Integrand size = 27, antiderivative size = 100 \[ \int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \arctan \left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}} \]
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)
Time = 0.12 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \arctan \left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}} \]
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[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b] ])/Sqrt[b]
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2544, 25, 359, 266, 827, 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.
\(\displaystyle \int \frac {1}{x \sqrt {\sqrt {a^2 x^2+b^2}+a x}} \, dx\) |
\(\Big \downarrow \) 2544 |
\(\displaystyle \int -\frac {\left (\sqrt {a^2 x^2+b^2}+a x\right )^2+b^2}{\left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2} \left (b^2-\left (\sqrt {a^2 x^2+b^2}+a x\right )^2\right )}d\left (\sqrt {a^2 x^2+b^2}+a x\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {b^2+\left (a x+\sqrt {b^2+a^2 x^2}\right )^2}{\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2} \left (b^2-\left (a x+\sqrt {b^2+a^2 x^2}\right )^2\right )}d\left (a x+\sqrt {b^2+a^2 x^2}\right )\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \frac {2}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}-2 \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (a x+\sqrt {b^2+a^2 x^2}\right )^2}d\left (a x+\sqrt {b^2+a^2 x^2}\right )\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}-4 \int \frac {a x+\sqrt {b^2+a^2 x^2}}{b^2-\left (a x+\sqrt {b^2+a^2 x^2}\right )^2}d\sqrt {a x+\sqrt {b^2+a^2 x^2}}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {2}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}-4 \left (\frac {1}{2} \int \frac {1}{b-a x-\sqrt {b^2+a^2 x^2}}d\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\frac {1}{2} \int \frac {1}{b+a x+\sqrt {b^2+a^2 x^2}}d\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {2}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}-4 \left (\frac {1}{2} \int \frac {1}{b-a x-\sqrt {b^2+a^2 x^2}}d\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\frac {\arctan \left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{2 \sqrt {b}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}-4 \left (\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{2 \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{2 \sqrt {b}}\right )\) |
2/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - 4*(-1/2*ArcTan[Sqrt[a*x + Sqrt[b^2 + a ^2*x^2]]/Sqrt[b]]/Sqrt[b] + ArcTanh[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b ]]/(2*Sqrt[b]))
3.15.5.3.1 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[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) 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]
\[\int \frac {1}{x \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]
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 \[ \int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\left [\frac {2 \, b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{\sqrt {b}}\right ) + b^{\frac {3}{2}} \log \left (\frac {b^{2} + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left ({\left (a x - b\right )} \sqrt {b} - \sqrt {a^{2} x^{2} + b^{2}} \sqrt {b}\right )} + \sqrt {a^{2} x^{2} + b^{2}} b}{x}\right ) - 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (a x - \sqrt {a^{2} x^{2} + b^{2}}\right )}}{b^{2}}, \frac {2 \, \sqrt {-b} b \arctan \left (\frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \sqrt {-b}}{b}\right ) - \sqrt {-b} b \log \left (-\frac {b^{2} + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left ({\left (a x + b\right )} \sqrt {-b} - \sqrt {a^{2} x^{2} + b^{2}} \sqrt {-b}\right )} - \sqrt {a^{2} x^{2} + b^{2}} b}{x}\right ) - 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (a x - \sqrt {a^{2} x^{2} + b^{2}}\right )}}{b^{2}}\right ] \]
[(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 + sqr t(a^2*x^2 + b^2))*sqrt(-b)/b) - sqrt(-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)) - 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]
\[ \int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {1}{x \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \]
\[ \int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} x} \,d x } \]
\[ \int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} x} \,d x } \]
Timed out. \[ \int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {1}{x\,\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \]