Integrand size = 20, antiderivative size = 53 \[ \int \frac {1+x}{\left (4+x^2\right ) \sqrt {9+x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {5} x}{2 \sqrt {9+x^2}}\right )}{2 \sqrt {5}}-\frac {\text {arctanh}\left (\frac {\sqrt {9+x^2}}{\sqrt {5}}\right )}{\sqrt {5}} \]
1/10*arctan(1/2*x*5^(1/2)/(x^2+9)^(1/2))*5^(1/2)-1/5*arctanh(1/5*(x^2+9)^( 1/2)*5^(1/2))*5^(1/2)
Time = 0.51 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.04 \[ \int \frac {1+x}{\left (4+x^2\right ) \sqrt {9+x^2}} \, dx=-\frac {\arctan \left (\frac {4+x^2-x \sqrt {9+x^2}}{2 \sqrt {5}}\right )+2 \text {arctanh}\left (\frac {\sqrt {9+x^2}}{\sqrt {5}}\right )}{2 \sqrt {5}} \]
-1/2*(ArcTan[(4 + x^2 - x*Sqrt[9 + x^2])/(2*Sqrt[5])] + 2*ArcTanh[Sqrt[9 + x^2]/Sqrt[5]])/Sqrt[5]
Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1343, 291, 216, 353, 73, 220}
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.
\(\displaystyle \int \frac {x+1}{\left (x^2+4\right ) \sqrt {x^2+9}} \, dx\) |
\(\Big \downarrow \) 1343 |
\(\displaystyle \int \frac {1}{\left (x^2+4\right ) \sqrt {x^2+9}}dx+\int \frac {x}{\left (x^2+4\right ) \sqrt {x^2+9}}dx\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \int \frac {x}{\left (x^2+4\right ) \sqrt {x^2+9}}dx+\int \frac {1}{\frac {5 x^2}{x^2+9}+4}d\frac {x}{\sqrt {x^2+9}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \int \frac {x}{\left (x^2+4\right ) \sqrt {x^2+9}}dx+\frac {\arctan \left (\frac {\sqrt {5} x}{2 \sqrt {x^2+9}}\right )}{2 \sqrt {5}}\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\left (x^2+4\right ) \sqrt {x^2+9}}dx^2+\frac {\arctan \left (\frac {\sqrt {5} x}{2 \sqrt {x^2+9}}\right )}{2 \sqrt {5}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \int \frac {1}{x^4-5}d\sqrt {x^2+9}+\frac {\arctan \left (\frac {\sqrt {5} x}{2 \sqrt {x^2+9}}\right )}{2 \sqrt {5}}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {5} x}{2 \sqrt {x^2+9}}\right )}{2 \sqrt {5}}-\frac {\text {arctanh}\left (\frac {\sqrt {x^2+9}}{\sqrt {5}}\right )}{\sqrt {5}}\) |
3.9.13.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((g_) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q _), x_Symbol] :> Simp[g Int[(a + c*x^2)^p*(d + f*x^2)^q, x], x] + Simp[h Int[x*(a + c*x^2)^p*(d + f*x^2)^q, x], x] /; FreeQ[{a, c, d, f, g, h, p, q}, x]
Time = 1.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {\arctan \left (\frac {x \sqrt {5}}{2 \sqrt {x^{2}+9}}\right ) \sqrt {5}}{10}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {x^{2}+9}\, \sqrt {5}}{5}\right ) \sqrt {5}}{5}\) | \(39\) |
trager | \(\operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right ) \ln \left (-\frac {-6400 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )^{5} x +1120 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )^{3} x +1440 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )^{3}-33 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right ) x +10 \sqrt {x^{2}+9}-198 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )}{80 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )^{2} x -3 x -8}\right )-16 \ln \left (-\frac {6400 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )^{5} x -1120 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )^{3} x +1440 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )^{3}+33 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right ) x +10 \sqrt {x^{2}+9}-198 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )}{80 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )^{2} x -3 x +8}\right ) \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )^{3}+\frac {6 \ln \left (-\frac {6400 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )^{5} x -1120 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )^{3} x +1440 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )^{3}+33 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right ) x +10 \sqrt {x^{2}+9}-198 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )}{80 \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )^{2} x -3 x +8}\right ) \operatorname {RootOf}\left (1280 \textit {\_Z}^{4}-96 \textit {\_Z}^{2}+5\right )}{5}\) | \(414\) |
1/10*arctan(1/2*x*5^(1/2)/(x^2+9)^(1/2))*5^(1/2)-1/5*arctanh(1/5*(x^2+9)^( 1/2)*5^(1/2))*5^(1/2)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.36 \[ \int \frac {1+x}{\left (4+x^2\right ) \sqrt {9+x^2}} \, dx=-\frac {1}{20} \, \sqrt {5} \sqrt {4 i + 3} \log \left (-\left (i - 2\right ) \, \sqrt {5} \sqrt {4 i + 3} - 5 \, x + 5 \, \sqrt {x^{2} + 9} - 10 i\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {4 i + 3} \log \left (\left (i - 2\right ) \, \sqrt {5} \sqrt {4 i + 3} - 5 \, x + 5 \, \sqrt {x^{2} + 9} - 10 i\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {-4 i + 3} \log \left (\left (i + 2\right ) \, \sqrt {5} \sqrt {-4 i + 3} - 5 \, x + 5 \, \sqrt {x^{2} + 9} + 10 i\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-4 i + 3} \log \left (-\left (i + 2\right ) \, \sqrt {5} \sqrt {-4 i + 3} - 5 \, x + 5 \, \sqrt {x^{2} + 9} + 10 i\right ) \]
-1/20*sqrt(5)*sqrt(4*I + 3)*log(-(I - 2)*sqrt(5)*sqrt(4*I + 3) - 5*x + 5*s qrt(x^2 + 9) - 10*I) + 1/20*sqrt(5)*sqrt(4*I + 3)*log((I - 2)*sqrt(5)*sqrt (4*I + 3) - 5*x + 5*sqrt(x^2 + 9) - 10*I) - 1/20*sqrt(5)*sqrt(-4*I + 3)*lo g((I + 2)*sqrt(5)*sqrt(-4*I + 3) - 5*x + 5*sqrt(x^2 + 9) + 10*I) + 1/20*sq rt(5)*sqrt(-4*I + 3)*log(-(I + 2)*sqrt(5)*sqrt(-4*I + 3) - 5*x + 5*sqrt(x^ 2 + 9) + 10*I)
\[ \int \frac {1+x}{\left (4+x^2\right ) \sqrt {9+x^2}} \, dx=\int \frac {x + 1}{\left (x^{2} + 4\right ) \sqrt {x^{2} + 9}}\, dx \]
\[ \int \frac {1+x}{\left (4+x^2\right ) \sqrt {9+x^2}} \, dx=\int { \frac {x + 1}{\sqrt {x^{2} + 9} {\left (x^{2} + 4\right )}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (38) = 76\).
Time = 0.34 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.32 \[ \int \frac {1+x}{\left (4+x^2\right ) \sqrt {9+x^2}} \, dx=-\frac {1}{10} \, \sqrt {5} \arctan \left (\frac {1}{2} \, x - \frac {1}{2} \, \sqrt {5} - \frac {1}{2} \, \sqrt {x^{2} + 9}\right ) - \frac {1}{10} \, \sqrt {5} \arctan \left (-\frac {1}{2} \, x - \frac {1}{2} \, \sqrt {5} + \frac {1}{2} \, \sqrt {x^{2} + 9}\right ) + \frac {1}{10} \, \sqrt {5} \log \left ({\left (x - \sqrt {x^{2} + 9}\right )}^{2} + 2 \, \sqrt {5} {\left (x - \sqrt {x^{2} + 9}\right )} + 9\right ) - \frac {1}{10} \, \sqrt {5} \log \left ({\left (x - \sqrt {x^{2} + 9}\right )}^{2} - 2 \, \sqrt {5} {\left (x - \sqrt {x^{2} + 9}\right )} + 9\right ) \]
-1/10*sqrt(5)*arctan(1/2*x - 1/2*sqrt(5) - 1/2*sqrt(x^2 + 9)) - 1/10*sqrt( 5)*arctan(-1/2*x - 1/2*sqrt(5) + 1/2*sqrt(x^2 + 9)) + 1/10*sqrt(5)*log((x - sqrt(x^2 + 9))^2 + 2*sqrt(5)*(x - sqrt(x^2 + 9)) + 9) - 1/10*sqrt(5)*log ((x - sqrt(x^2 + 9))^2 - 2*sqrt(5)*(x - sqrt(x^2 + 9)) + 9)
Time = 20.64 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.26 \[ \int \frac {1+x}{\left (4+x^2\right ) \sqrt {9+x^2}} \, dx=\sqrt {5}\,\left (\ln \left (x-2{}\mathrm {i}\right )-\ln \left (\sqrt {5}\,\sqrt {x^2+9}+9+x\,2{}\mathrm {i}\right )\right )\,\left (\frac {1}{10}-\frac {1}{20}{}\mathrm {i}\right )+\sqrt {5}\,\left (\ln \left (x+2{}\mathrm {i}\right )-\ln \left (\sqrt {5}\,\sqrt {x^2+9}+9-x\,2{}\mathrm {i}\right )\right )\,\left (\frac {1}{10}+\frac {1}{20}{}\mathrm {i}\right ) \]