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