3.11.47 \(\int \frac {1+x^4}{\sqrt {-x+x^3} (-1+x^4)} \, dx\)

Optimal. Leaf size=79 \[ \frac {\sqrt {x^3-x}}{1-x^2}-\frac {1}{4} \tan ^{-1}\left (\frac {2 \sqrt {x^3-x}}{x^2-2 x-1}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {\frac {x^2}{2}+x-\frac {1}{2}}{\sqrt {x^3-x}}\right ) \]

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Rubi [C]  time = 0.68, antiderivative size = 119, normalized size of antiderivative = 1.51, number of steps used = 18, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2056, 6715, 6725, 222, 1404, 414, 523, 409, 1211, 1699, 206, 203} \begin {gather*} -\frac {x}{\sqrt {x^3-x}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {x^2-1} \sqrt {x} \tan ^{-1}\left (\frac {(1+i) \sqrt {x}}{\sqrt {x^2-1}}\right )}{\sqrt {x^3-x}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {x^2-1} \sqrt {x} \tanh ^{-1}\left (\frac {(1+i) \sqrt {x}}{\sqrt {x^2-1}}\right )}{\sqrt {x^3-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(a*b), 2]}, Simp[(Sqrt[-a + q*x^2]*Sqrt[(a + q*x^2
)/q]*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2])/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4]), x] /; IntegerQ[q]]
/; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]

Rule 409

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-(d/c), 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-(d/c), 2]*x^2)), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 1211

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[1/(2*d), Int[1/Sqrt[a + c*x^4], x],
x] + Dist[1/(2*d), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d
^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]

Rule 1404

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(d + e*x^n)^(p + q)*(a/d
+ (c*x^n)/e)^p, x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]

Rule 1699

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

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]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1+x^4}{\sqrt {x} \sqrt {-1+x^2} \left (-1+x^4\right )} \, dx}{\sqrt {-x+x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1+x^8}{\sqrt {-1+x^4} \left (-1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {-1+x^4}}+\frac {2}{\sqrt {-1+x^4} \left (-1+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}+\frac {\left (4 \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4} \left (-1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}+\frac {\left (4 \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^4\right )^{3/2} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=-\frac {x}{\sqrt {-x+x^3}}+\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {-3-x^4}{\sqrt {-1+x^4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=-\frac {x}{\sqrt {-x+x^3}}+\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=-\frac {x}{\sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {-x+x^3}}+\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-i x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=-\frac {x}{\sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {-x+x^3}}+\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}-2 \frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1-i x^2}{\left (1+i x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1+i x^2}{\left (1-i x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^3}}\\ &=-\frac {x}{\sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 i x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-1+x^2}}\right )}{2 \sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+2 i x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-1+x^2}}\right )}{2 \sqrt {-x+x^3}}\\ &=-\frac {x}{\sqrt {-x+x^3}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {x} \sqrt {-1+x^2} \tan ^{-1}\left (\frac {(1+i) \sqrt {x}}{\sqrt {-1+x^2}}\right )}{\sqrt {-x+x^3}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {x} \sqrt {-1+x^2} \tanh ^{-1}\left (\frac {(1+i) \sqrt {x}}{\sqrt {-1+x^2}}\right )}{\sqrt {-x+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.25, size = 63, normalized size = 0.80 \begin {gather*} \frac {x \left (x^2-1\right ) F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};x^2,-x^2\right )}{\sqrt {1-x^2} \sqrt {x \left (x^2-1\right )}}-\frac {x}{\sqrt {x \left (x^2-1\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(x/Sqrt[x*(-1 + x^2)]) + (x*(-1 + x^2)*AppellF1[1/4, -1/2, 1, 5/4, x^2, -x^2])/(Sqrt[1 - x^2]*Sqrt[x*(-1 + x^
2)])

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IntegrateAlgebraic [A]  time = 0.29, size = 79, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-x+x^3}}{1-x^2}-\frac {1}{4} \tan ^{-1}\left (\frac {2 \sqrt {-x+x^3}}{-1-2 x+x^2}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {-\frac {1}{2}+x+\frac {x^2}{2}}{\sqrt {-x+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^4)/(Sqrt[-x + x^3]*(-1 + x^4)),x]

[Out]

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

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fricas [A]  time = 0.58, size = 105, normalized size = 1.33 \begin {gather*} \frac {2 \, {\left (x^{2} - 1\right )} \arctan \left (\frac {x^{2} - 2 \, x - 1}{2 \, \sqrt {x^{3} - x}}\right ) + {\left (x^{2} - 1\right )} \log \left (\frac {x^{4} + 8 \, x^{3} + 2 \, x^{2} - 4 \, \sqrt {x^{3} - x} {\left (x^{2} + 2 \, x - 1\right )} - 8 \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) - 8 \, \sqrt {x^{3} - x}}{8 \, {\left (x^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {x^{3} - x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.51, size = 223, normalized size = 2.82

method	result	size
risch	\(-\frac {x}{\sqrt {x \left (x^{2}-1\right )}}+\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}+\frac {i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}\)	\(223\)
elliptic	\(-\frac {x}{\sqrt {x \left (x^{2}-1\right )}}+\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}+\frac {i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}\)	\(223\)
default	\(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}+\frac {i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}+\frac {x^{2}-x}{2 \sqrt {\left (1+x \right ) \left (x^{2}-x \right )}}-\frac {x^{2}+x}{2 \sqrt {\left (-1+x \right ) \left (x^{2}+x \right )}}\)	\(251\)
trager	\(-\frac {\sqrt {x^{3}-x}}{x^{2}-1}+9 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) \ln \left (\frac {-150336 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x^{2}+225504 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x +2826 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x^{2}+7200 \sqrt {x^{3}-x}\, \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+150336 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2}-10764 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x -8 x^{2}-125 \sqrt {x^{3}-x}-2826 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+112 x +8}{\left (72 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x -144 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )-3 x +1\right )^{2}}\right )+\frac {\ln \left (-\frac {300672 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x^{2}-451008 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x -11052 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x^{2}+14400 \sqrt {x^{3}-x}\, \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )-300672 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2}+3528 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x +91 x^{2}-150 \sqrt {x^{3}-x}+11052 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+26 x -91}{\left (72 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x -144 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+x +3\right )^{2}}\right )}{4}-9 \ln \left (-\frac {300672 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x^{2}-451008 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x -11052 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x^{2}+14400 \sqrt {x^{3}-x}\, \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )-300672 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2}+3528 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x +91 x^{2}-150 \sqrt {x^{3}-x}+11052 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+26 x -91}{\left (72 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x -144 \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+x +3\right )^{2}}\right ) \RootOf \left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )\)	\(559\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x*(x^2-1))^(1/2)+1/2*(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)*EllipticF((1+x)^(1/2),1/2*2^(1/2))
-1/4*(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)*EllipticPi((1+x)^(1/2),1/2-1/2*I,1/2*2^(1/2))-1/4*I*(1
+x)^(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)*EllipticPi((1+x)^(1/2),1/2-1/2*I,1/2*2^(1/2))-1/4*(1+x)^(1/2)
*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)*EllipticPi((1+x)^(1/2),1/2+1/2*I,1/2*2^(1/2))+1/4*I*(1+x)^(1/2)*(2-2*x
)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)*EllipticPi((1+x)^(1/2),1/2+1/2*I,1/2*2^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {x^{3} - x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.76, size = 232, normalized size = 2.94 \begin {gather*} \frac {\sqrt {-x}\,\left (\frac {\sin \left (2\,\mathrm {asin}\left (\sqrt {-x}\right )\right )}{4\,\sqrt {1-x}}+\frac {\mathrm {E}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{2}\right )\,\sqrt {1-x}\,\sqrt {x+1}}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (-\mathrm {i};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (1{}\mathrm {i};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}-\frac {2\,\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\left (\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )-\frac {\mathrm {E}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{2}+\frac {\sqrt {-x}\,\sqrt {1-x}}{2\,\sqrt {x+1}}\right )}{\sqrt {x^3-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi(-1i, asin((-x)^(1/2)), -1))/(x^3 - x)^(1/2) - (2*(-x)^(1/2)
*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticF(asin((-x)^(1/2)), -1))/(x^3 - x)^(1/2) + ((-x)^(1/2)*(1 - x)^(1/2)*(x +
1)^(1/2)*ellipticPi(1i, asin((-x)^(1/2)), -1))/(x^3 - x)^(1/2) + ((-x)^(1/2)*(sin(2*asin((-x)^(1/2)))/(4*(1 -
x)^(1/2)) + ellipticE(asin((-x)^(1/2)), -1)/2)*(1 - x)^(1/2)*(x + 1)^(1/2))/(x^3 - x)^(1/2) + ((-x)^(1/2)*(1
- x)^(1/2)*(x + 1)^(1/2)*(ellipticF(asin((-x)^(1/2)), -1) - ellipticE(asin((-x)^(1/2)), -1)/2 + ((-x)^(1/2)*(1
- x)^(1/2))/(2*(x + 1)^(1/2))))/(x^3 - x)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{\sqrt {x \left (x - 1\right ) \left (x + 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**3-x)**(1/2)/(x**4-1),x)

[Out]

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

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