3.8.20 \(\int \frac {1}{(1+x) \sqrt [6]{1+x^2}} \, dx\) [720]

3.8.20.1 Optimal result

Integrand size = 15, antiderivative size = 203 \[ \int \frac {1}{(1+x) \sqrt [6]{1+x^2}} \, dx=x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{6},\frac {3}{2},x^2,-x^2\right )-\frac {\sqrt {3} \arctan \left (\frac {1-2^{5/6} \sqrt [6]{1+x^2}}{\sqrt {3}}\right )}{2 \sqrt [6]{2}}+\frac {\sqrt {3} \arctan \left (\frac {1+2^{5/6} \sqrt [6]{1+x^2}}{\sqrt {3}}\right )}{2 \sqrt [6]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{1+x^2}}{\sqrt [6]{2}}\right )}{\sqrt [6]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [6]{2} \sqrt [6]{1+x^2}+\sqrt [3]{1+x^2}\right )}{4 \sqrt [6]{2}}-\frac {\log \left (\sqrt [3]{2}+\sqrt [6]{2} \sqrt [6]{1+x^2}+\sqrt [3]{1+x^2}\right )}{4 \sqrt [6]{2}} \]

output

x*AppellF1(1/2,1,1/6,3/2,x^2,-x^2)-1/2*arctanh(1/2*(x^2+1)^(1/6)*2^(5/6))* 
2^(5/6)+1/8*ln(2^(1/3)-2^(1/6)*(x^2+1)^(1/6)+(x^2+1)^(1/3))*2^(5/6)-1/8*ln 
(2^(1/3)+2^(1/6)*(x^2+1)^(1/6)+(x^2+1)^(1/3))*2^(5/6)-1/4*arctan(1/3*(1-(x 
^2+1)^(1/6)*2^(5/6))*3^(1/2))*3^(1/2)*2^(5/6)+1/4*arctan(1/3*(1+(x^2+1)^(1 
/6)*2^(5/6))*3^(1/2))*3^(1/2)*2^(5/6)
 

3.8.20.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 17.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.35 \[ \int \frac {1}{(1+x) \sqrt [6]{1+x^2}} \, dx=-\frac {3 \sqrt [6]{\frac {-i+x}{1+x}} \sqrt [6]{\frac {i+x}{1+x}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{6},\frac {1}{6},\frac {4}{3},\frac {1-i}{1+x},\frac {1+i}{1+x}\right )}{\sqrt [6]{1+x^2}} \]

input

Integrate[1/((1 + x)*(1 + x^2)^(1/6)),x]
 

output

(-3*((-I + x)/(1 + x))^(1/6)*((I + x)/(1 + x))^(1/6)*AppellF1[1/3, 1/6, 1/ 
6, 4/3, (1 - I)/(1 + x), (1 + I)/(1 + x)])/(1 + x^2)^(1/6)
 

3.8.20.3 Rubi [A] (warning: unable to verify)

Time = 0.39 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.94, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.867, Rules used = {504, 333, 353, 73, 825, 27, 219, 1142, 25, 27, 1082, 217, 1103}

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.

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

\(\Big \downarrow \) 504

\(\displaystyle \int \frac {1}{\left (1-x^2\right ) \sqrt [6]{x^2+1}}dx-\int \frac {x}{\left (1-x^2\right ) \sqrt [6]{x^2+1}}dx\)

\(\Big \downarrow \) 333

\(\displaystyle x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{6},\frac {3}{2},x^2,-x^2\right )-\int \frac {x}{\left (1-x^2\right ) \sqrt [6]{x^2+1}}dx\)

\(\Big \downarrow \) 353

\(\displaystyle x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{6},\frac {3}{2},x^2,-x^2\right )-\frac {1}{2} \int \frac {1}{\left (1-x^2\right ) \sqrt [6]{x^2+1}}dx^2\)

\(\Big \downarrow \) 73

\(\displaystyle x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{6},\frac {3}{2},x^2,-x^2\right )-3 \int \frac {x^8}{2-x^{12}}d\sqrt [6]{x^2+1}\)

\(\Big \downarrow \) 825

\(\displaystyle x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{6},\frac {3}{2},x^2,-x^2\right )-3 \left (\frac {1}{3} \int \frac {1}{\sqrt [3]{2}-x^4}d\sqrt [6]{x^2+1}+\frac {\int -\frac {\sqrt [6]{x^2+1}+\sqrt [6]{2}}{2 \left (x^4-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}\right )}d\sqrt [6]{x^2+1}}{3 \sqrt [6]{2}}+\frac {\int -\frac {\sqrt [6]{2}-\sqrt [6]{x^2+1}}{2 \left (x^4+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}\right )}d\sqrt [6]{x^2+1}}{3 \sqrt [6]{2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{6},\frac {3}{2},x^2,-x^2\right )-3 \left (\frac {1}{3} \int \frac {1}{\sqrt [3]{2}-x^4}d\sqrt [6]{x^2+1}-\frac {\int \frac {\sqrt [6]{x^2+1}+\sqrt [6]{2}}{x^4-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{6 \sqrt [6]{2}}-\frac {\int \frac {\sqrt [6]{2}-\sqrt [6]{x^2+1}}{x^4+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{6 \sqrt [6]{2}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{6},\frac {3}{2},x^2,-x^2\right )-3 \left (-\frac {\int \frac {\sqrt [6]{x^2+1}+\sqrt [6]{2}}{x^4-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{6 \sqrt [6]{2}}-\frac {\int \frac {\sqrt [6]{2}-\sqrt [6]{x^2+1}}{x^4+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{6 \sqrt [6]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{x^2+1}}{\sqrt [6]{2}}\right )}{3 \sqrt [6]{2}}\right )\)

\(\Big \downarrow \) 1142

\(\displaystyle x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{6},\frac {3}{2},x^2,-x^2\right )-3 \left (-\frac {\frac {3 \int \frac {1}{x^4-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{2^{5/6}}+\frac {1}{2} \int -\frac {\sqrt [6]{2} \left (1-2^{5/6} \sqrt [6]{x^2+1}\right )}{x^4-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{6 \sqrt [6]{2}}-\frac {\frac {3 \int \frac {1}{x^4+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{2^{5/6}}-\frac {1}{2} \int \frac {\sqrt [6]{2} \left (2^{5/6} \sqrt [6]{x^2+1}+1\right )}{x^4+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{6 \sqrt [6]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{x^2+1}}{\sqrt [6]{2}}\right )}{3 \sqrt [6]{2}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{6},\frac {3}{2},x^2,-x^2\right )-3 \left (-\frac {\frac {3 \int \frac {1}{x^4-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{2^{5/6}}-\frac {1}{2} \int \frac {\sqrt [6]{2} \left (1-2^{5/6} \sqrt [6]{x^2+1}\right )}{x^4-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{6 \sqrt [6]{2}}-\frac {\frac {3 \int \frac {1}{x^4+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{2^{5/6}}-\frac {1}{2} \int \frac {\sqrt [6]{2} \left (2^{5/6} \sqrt [6]{x^2+1}+1\right )}{x^4+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{6 \sqrt [6]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{x^2+1}}{\sqrt [6]{2}}\right )}{3 \sqrt [6]{2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{6},\frac {3}{2},x^2,-x^2\right )-3 \left (-\frac {\frac {3 \int \frac {1}{x^4-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{2^{5/6}}-\frac {\int \frac {1-2^{5/6} \sqrt [6]{x^2+1}}{x^4-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{2^{5/6}}}{6 \sqrt [6]{2}}-\frac {\frac {3 \int \frac {1}{x^4+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{2^{5/6}}-\frac {\int \frac {2^{5/6} \sqrt [6]{x^2+1}+1}{x^4+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{2^{5/6}}}{6 \sqrt [6]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{x^2+1}}{\sqrt [6]{2}}\right )}{3 \sqrt [6]{2}}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{6},\frac {3}{2},x^2,-x^2\right )-3 \left (-\frac {3 \int \frac {1}{-x^4-3}d\left (1-2^{5/6} \sqrt [6]{x^2+1}\right )-\frac {\int \frac {1-2^{5/6} \sqrt [6]{x^2+1}}{x^4-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{2^{5/6}}}{6 \sqrt [6]{2}}-\frac {-3 \int \frac {1}{-x^4-3}d\left (2^{5/6} \sqrt [6]{x^2+1}+1\right )-\frac {\int \frac {2^{5/6} \sqrt [6]{x^2+1}+1}{x^4+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{2^{5/6}}}{6 \sqrt [6]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{x^2+1}}{\sqrt [6]{2}}\right )}{3 \sqrt [6]{2}}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{6},\frac {3}{2},x^2,-x^2\right )-3 \left (-\frac {-\frac {\int \frac {1-2^{5/6} \sqrt [6]{x^2+1}}{x^4-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{2^{5/6}}-\sqrt {3} \arctan \left (\frac {1-2^{5/6} \sqrt [6]{x^2+1}}{\sqrt {3}}\right )}{6 \sqrt [6]{2}}-\frac {\sqrt {3} \arctan \left (\frac {2^{5/6} \sqrt [6]{x^2+1}+1}{\sqrt {3}}\right )-\frac {\int \frac {2^{5/6} \sqrt [6]{x^2+1}+1}{x^4+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}}d\sqrt [6]{x^2+1}}{2^{5/6}}}{6 \sqrt [6]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{x^2+1}}{\sqrt [6]{2}}\right )}{3 \sqrt [6]{2}}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{6},\frac {3}{2},x^2,-x^2\right )-3 \left (-\frac {\frac {1}{2} \log \left (x^4-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}\right )-\sqrt {3} \arctan \left (\frac {1-2^{5/6} \sqrt [6]{x^2+1}}{\sqrt {3}}\right )}{6 \sqrt [6]{2}}-\frac {\sqrt {3} \arctan \left (\frac {2^{5/6} \sqrt [6]{x^2+1}+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (x^4+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}\right )}{6 \sqrt [6]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{x^2+1}}{\sqrt [6]{2}}\right )}{3 \sqrt [6]{2}}\right )\)

input

Int[1/((1 + x)*(1 + x^2)^(1/6)),x]
 

output

x*AppellF1[1/2, 1, 1/6, 3/2, x^2, -x^2] - 3*(ArcTanh[(1 + x^2)^(1/6)/2^(1/ 
6)]/(3*2^(1/6)) - (-(Sqrt[3]*ArcTan[(1 - 2^(5/6)*(1 + x^2)^(1/6))/Sqrt[3]] 
) + Log[2^(1/3) + x^4 - 2^(1/6)*(1 + x^2)^(1/6)]/2)/(6*2^(1/6)) - (Sqrt[3] 
*ArcTan[(1 + 2^(5/6)*(1 + x^2)^(1/6))/Sqrt[3]] - Log[2^(1/3) + x^4 + 2^(1/ 
6)*(1 + x^2)^(1/6)]/2)/(6*2^(1/6)))
 

3.8.20.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[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 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 353

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

rule 504

Int[((a_) + (b_.)*(x_)^2)^(p_)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[c   I 
nt[(a + b*x^2)^p/(c^2 - d^2*x^2), x], x] - Simp[d   Int[x*((a + b*x^2)^p/(c 
^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, p}, x]
 

rule 825

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator 
[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k 
*m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]*x + 
s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 
 + 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m))   Int[1/ 
(r^2 - s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m))   Sum[u, {k, 1, (n - 2)/4}], 
x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1 
] && NegQ[a/b]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

3.8.20.4 Maple [F]

\[\int \frac {1}{\left (1+x \right ) \left (x^{2}+1\right )^{\frac {1}{6}}}d x\]

input

int(1/(1+x)/(x^2+1)^(1/6),x)
 

output

int(1/(1+x)/(x^2+1)^(1/6),x)
 

3.8.20.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{(1+x) \sqrt [6]{1+x^2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate(1/(1+x)/(x^2+1)^(1/6),x, algorithm="fricas")
 

output

Exception raised: TypeError >>  Error detected within library code:   Not 
integrable (provided residues have no relations)
 

3.8.20.6 Sympy [F]

\[ \int \frac {1}{(1+x) \sqrt [6]{1+x^2}} \, dx=\int \frac {1}{\left (x + 1\right ) \sqrt [6]{x^{2} + 1}}\, dx \]

input

integrate(1/(1+x)/(x**2+1)**(1/6),x)
 

output

Integral(1/((x + 1)*(x**2 + 1)**(1/6)), x)
 

3.8.20.7 Maxima [F]

\[ \int \frac {1}{(1+x) \sqrt [6]{1+x^2}} \, dx=\int { \frac {1}{{\left (x^{2} + 1\right )}^{\frac {1}{6}} {\left (x + 1\right )}} \,d x } \]

input

integrate(1/(1+x)/(x^2+1)^(1/6),x, algorithm="maxima")
 

output

integrate(1/((x^2 + 1)^(1/6)*(x + 1)), x)
 

3.8.20.8 Giac [F]

\[ \int \frac {1}{(1+x) \sqrt [6]{1+x^2}} \, dx=\int { \frac {1}{{\left (x^{2} + 1\right )}^{\frac {1}{6}} {\left (x + 1\right )}} \,d x } \]

input

integrate(1/(1+x)/(x^2+1)^(1/6),x, algorithm="giac")
 

output

integrate(1/((x^2 + 1)^(1/6)*(x + 1)), x)
 

3.8.20.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(1+x) \sqrt [6]{1+x^2}} \, dx=\int \frac {1}{{\left (x^2+1\right )}^{1/6}\,\left (x+1\right )} \,d x \]

input

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

output

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