Integrand size = 21, antiderivative size = 82 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=2 \sqrt {x+\sqrt {a^2+x^2}}-2 \sqrt {a} \arctan \left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right )-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right ) \]
-2*arctan((x+(a^2+x^2)^(1/2))^(1/2)/a^(1/2))*a^(1/2)-2*arctanh((x+(a^2+x^2 )^(1/2))^(1/2)/a^(1/2))*a^(1/2)+2*(x+(a^2+x^2)^(1/2))^(1/2)
Time = 0.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=2 \sqrt {x+\sqrt {a^2+x^2}}-2 \sqrt {a} \arctan \left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right )-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right ) \]
2*Sqrt[x + Sqrt[a^2 + x^2]] - 2*Sqrt[a]*ArcTan[Sqrt[x + Sqrt[a^2 + x^2]]/S qrt[a]] - 2*Sqrt[a]*ArcTanh[Sqrt[x + Sqrt[a^2 + x^2]]/Sqrt[a]]
Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2544, 25, 363, 266, 756, 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.
| \(\displaystyle \int \frac {\sqrt {\sqrt {a^2+x^2}+x}}{x} \, dx\) |
| \(\Big \downarrow \) 2544 |
| \(\displaystyle \int -\frac {\left (\sqrt {a^2+x^2}+x\right )^2+a^2}{\sqrt {\sqrt {a^2+x^2}+x} \left (a^2-\left (\sqrt {a^2+x^2}+x\right )^2\right )}d\left (\sqrt {a^2+x^2}+x\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {a^2+\left (x+\sqrt {a^2+x^2}\right )^2}{\sqrt {x+\sqrt {a^2+x^2}} \left (a^2-\left (x+\sqrt {a^2+x^2}\right )^2\right )}d\left (x+\sqrt {a^2+x^2}\right )\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle 2 \sqrt {\sqrt {a^2+x^2}+x}-2 a^2 \int \frac {1}{\sqrt {x+\sqrt {a^2+x^2}} \left (a^2-\left (x+\sqrt {a^2+x^2}\right )^2\right )}d\left (x+\sqrt {a^2+x^2}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle 2 \sqrt {\sqrt {a^2+x^2}+x}-4 a^2 \int \frac {1}{a^2-\left (x+\sqrt {a^2+x^2}\right )^2}d\sqrt {x+\sqrt {a^2+x^2}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 2 \sqrt {\sqrt {a^2+x^2}+x}-4 a^2 \left (\frac {\int \frac {1}{a-x-\sqrt {a^2+x^2}}d\sqrt {x+\sqrt {a^2+x^2}}}{2 a}+\frac {\int \frac {1}{a+x+\sqrt {a^2+x^2}}d\sqrt {x+\sqrt {a^2+x^2}}}{2 a}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 2 \sqrt {\sqrt {a^2+x^2}+x}-4 a^2 \left (\frac {\int \frac {1}{a-x-\sqrt {a^2+x^2}}d\sqrt {x+\sqrt {a^2+x^2}}}{2 a}+\frac {\arctan \left (\frac {\sqrt {\sqrt {a^2+x^2}+x}}{\sqrt {a}}\right )}{2 a^{3/2}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \sqrt {\sqrt {a^2+x^2}+x}-4 a^2 \left (\frac {\arctan \left (\frac {\sqrt {\sqrt {a^2+x^2}+x}}{\sqrt {a}}\right )}{2 a^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+x^2}+x}}{\sqrt {a}}\right )}{2 a^{3/2}}\right )\) |
2*Sqrt[x + Sqrt[a^2 + x^2]] - 4*a^2*(ArcTan[Sqrt[x + Sqrt[a^2 + x^2]]/Sqrt [a]]/(2*a^(3/2)) + ArcTanh[Sqrt[x + Sqrt[a^2 + x^2]]/Sqrt[a]]/(2*a^(3/2)))
3.1.28.3.1 Defintions of rubi rules used
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])
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])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
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]
Int[((g_.) + (h_.)*(x_))^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^ 2])^(n_.), x_Symbol] :> Simp[1/(2^(m + 1)*e^(m + 1)) Subst[Int[x^(n - m - 2)*(a*f^2 + x^2)*((-a)*f^2*h + 2*e*g*x + h*x^2)^m, x], x, e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && I ntegerQ[m]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.30
| method | result | size |
| meijerg | \(2 \sqrt {2}\, \sqrt {x}\, {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (-\frac {1}{4},-\frac {1}{4},\frac {1}{4};\frac {1}{2},\frac {3}{4};-\frac {a^{2}}{x^{2}}\right )\) | \(25\) |
Time = 0.26 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.63 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=\left [-2 \, \sqrt {a} \arctan \left (\frac {\sqrt {x + \sqrt {a^{2} + x^{2}}}}{\sqrt {a}}\right ) + \sqrt {a} \log \left (\frac {a^{2} + \sqrt {a^{2} + x^{2}} a - {\left ({\left (a - x\right )} \sqrt {a} + \sqrt {a^{2} + x^{2}} \sqrt {a}\right )} \sqrt {x + \sqrt {a^{2} + x^{2}}}}{x}\right ) + 2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}}, 2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {x + \sqrt {a^{2} + x^{2}}}}{a}\right ) + \sqrt {-a} \log \left (-\frac {a^{2} - \sqrt {a^{2} + x^{2}} a + {\left (\sqrt {-a} {\left (a + x\right )} - \sqrt {a^{2} + x^{2}} \sqrt {-a}\right )} \sqrt {x + \sqrt {a^{2} + x^{2}}}}{x}\right ) + 2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}}\right ] \]
[-2*sqrt(a)*arctan(sqrt(x + sqrt(a^2 + x^2))/sqrt(a)) + sqrt(a)*log((a^2 + sqrt(a^2 + x^2)*a - ((a - x)*sqrt(a) + sqrt(a^2 + x^2)*sqrt(a))*sqrt(x + sqrt(a^2 + x^2)))/x) + 2*sqrt(x + sqrt(a^2 + x^2)), 2*sqrt(-a)*arctan(sqrt (-a)*sqrt(x + sqrt(a^2 + x^2))/a) + sqrt(-a)*log(-(a^2 - sqrt(a^2 + x^2)*a + (sqrt(-a)*(a + x) - sqrt(a^2 + x^2)*sqrt(-a))*sqrt(x + sqrt(a^2 + x^2)) )/x) + 2*sqrt(x + sqrt(a^2 + x^2))]
Result contains complex when optimal does not.
Time = 1.76 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=\frac {\sqrt {x} \Gamma ^{2}\left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4}, \frac {1}{4} \\ \frac {1}{2}, \frac {3}{4} \end {matrix}\middle | {\frac {a^{2} e^{i \pi }}{x^{2}}} \right )}}{8 \pi \Gamma \left (\frac {3}{4}\right )} \]
sqrt(x)*gamma(-1/4)**2*gamma(1/4)*hyper((-1/4, -1/4, 1/4), (1/2, 3/4), a** 2*exp_polar(I*pi)/x**2)/(8*pi*gamma(3/4))
\[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=\int { \frac {\sqrt {x + \sqrt {a^{2} + x^{2}}}}{x} \,d x } \]
\[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=\int { \frac {\sqrt {x + \sqrt {a^{2} + x^{2}}}}{x} \,d x } \]
Timed out. \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=\int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \,d x \]
Time = 0.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=\sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {a}\, \sqrt {\sqrt {a^{2}+x^{2}}+x}\, \sqrt {a^{2}+x^{2}}-\sqrt {a}\, \sqrt {\sqrt {a^{2}+x^{2}}+x}\, a -\sqrt {a}\, \sqrt {\sqrt {a^{2}+x^{2}}+x}\, x}{2 a^{2}}\right )+2 \sqrt {\sqrt {a^{2}+x^{2}}+x}+\sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {a^{2}+x^{2}}+x}-\sqrt {a}\right )-\sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {a^{2}+x^{2}}+x}+\sqrt {a}\right ) \]
sqrt(a)*atan((sqrt(a)*sqrt(sqrt(a**2 + x**2) + x)*sqrt(a**2 + x**2) - sqrt (a)*sqrt(sqrt(a**2 + x**2) + x)*a - sqrt(a)*sqrt(sqrt(a**2 + x**2) + x)*x) /(2*a**2)) + 2*sqrt(sqrt(a**2 + x**2) + x) + sqrt(a)*log(sqrt(sqrt(a**2 + x**2) + x) - sqrt(a)) - sqrt(a)*log(sqrt(sqrt(a**2 + x**2) + x) + sqrt(a))