3.10.74 \(\int \frac {-1-x+x^4}{x \sqrt [4]{1+x^4}} \, dx\) [974]

Optimal. Leaf size=74 \[ \frac {1}{3} \left (1+x^4\right )^{3/4}-\frac {1}{2} \text {ArcTan}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \text {ArcTan}\left (\sqrt [4]{1+x^4}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{1+x^4}\right ) \]

[Out]

1/3*(x^4+1)^(3/4)-1/2*arctan(x/(x^4+1)^(1/4))-1/2*arctan((x^4+1)^(1/4))-1/2*arctanh(x/(x^4+1)^(1/4))+1/2*arcta
nh((x^4+1)^(1/4))

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Rubi [A]
time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1847, 246, 218, 212, 209, 457, 81, 65, 304} \begin {gather*} -\frac {1}{2} \text {ArcTan}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \text {ArcTan}\left (\sqrt [4]{x^4+1}\right )+\frac {1}{3} \left (x^4+1\right )^{3/4}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{x^4+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + x^4)^(3/4)/3 - ArcTan[x/(1 + x^4)^(1/4)]/2 - ArcTan[(1 + x^4)^(1/4)]/2 - ArcTanh[x/(1 + x^4)^(1/4)]/2 + A
rcTanh[(1 + x^4)^(1/4)]/2

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 209

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

Rule 212

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

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 457

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

Rule 1847

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[
Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {
j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

Rubi steps

\begin {align*} \int \frac {-1-x+x^4}{x \sqrt [4]{1+x^4}} \, dx &=\int \left (-\frac {1}{\sqrt [4]{1+x^4}}+\frac {-1+x^4}{x \sqrt [4]{1+x^4}}\right ) \, dx\\ &=-\int \frac {1}{\sqrt [4]{1+x^4}} \, dx+\int \frac {-1+x^4}{x \sqrt [4]{1+x^4}} \, dx\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {-1+x}{x \sqrt [4]{1+x}} \, dx,x,x^4\right )-\text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {1}{3} \left (1+x^4\right )^{3/4}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,x^4\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {1}{3} \left (1+x^4\right )^{3/4}-\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^4}\right )\\ &=\frac {1}{3} \left (1+x^4\right )^{3/4}-\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^4}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^4}\right )\\ &=\frac {1}{3} \left (1+x^4\right )^{3/4}-\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \tan ^{-1}\left (\sqrt [4]{1+x^4}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{1+x^4}\right )\\ \end {align*}

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Mathematica [A]
time = 5.26, size = 68, normalized size = 0.92 \begin {gather*} \frac {1}{6} \left (2 \left (1+x^4\right )^{3/4}-3 \text {ArcTan}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-3 \text {ArcTan}\left (\sqrt [4]{1+x^4}\right )-3 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+3 \tanh ^{-1}\left (\sqrt [4]{1+x^4}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 3.
time = 2.51, size = 90, normalized size = 1.22

method	result	size
meijerg	\(-\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\frac {\pi \sqrt {2}\, x^{4} \hypergeom \left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], -x^{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+4 \ln \left (x \right )\right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{8 \pi }+\frac {x^{4} \hypergeom \left (\left [\frac {1}{4}, 1\right ], \left [2\right ], -x^{4}\right )}{4}-x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], -x^{4}\right )\)	\(90\)
trager	\(\frac {\left (x^{4}+1\right )^{\frac {3}{4}}}{3}+\frac {\ln \left (\frac {\left (x^{4}+1\right )^{\frac {3}{4}} x -\sqrt {x^{4}+1}\, x^{2}+x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-x^{4}-\left (x^{4}+1\right )^{\frac {3}{4}}+\sqrt {x^{4}+1}\, x -x^{2} \left (x^{4}+1\right )^{\frac {1}{4}}+x^{3}-\sqrt {x^{4}+1}+x \left (x^{4}+1\right )^{\frac {1}{4}}-\left (x^{4}+1\right )^{\frac {1}{4}}+x -1}{x^{2}}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+\left (x^{4}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x -x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-\left (x^{4}+1\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}+x^{2} \left (x^{4}+1\right )^{\frac {1}{4}}-x \left (x^{4}+1\right )^{\frac {1}{4}}-x \RootOf \left (\textit {\_Z}^{2}+1\right )+\left (x^{4}+1\right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{2}}\right )}{2}\)	\(275\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/Pi*2^(1/2)*GAMMA(3/4)*(-1/4*Pi*2^(1/2)/GAMMA(3/4)*x^4*hypergeom([1,1,5/4],[2,2],-x^4)+(-3*ln(2)-1/2*Pi+4*
ln(x))*Pi*2^(1/2)/GAMMA(3/4))+1/4*x^4*hypergeom([1/4,1],[2],-x^4)-x*hypergeom([1/4,1/4],[5/4],-x^4)

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Maxima [A]
time = 0.46, size = 90, normalized size = 1.22 \begin {gather*} \frac {1}{3} \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} - \frac {1}{2} \, \arctan \left ({\left (x^{4} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{4} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{4}} - 1\right ) - \frac {1}{4} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {1}{4} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (54) = 108\).
time = 4.78, size = 123, normalized size = 1.66 \begin {gather*} \frac {1}{3} \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} - \frac {1}{2} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )} - {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{2} - x\right )}}{x^{4} - x^{2} + 1}\right ) + \frac {1}{2} \, \log \left (-\frac {x^{4} - x^{3} - {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )} + \sqrt {x^{4} + 1} {\left (x^{2} - x + 1\right )} - {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{3} - x^{2} + x - 1\right )} - x + 1}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 2.49, size = 66, normalized size = 0.89 \begin {gather*} - \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {\left (x^{4} + 1\right )^{\frac {3}{4}}}{3} + \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 x \Gamma \left (\frac {5}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*gamma(1/4)*hyper((1/4, 1/4), (5/4,), x**4*exp_polar(I*pi))/(4*gamma(5/4)) + (x**4 + 1)**(3/4)/3 + gamma(1/4
)*hyper((1/4, 1/4), (5/4,), exp_polar(I*pi)/x**4)/(4*x*gamma(5/4))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.24, size = 43, normalized size = 0.58 \begin {gather*} \frac {\mathrm {atanh}\left ({\left (x^4+1\right )}^{1/4}\right )}{2}-\frac {\mathrm {atan}\left ({\left (x^4+1\right )}^{1/4}\right )}{2}-x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ -x^4\right )+\frac {{\left (x^4+1\right )}^{3/4}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atanh((x^4 + 1)^(1/4))/2 - atan((x^4 + 1)^(1/4))/2 - x*hypergeom([1/4, 1/4], 5/4, -x^4) + (x^4 + 1)^(3/4)/3

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