\(\int \frac {1+x^3}{(1-x^4) \sqrt [4]{1+x^4}} \, dx\) [590]

Optimal result

Integrand size = 24, antiderivative size = 103 \[ \int \frac {1+x^3}{\left (1-x^4\right ) \sqrt [4]{1+x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\arctan \left (\frac {\sqrt [4]{1+x^4}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1+x^4}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1913, 385, 218, 212, 209, 455, 65, 304} \[ \int \frac {1+x^3}{\left (1-x^4\right ) \sqrt [4]{1+x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}-\frac {\arctan \left (\frac {\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}} \]

[In]

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

[Out]

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

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 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 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 385

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

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

Rule 1913

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

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

Mathematica [C] (verified)

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

Time = 10.54 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90 \[ \int \frac {1+x^3}{\left (1-x^4\right ) \sqrt [4]{1+x^4}} \, dx=\frac {1}{4} x^4 \operatorname {AppellF1}\left (1,\frac {1}{4},1,2,-x^4,x^4\right )+\frac {2 \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )-\log \left (1-\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )+\log \left (1+\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}} \]

[In]

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

[Out]

(x^4*AppellF1[1, 1/4, 1, 2, -x^4, x^4])/4 + (2*ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)] - Log[1 - (2^(1/4)*x)/(1 +
x^4)^(1/4)] + Log[1 + (2^(1/4)*x)/(1 + x^4)^(1/4)])/(4*2^(1/4))

Maple [F(-1)]

Timed out.

\[\int \frac {x^{3}+1}{\left (-x^{4}+1\right ) \left (x^{4}+1\right )^{\frac {1}{4}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1+x^3}{\left (1-x^4\right ) \sqrt [4]{1+x^4}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

Sympy [F]

\[ \int \frac {1+x^3}{\left (1-x^4\right ) \sqrt [4]{1+x^4}} \, dx=- \int \left (- \frac {x}{x^{3} \sqrt [4]{x^{4} + 1} - x^{2} \sqrt [4]{x^{4} + 1} + x \sqrt [4]{x^{4} + 1} - \sqrt [4]{x^{4} + 1}}\right )\, dx - \int \frac {x^{2}}{x^{3} \sqrt [4]{x^{4} + 1} - x^{2} \sqrt [4]{x^{4} + 1} + x \sqrt [4]{x^{4} + 1} - \sqrt [4]{x^{4} + 1}}\, dx - \int \frac {1}{x^{3} \sqrt [4]{x^{4} + 1} - x^{2} \sqrt [4]{x^{4} + 1} + x \sqrt [4]{x^{4} + 1} - \sqrt [4]{x^{4} + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1+x^3}{\left (1-x^4\right ) \sqrt [4]{1+x^4}} \, dx=\int { -\frac {x^{3} + 1}{{\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1+x^3}{\left (1-x^4\right ) \sqrt [4]{1+x^4}} \, dx=\int { -\frac {x^{3} + 1}{{\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1+x^3}{\left (1-x^4\right ) \sqrt [4]{1+x^4}} \, dx=\int -\frac {x^3+1}{\left (x^4-1\right )\,{\left (x^4+1\right )}^{1/4}} \,d x \]

[In]

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

[Out]

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