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

3.6.90.1 Optimal result

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

output

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)
 

3.6.90.2 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 \[ \int \frac {1+x^3}{\left (1-x^4\right ) \sqrt [4]{1+x^4}} \, dx=\frac {1}{4} x^4 \operatorname {AppellF1}\left (1,\frac {1}{4},1,2,-x^4,x^4\right )+\frac {2 \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )-\log \left (1-\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )+\log \left (1+\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}} \]

input

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

output

(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))
 

3.6.90.3 Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2438, 902, 756, 216, 219, 946, 73, 827, 216, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2438

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

\(\Big \downarrow \) 902

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

\(\displaystyle \int \frac {x^3}{\left (1-x^4\right ) \sqrt [4]{x^4+1}}dx+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}\)

\(\Big \downarrow \) 946

\(\displaystyle \frac {1}{4} \int \frac {1}{\left (1-x^4\right ) \sqrt [4]{x^4+1}}dx^4+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \int \frac {x^8}{2-x^{16}}d\sqrt [4]{x^4+1}+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {2}-x^8}d\sqrt [4]{x^4+1}-\frac {1}{2} \int \frac {1}{x^8+\sqrt {2}}d\sqrt [4]{x^4+1}+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {2}-x^8}d\sqrt [4]{x^4+1}+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}-\frac {\arctan \left (\frac {\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}-\frac {\arctan \left (\frac {\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}\)

input

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

output

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)) + A 
rcTanh[(1 + x^4)^(1/4)/2^(1/4)]/(2*2^(1/4))
 

3.6.90.3.1 Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 216

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

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 756

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

rule 827

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

rule 902

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su 
bst[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 946

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[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 2438

Int[((A_) + (B_.)*(x_)^(m_.))*((a_.) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.) 
*(x_)^(n_))^(q_.), x_Symbol] :> Simp[A   Int[(a + b*x^n)^p*(c + d*x^n)^q, x 
], x] + Simp[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]
 

3.6.90.4 Maple [F(-1)]

Timed out.

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

input

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

output

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

3.6.90.5 Fricas [F(-2)]

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

input

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

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (residue poly has multiple non-linear fac 
tors)
 

3.6.90.6 Sympy [F]

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

input

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

output

-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)
 

3.6.90.7 Maxima [F]

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

input

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

output

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

3.6.90.8 Giac [F]

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

input

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

output

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

3.6.90.9 Mupad [F(-1)]

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

input

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

output

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