\(\int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\) [1405]

Optimal result

Integrand size = 27, antiderivative size = 100 \[ \int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \arctan \left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2144, 464, 335, 304, 209, 212} \[ \int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}}+\frac {2}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}} \]

[In]

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

[Out]

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

Rule 209

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 2144

Int[((g_.) + (h_.)*(x_))^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dist[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] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \left (-b^2+x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right ) \\ & = \frac {2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+2 \text {Subst}\left (\int \frac {\sqrt {x}}{-b^2+x^2} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right ) \\ & = \frac {2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+4 \text {Subst}\left (\int \frac {x^2}{-b^2+x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right ) \\ & = \frac {2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}-2 \text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )+2 \text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right ) \\ & = \frac {2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \arctan \left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \arctan \left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (80) = 160\).

Time = 0.29 (sec) , antiderivative size = 321, normalized size of antiderivative = 3.21 \[ \int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\left [\frac {2 \, b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{\sqrt {b}}\right ) + b^{\frac {3}{2}} \log \left (\frac {b^{2} + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left ({\left (a x - b\right )} \sqrt {b} - \sqrt {a^{2} x^{2} + b^{2}} \sqrt {b}\right )} + \sqrt {a^{2} x^{2} + b^{2}} b}{x}\right ) - 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (a x - \sqrt {a^{2} x^{2} + b^{2}}\right )}}{b^{2}}, \frac {2 \, \sqrt {-b} b \arctan \left (\frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \sqrt {-b}}{b}\right ) - \sqrt {-b} b \log \left (-\frac {b^{2} + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left ({\left (a x + b\right )} \sqrt {-b} - \sqrt {a^{2} x^{2} + b^{2}} \sqrt {-b}\right )} - \sqrt {a^{2} x^{2} + b^{2}} b}{x}\right ) - 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (a x - \sqrt {a^{2} x^{2} + b^{2}}\right )}}{b^{2}}\right ] \]

[In]

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

[Out]

[(2*b^(3/2)*arctan(sqrt(a*x + sqrt(a^2*x^2 + b^2))/sqrt(b)) + b^(3/2)*log((b^2 + sqrt(a*x + sqrt(a^2*x^2 + b^2
))*((a*x - b)*sqrt(b) - sqrt(a^2*x^2 + b^2)*sqrt(b)) + sqrt(a^2*x^2 + b^2)*b)/x) - 2*sqrt(a*x + sqrt(a^2*x^2 +
b^2))*(a*x - sqrt(a^2*x^2 + b^2)))/b^2, (2*sqrt(-b)*b*arctan(sqrt(a*x + sqrt(a^2*x^2 + b^2))*sqrt(-b)/b) - sq
rt(-b)*b*log(-(b^2 + sqrt(a*x + sqrt(a^2*x^2 + b^2))*((a*x + b)*sqrt(-b) - sqrt(a^2*x^2 + b^2)*sqrt(-b)) - sqr
t(a^2*x^2 + b^2)*b)/x) - 2*sqrt(a*x + sqrt(a^2*x^2 + b^2))*(a*x - sqrt(a^2*x^2 + b^2)))/b^2]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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