\(\int \frac {x^7 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx\) [49]

Optimal result

Integrand size = 30, antiderivative size = 94 \[ \int \frac {x^7 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=-\frac {16}{35 \sqrt {x^2+\sqrt {1+x^4}}}+\frac {6 x^4}{35 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {8}{35} x^2 \sqrt {x^2+\sqrt {1+x^4}}+\frac {1}{7} x^6 \sqrt {x^2+\sqrt {1+x^4}} \] Output:

-16/35/(x^2+(x^4+1)^(1/2))^(1/2)+6/35*x^4/(x^2+(x^4+1)^(1/2))^(1/2)-8/35*x 
^2*(x^2+(x^4+1)^(1/2))^(1/2)+1/7*x^6*(x^2+(x^4+1)^(1/2))^(1/2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.85 \[ \int \frac {x^7 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {-16-50 x^4-5 x^8+20 x^{12}-40 x^2 \sqrt {1+x^4}-15 x^6 \sqrt {1+x^4}+20 x^{10} \sqrt {1+x^4}}{35 \left (x^2+\sqrt {1+x^4}\right )^{5/2}} \] Input:

Integrate[(x^7*Sqrt[x^2 + Sqrt[1 + x^4]])/Sqrt[1 + x^4],x]
 

Output:

(-16 - 50*x^4 - 5*x^8 + 20*x^12 - 40*x^2*Sqrt[1 + x^4] - 15*x^6*Sqrt[1 + x 
^4] + 20*x^10*Sqrt[1 + x^4])/(35*(x^2 + Sqrt[1 + x^4])^(5/2))
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.56, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2558, 243, 53, 2009}

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.

\(\displaystyle \int \frac {x^7 \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}} \, dx\)

\(\Big \downarrow \) 2558

\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {x^7}{\sqrt {1-i x^2}}dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {x^7}{\sqrt {i x^2+1}}dx\)

\(\Big \downarrow \) 243

\(\displaystyle \left (\frac {1}{4}-\frac {i}{4}\right ) \int \frac {x^6}{\sqrt {1-i x^2}}dx^2+\left (\frac {1}{4}+\frac {i}{4}\right ) \int \frac {x^6}{\sqrt {i x^2+1}}dx^2\)

\(\Big \downarrow \) 53

\(\displaystyle \left (\frac {1}{4}-\frac {i}{4}\right ) \int \left (-i \left (1-i x^2\right )^{5/2}+3 i \left (1-i x^2\right )^{3/2}-3 i \sqrt {1-i x^2}+\frac {i}{\sqrt {1-i x^2}}\right )dx^2+\left (\frac {1}{4}+\frac {i}{4}\right ) \int \left (i \left (i x^2+1\right )^{5/2}-3 i \left (i x^2+1\right )^{3/2}+3 i \sqrt {i x^2+1}-\frac {i}{\sqrt {i x^2+1}}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \left (\frac {1}{4}-\frac {i}{4}\right ) \left (\frac {2}{7} \left (1-i x^2\right )^{7/2}-\frac {6}{5} \left (1-i x^2\right )^{5/2}+2 \left (1-i x^2\right )^{3/2}-2 \sqrt {1-i x^2}\right )+\left (\frac {1}{4}+\frac {i}{4}\right ) \left (\frac {2}{7} \left (1+i x^2\right )^{7/2}-\frac {6}{5} \left (1+i x^2\right )^{5/2}+2 \left (1+i x^2\right )^{3/2}-2 \sqrt {1+i x^2}\right )\)

Input:

Int[(x^7*Sqrt[x^2 + Sqrt[1 + x^4]])/Sqrt[1 + x^4],x]
 

Output:

(1/4 - I/4)*(-2*Sqrt[1 - I*x^2] + 2*(1 - I*x^2)^(3/2) - (6*(1 - I*x^2)^(5/ 
2))/5 + (2*(1 - I*x^2)^(7/2))/7) + (1/4 + I/4)*(-2*Sqrt[1 + I*x^2] + 2*(1 
+ I*x^2)^(3/2) - (6*(1 + I*x^2)^(5/2))/5 + (2*(1 + I*x^2)^(7/2))/7)
 

Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2558

Int[(((c_.) + (d_.)*(x_))^(m_.)*Sqrt[(b_.)*(x_)^2 + Sqrt[(a_) + (e_.)*(x_)^ 
4]])/Sqrt[(a_) + (e_.)*(x_)^4], x_Symbol] :> Simp[(1 - I)/2   Int[(c + d*x) 
^m/Sqrt[Sqrt[a] - I*b*x^2], x], x] + Simp[(1 + I)/2   Int[(c + d*x)^m/Sqrt[ 
Sqrt[a] + I*b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[e, b^2] && G 
tQ[a, 0]
 

Maple [F]

\[\int \frac {x^{7} \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {x^{4}+1}}d x\]

Input:

int(x^7*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x)
 

Output:

int(x^7*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43 \[ \int \frac {x^7 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=-\frac {1}{35} \, {\left (x^{6} - 8 \, x^{2} - 2 \, {\left (3 \, x^{4} - 8\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \] Input:

integrate(x^7*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x, algorithm="fricas 
")
 

Output:

-1/35*(x^6 - 8*x^2 - 2*(3*x^4 - 8)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1) 
)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2011 vs. \(2 (82) = 164\).

Time = 1.85 (sec) , antiderivative size = 2011, normalized size of antiderivative = 21.39 \[ \int \frac {x^7 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\text {Too large to display} \] Input:

integrate(x**7*(x**2+(x**4+1)**(1/2))**(1/2)/(x**4+1)**(1/2),x)
                                                                                    
                                                                                    
 

Output:

-20*sqrt(2)*x**14*gamma(3/4)/(70*x**4*sqrt(x**4 + 1)*sqrt(sqrt(x**4 + 1) + 
 1)*gamma(-1/4) + 210*x**4*sqrt(sqrt(x**4 + 1) + 1)*gamma(-1/4) + 280*sqrt 
(x**4 + 1)*sqrt(sqrt(x**4 + 1) + 1)*gamma(-1/4) + 280*sqrt(sqrt(x**4 + 1) 
+ 1)*gamma(-1/4)) + 5*sqrt(2)*x**12*sqrt(x**4 + 1)*gamma(-1/4)/(70*x**4*sq 
rt(x**4 + 1)*sqrt(sqrt(x**4 + 1) + 1)*gamma(-1/4) + 210*x**4*sqrt(sqrt(x** 
4 + 1) + 1)*gamma(-1/4) + 280*sqrt(x**4 + 1)*sqrt(sqrt(x**4 + 1) + 1)*gamm 
a(-1/4) + 280*sqrt(sqrt(x**4 + 1) + 1)*gamma(-1/4)) + 21*sqrt(2)*x**12*gam 
ma(-1/4)/(70*x**4*sqrt(x**4 + 1)*sqrt(sqrt(x**4 + 1) + 1)*gamma(-1/4) + 21 
0*x**4*sqrt(sqrt(x**4 + 1) + 1)*gamma(-1/4) + 280*sqrt(x**4 + 1)*sqrt(sqrt 
(x**4 + 1) + 1)*gamma(-1/4) + 280*sqrt(sqrt(x**4 + 1) + 1)*gamma(-1/4)) - 
56*sqrt(2)*x**10*sqrt(x**4 + 1)*gamma(3/4)/(70*x**4*sqrt(x**4 + 1)*sqrt(sq 
rt(x**4 + 1) + 1)*gamma(-1/4) + 210*x**4*sqrt(sqrt(x**4 + 1) + 1)*gamma(-1 
/4) + 280*sqrt(x**4 + 1)*sqrt(sqrt(x**4 + 1) + 1)*gamma(-1/4) + 280*sqrt(s 
qrt(x**4 + 1) + 1)*gamma(-1/4)) - 56*sqrt(2)*x**10*gamma(3/4)/(70*x**4*sqr 
t(x**4 + 1)*sqrt(sqrt(x**4 + 1) + 1)*gamma(-1/4) + 210*x**4*sqrt(sqrt(x**4 
 + 1) + 1)*gamma(-1/4) + 280*sqrt(x**4 + 1)*sqrt(sqrt(x**4 + 1) + 1)*gamma 
(-1/4) + 280*sqrt(sqrt(x**4 + 1) + 1)*gamma(-1/4)) + 36*sqrt(2)*x**8*sqrt( 
x**4 + 1)*gamma(-1/4)/(70*x**4*sqrt(x**4 + 1)*sqrt(sqrt(x**4 + 1) + 1)*gam 
ma(-1/4) + 210*x**4*sqrt(sqrt(x**4 + 1) + 1)*gamma(-1/4) + 280*sqrt(x**4 + 
 1)*sqrt(sqrt(x**4 + 1) + 1)*gamma(-1/4) + 280*sqrt(sqrt(x**4 + 1) + 1)...
 

Maxima [F]

\[ \int \frac {x^7 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} x^{7}}{\sqrt {x^{4} + 1}} \,d x } \] Input:

integrate(x^7*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x, algorithm="maxima 
")
 

Output:

integrate(sqrt(x^2 + sqrt(x^4 + 1))*x^7/sqrt(x^4 + 1), x)
 

Giac [F]

\[ \int \frac {x^7 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} x^{7}}{\sqrt {x^{4} + 1}} \,d x } \] Input:

integrate(x^7*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(x^2 + sqrt(x^4 + 1))*x^7/sqrt(x^4 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^7 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int \frac {x^7\,\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}} \,d x \] Input:

int((x^7*((x^4 + 1)^(1/2) + x^2)^(1/2))/(x^4 + 1)^(1/2),x)
 

Output:

int((x^7*((x^4 + 1)^(1/2) + x^2)^(1/2))/(x^4 + 1)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^7 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, x^{7}}{x^{4}+1}d x \] Input:

int(x^7*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x)
 

Output:

int((sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1)*x**7)/(x**4 + 1),x)