\(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx\) [3099]

Optimal result

Integrand size = 25, antiderivative size = 553 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx=\frac {\sqrt {x^2+\sqrt {1+x^4}}}{a}+\frac {\left (1+a^4+\sqrt {1+a^4}\right ) \arctan \left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1-\sqrt {1+a^4}}}\right )}{a^2 \sqrt {1+a^4} \sqrt {-1-\sqrt {1+a^4}}}+\frac {\left (-1-a^4+\sqrt {1+a^4}\right ) \arctan \left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1+\sqrt {1+a^4}}}\right )}{a^2 \sqrt {1+a^4} \sqrt {-1+\sqrt {1+a^4}}}+\frac {\left (\sqrt {2} \sqrt {-a^2-\sqrt {1+a^4}}-\sqrt {2} a^2 \sqrt {-a^2-\sqrt {1+a^4}}+\sqrt {2} \sqrt {1+a^4} \sqrt {-a^2-\sqrt {1+a^4}}\right ) \arctan \left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-a^2-\sqrt {1+a^4}} \left (-1+x^2+\sqrt {1+x^4}\right )}\right )}{2 a^2}+\frac {\left (-\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}}+\sqrt {2} a^2 \sqrt {-a^2+\sqrt {1+a^4}}+\sqrt {2} \sqrt {1+a^4} \sqrt {-a^2+\sqrt {1+a^4}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{2 a^2}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{a^2} \]

[Out]

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

Rubi [F]

\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx=\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 4.63 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx=\frac {2 a \sqrt {x^2+\sqrt {1+x^4}}+\frac {2 \left (1+a^4+\sqrt {1+a^4}\right ) \arctan \left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1-\sqrt {1+a^4}}}\right )}{\sqrt {1+a^4} \sqrt {-1-\sqrt {1+a^4}}}+\frac {2 \left (-1-a^4+\sqrt {1+a^4}\right ) \arctan \left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1+\sqrt {1+a^4}}}\right )}{\sqrt {1+a^4} \sqrt {-1+\sqrt {1+a^4}}}+\sqrt {2} \sqrt {-a^2-\sqrt {1+a^4}} \left (1-a^2+\sqrt {1+a^4}\right ) \arctan \left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-a^2-\sqrt {1+a^4}} \left (-1+x^2+\sqrt {1+x^4}\right )}\right )+\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}} \left (-1+a^2+\sqrt {1+a^4}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{2 a^2} \]

[In]

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

[Out]

(2*a*Sqrt[x^2 + Sqrt[1 + x^4]] + (2*(1 + a^4 + Sqrt[1 + a^4])*ArcTan[(a*Sqrt[x^2 + Sqrt[1 + x^4]])/Sqrt[-1 - S
qrt[1 + a^4]]])/(Sqrt[1 + a^4]*Sqrt[-1 - Sqrt[1 + a^4]]) + (2*(-1 - a^4 + Sqrt[1 + a^4])*ArcTan[(a*Sqrt[x^2 +
Sqrt[1 + x^4]])/Sqrt[-1 + Sqrt[1 + a^4]]])/(Sqrt[1 + a^4]*Sqrt[-1 + Sqrt[1 + a^4]]) + Sqrt[2]*Sqrt[-a^2 - Sqrt
[1 + a^4]]*(1 - a^2 + Sqrt[1 + a^4])*ArcTan[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[-a^2 - Sqrt[1 + a^4]]*
(-1 + x^2 + Sqrt[1 + x^4]))] + Sqrt[2]*Sqrt[-a^2 + Sqrt[1 + a^4]]*(-1 + a^2 + Sqrt[1 + a^4])*ArcTan[(Sqrt[2]*S
qrt[-a^2 + Sqrt[1 + a^4]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]
*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/(2*a^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx=\int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{a x + 1}\, dx \]

[In]

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

[Out]

Integral(sqrt(x**2 + sqrt(x**4 + 1))/(a*x + 1), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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