 - 16*x^7 - 2*x^5 - 16*x^3 + x) - 4*2^(1/4)*sqrt(x^6 + x^2)*(x
^5 - 6*x^3 + x) + 8*(x^6 + 2*x^4 + x^2)*(x^6 + x^2)^(1/4))*sqrt(-(4*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 - sqrt(2)*(x
^5 + 2*x^3 + x) - 8*sqrt(x^6 + x^2)*x + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) - 8*2^(1/4)*(3*x^6 - 2
*x^4 + 3*x^2)*(x^6 + x^2)^(1/4) + 2*x)/(x^9 - 28*x^7 + 6*x^5 - 28*x^3 + x)) - 1/16*2^(1/4)*log(8*(4*2^(3/4)*(x
^6 + x^2)^(1/4)*x^2 + sqrt(2)*(x^5 + 2*x^3 + x) + 8*sqrt(x^6 + x^2)*x + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*
x^3 + x)) + 1/16*2^(1/4)*log(-8*(4*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 - sqrt(2)*(x^5 + 2*x^3 + x) - 8*sqrt(x^6 + x^
2)*x + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 19.20, size = 633, normalized size = 3.91

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} \sqrt {x^{6}+x^{2}}\, x +\RootOf \left (\textit {\_Z}^{4}-8\right ) x^{5}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{3}-4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}-8\right ) x}{\left (-1+x \right )^{2} \left (1+x \right )^{2} x}\right )}{8}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \sqrt {x^{6}+x^{2}}\, x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{5}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{\left (-1+x \right )^{2} \left (1+x \right )^{2} x}\right )}{8}+\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}-8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (x^{2}+1\right )}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{3}}{16}+\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}-8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (x^{2}+1\right )}\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2}}{16}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{5}-\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{5}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{3}+8 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{2}-8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \sqrt {x^{6}+x^{2}}\, x -8 \sqrt {x^{6}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}-8\right ) x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x -\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x +16 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{x \left (x^{2}+1\right )^{2}}\right )}{16}\)	\(633\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*RootOf(_Z^4-8)*ln(-(RootOf(_Z^4-8)^3*(x^6+x^2)^(1/2)*x+RootOf(_Z^4-8)*x^5-2*RootOf(_Z^4-8)^2*(x^6+x^2)^(1/
4)*x^2+2*RootOf(_Z^4-8)*x^3-4*(x^6+x^2)^(3/4)+RootOf(_Z^4-8)*x)/(-1+x)^2/(1+x)^2/x)-1/8*RootOf(_Z^2+RootOf(_Z^
4-8)^2)*ln((-RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*(x^6+x^2)^(1/2)*x+RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^
5-2*RootOf(_Z^4-8)^2*(x^6+x^2)^(1/4)*x^2+2*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^3+4*(x^6+x^2)^(3/4)+RootOf(_Z^2+Roo
tOf(_Z^4-8)^2)*x)/(-1+x)^2/(1+x)^2/x)+1/16*ln(-(RootOf(_Z^4-8)^2*x^2-2*RootOf(_Z^4-8)*(x^6+x^2)^(1/4)*x+2*(x^6
+x^2)^(1/2))/x/(x^2+1))*RootOf(_Z^4-8)^3+1/16*ln(-(RootOf(_Z^4-8)^2*x^2-2*RootOf(_Z^4-8)*(x^6+x^2)^(1/4)*x+2*(
x^6+x^2)^(1/2))/x/(x^2+1))*RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2-1/16*RootOf(_Z^2+RootOf(_Z^4-8)^2)*R
ootOf(_Z^4-8)^2*ln(-(RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*x^5-RootOf(_Z^4-8)^3*x^5-2*RootOf(_Z^2+Roo
tOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*x^3+2*RootOf(_Z^4-8)^3*x^3+8*RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)*(x^6
+x^2)^(1/4)*x^2+RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*x-8*RootOf(_Z^2+RootOf(_Z^4-8)^2)*(x^6+x^2)^(1/
2)*x-RootOf(_Z^4-8)^3*x-8*(x^6+x^2)^(1/2)*RootOf(_Z^4-8)*x+16*(x^6+x^2)^(3/4))/x/(x^2+1)^2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4+1}{{\left (x^6+x^2\right )}^{1/4}\,\left (x^4-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________