∫ −1+x4 ___________________________(1+x4 ) 4√____

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

Optimal. Leaf size=452 \[ \frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (-2 x^2+2^{3/4} \sqrt {2+\sqrt {2}} \sqrt [4]{x^6-x^2} x-\sqrt {2} \sqrt {x^6-x^2}\right )-\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (2 \sqrt {2-\sqrt {2}} x^2+2 \sqrt [4]{2} \sqrt [4]{x^6-x^2} x+\sqrt {4-2 \sqrt {2}} \sqrt {x^6-x^2}\right )-\frac {1}{4} \sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x}{2^{3/4} \sqrt [4]{x^6-x^2}-\sqrt {2+\sqrt {2}} x}\right )-\frac {1}{4} \sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x}{2^{3/4} \sqrt [4]{x^6-x^2}+\sqrt {2+\sqrt {2}} x}\right )-\frac {1}{4} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{x^6-x^2}}{\sqrt {2} \sqrt {x^6-x^2}-2 x^2}\right )-\frac {1}{4} \sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{2} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {x^6-x^2}}{\sqrt [4]{2} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{x^6-x^2}}\right ) \]

________________________________________________________________________________________

Rubi [C] time = 0.11, antiderivative size = 46, normalized size of antiderivative = 0.10, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2056, 466, 430, 429} \begin {gather*} -\frac {2 x \sqrt [4]{1-x^4} 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]

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, -3/4, 1, 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 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F

racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,

p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && !(IntegerQ[p] || GtQ[a, 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 {\left (2 \sqrt {x} \left (-1+x^4\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^8\right )^{3/4}}{1+x^8} \, dx,x,\sqrt {x}\right )}{\left (1-x^4\right )^{3/4} \sqrt [4]{-x^2+x^6}}\\ &=-\frac {2 x \sqrt [4]{1-x^4} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};x^4,-x^4\right )}{\sqrt [4]{-x^2+x^6}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] time = 0.03, size = 48, normalized size = 0.11 \begin {gather*} \frac {2 \left (x^2 \left (x^4-1\right )\right )^{3/4} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};x^4,-x^4\right )}{x \left (1-x^4\right )^{3/4}} \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^2*(-1 + x^4))^(3/4)*AppellF1[1/8, -3/4, 1, 9/8, x^4, -x^4])/(x*(1 - x^4)^(3/4))

________________________________________________________________________________________

IntegrateAlgebraic [C] time = 0.78, size = 153, normalized size = 0.34 \begin {gather*} -\frac {1}{2} \sqrt {-\frac {1}{2}+\frac {i}{2}} \tan ^{-1}\left (\frac {\sqrt {-1-i} x}{\sqrt [4]{-x^2+x^6}}\right )-\frac {1}{2} \sqrt {-\frac {1}{2}-\frac {i}{2}} \tan ^{-1}\left (\frac {\sqrt {-1+i} x}{\sqrt [4]{-x^2+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2}+\frac {i}{2}} \tan ^{-1}\left (\frac {\sqrt {1-i} x}{\sqrt [4]{-x^2+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2}-\frac {i}{2}} \tan ^{-1}\left (\frac {\sqrt {1+i} x}{\sqrt [4]{-x^2+x^6}}\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

rcTan[(Sqrt[1 + I]*x)/(-x^2 + x^6)^(1/4)])/2

________________________________________________________________________________________

fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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="fricas")

[Out]

Timed out

________________________________________________________________________________________

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 = 99.30, size = 2857, normalized size = 6.32


method	result	size

trager	\(\text {Expression too large to display}\)	\(2857\)

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]

1/4*RootOf(2*_Z^4-2*_Z^2+1)*ln(-(-12032*RootOf(2*_Z^4-2*_Z^2+1)^5*x^5+45120*RootOf(2*_Z^4-2*_Z^2+1)^5*x^3-6464

*RootOf(2*_Z^4-2*_Z^2+1)^3*x^5-21964*(x^6-x^2)^(1/2)*RootOf(2*_Z^4-2*_Z^2+1)^3*x+12032*RootOf(2*_Z^4-2*_Z^2+1)

^5*x-30092*RootOf(2*_Z^4-2*_Z^2+1)^3*x^3-525*RootOf(2*_Z^4-2*_Z^2+1)*x^5+36992*(x^6-x^2)^(3/4)*RootOf(2*_Z^4-2

*_Z^2+1)^2+21964*(x^6-x^2)^(1/4)*RootOf(2*_Z^4-2*_Z^2+1)^2*x^2-15028*(x^6-x^2)^(1/2)*RootOf(2*_Z^4-2*_Z^2+1)*x

+6464*RootOf(2*_Z^4-2*_Z^2+1)^3*x-3450*RootOf(2*_Z^4-2*_Z^2+1)*x^3-10982*(x^6-x^2)^(3/4)+15028*(x^6-x^2)^(1/4)

*x^2+525*RootOf(2*_Z^4-2*_Z^2+1)*x)/(2*RootOf(2*_Z^4-2*_Z^2+1)^2*x^2-8*RootOf(2*_Z^4-2*_Z^2+1)^2-5*x^2+3)^2/x)

-1/4*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*ln((-12032*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^

4-2*_Z^2+1)^4*x^5+45120*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^2+1)^4*x^3+30528*RootOf(Ro

otOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^2+1)^2*x^5+21964*RootOf(2*_Z^4-2*_Z^2+1)^2*(x^6-x^2)^(1/2)*

RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*x+12032*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^2

+1)^4*x-60148*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^2+1)^2*x^3-19021*RootOf(RootOf(2*_Z^

4-2*_Z^2+1)^2+_Z^2-1)*x^5+36992*(x^6-x^2)^(3/4)*RootOf(2*_Z^4-2*_Z^2+1)^2+21964*(x^6-x^2)^(1/4)*RootOf(2*_Z^4-

2*_Z^2+1)^2*x^2-36992*(x^6-x^2)^(1/2)*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*x-30528*RootOf(RootOf(2*_Z^4-2*

_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^2+1)^2*x+11578*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*x^3-26010*(x^6-x^

2)^(3/4)-36992*(x^6-x^2)^(1/4)*x^2+19021*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*x)/(2*RootOf(2*_Z^4-2*_Z^2+1

)^2*x^2-8*RootOf(2*_Z^4-2*_Z^2+1)^2+3*x^2+5)^2/x)-1/2*RootOf(2*_Z^4-2*_Z^2+1)^2*RootOf(RootOf(2*_Z^4-2*_Z^2+1)

^2+_Z^2-1)*ln((14432*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^2+1)^4*x^5-54120*RootOf(RootO

f(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^2+1)^4*x^3-21946*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*Root

Of(2*_Z^4-2*_Z^2+1)^2*x^5+52020*RootOf(2*_Z^4-2*_Z^2+1)^2*(x^6-x^2)^(1/2)*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^

2-1)*x-14432*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^2+1)^4*x+17128*RootOf(RootOf(2*_Z^4-2

*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^2+1)^2*x^3+1725*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*x^5+36992*(x^6-

x^2)^(3/4)*RootOf(2*_Z^4-2*_Z^2+1)^2-21964*(x^6-x^2)^(1/4)*RootOf(2*_Z^4-2*_Z^2+1)^2*x^2-15028*(x^6-x^2)^(1/2)

*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*x+21946*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^

2+1)^2*x-1050*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*x^3-26010*(x^6-x^2)^(3/4)+36992*(x^6-x^2)^(1/4)*x^2-172

5*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*x)/(2*RootOf(2*_Z^4-2*_Z^2+1)^2*x^2-8*RootOf(2*_Z^4-2*_Z^2+1)^2+3*x

^2+5)^2/x)+1/4*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*ln((14432*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*Roo

tOf(2*_Z^4-2*_Z^2+1)^4*x^5-54120*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^2+1)^4*x^3-21946*

RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^2+1)^2*x^5+52020*RootOf(2*_Z^4-2*_Z^2+1)^2*(x^6-x^

2)^(1/2)*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*x-14432*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z

^4-2*_Z^2+1)^4*x+17128*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^2+1)^2*x^3+1725*RootOf(Root

Of(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*x^5+36992*(x^6-x^2)^(3/4)*RootOf(2*_Z^4-2*_Z^2+1)^2-21964*(x^6-x^2)^(1/4)*RootOf

(2*_Z^4-2*_Z^2+1)^2*x^2-15028*(x^6-x^2)^(1/2)*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*x+21946*RootOf(RootOf(2

*_Z^4-2*_Z^2+1)^2+_Z^2-1)*RootOf(2*_Z^4-2*_Z^2+1)^2*x-1050*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*x^3-26010*

(x^6-x^2)^(3/4)+36992*(x^6-x^2)^(1/4)*x^2-1725*RootOf(RootOf(2*_Z^4-2*_Z^2+1)^2+_Z^2-1)*x)/(2*RootOf(2*_Z^4-2*

_Z^2+1)^2*x^2-8*RootOf(2*_Z^4-2*_Z^2+1)^2+3*x^2+5)^2/x)-1/2*RootOf(2*_Z^4-2*_Z^2+1)^3*ln(-(14432*RootOf(2*_Z^4

-2*_Z^2+1)^5*x^5-54120*RootOf(2*_Z^4-2*_Z^2+1)^5*x^3-6918*RootOf(2*_Z^4-2*_Z^2+1)^3*x^5-52020*(x^6-x^2)^(1/2)*

RootOf(2*_Z^4-2*_Z^2+1)^3*x-14432*RootOf(2*_Z^4-2*_Z^2+1)^5*x+91112*RootOf(2*_Z^4-2*_Z^2+1)^3*x^3-5789*RootOf(

2*_Z^4-2*_Z^2+1)*x^5+36992*(x^6-x^2)^(3/4)*RootOf(2*_Z^4-2*_Z^2+1)^2-21964*(x^6-x^2)^(1/4)*RootOf(2*_Z^4-2*_Z^

2+1)^2*x^2+36992*(x^6-x^2)^(1/2)*RootOf(2*_Z^4-2*_Z^2+1)*x+6918*RootOf(2*_Z^4-2*_Z^2+1)^3*x-38042*RootOf(2*_Z^

4-2*_Z^2+1)*x^3-10982*(x^6-x^2)^(3/4)-15028*(x^6-x^2)^(1/4)*x^2+5789*RootOf(2*_Z^4-2*_Z^2+1)*x)/(2*RootOf(2*_Z

^4-2*_Z^2+1)^2*x^2-8*RootOf(2*_Z^4-2*_Z^2+1)^2-5*x^2+3)^2/x)+1/4*RootOf(2*_Z^4-2*_Z^2+1)*ln(-(14432*RootOf(2*_

Z^4-2*_Z^2+1)^5*x^5-54120*RootOf(2*_Z^4-2*_Z^2+1)^5*x^3-6918*RootOf(2*_Z^4-2*_Z^2+1)^3*x^5-52020*(x^6-x^2)^(1/

2)*RootOf(2*_Z^4-2*_Z^2+1)^3*x-14432*RootOf(2*_Z^4-2*_Z^2+1)^5*x+91112*RootOf(2*_Z^4-2*_Z^2+1)^3*x^3-5789*Root

Of(2*_Z^4-2*_Z^2+1)*x^5+36992*(x^6-x^2)^(3/4)*RootOf(2*_Z^4-2*_Z^2+1)^2-21964*(x^6-x^2)^(1/4)*RootOf(2*_Z^4-2*

_Z^2+1)^2*x^2+36992*(x^6-x^2)^(1/2)*RootOf(2*_Z^4-2*_Z^2+1)*x+6918*RootOf(2*_Z^4-2*_Z^2+1)^3*x-38042*RootOf(2*

_Z^4-2*_Z^2+1)*x^3-10982*(x^6-x^2)^(3/4)-15028*(x^6-x^2)^(1/4)*x^2+5789*RootOf(2*_Z^4-2*_Z^2+1)*x)/(2*RootOf(2

*_Z^4-2*_Z^2+1)^2*x^2-8*RootOf(2*_Z^4-2*_Z^2+1)^2-5*x^2+3)^2/x)

________________________________________________________________________________________

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 = -0.00 \begin {gather*} \int \frac {x^4-1}{\left (x^4+1\right )\,{\left (x^6-x^2\right )}^{1/4}} \,d x \end {gather*}

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)

[Out]

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

________________________________________________________________________________________

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

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]