Integrand size = 12, antiderivative size = 120 \[ \int \arctan \left (x \sqrt {1+x^2}\right ) \, dx=x \arctan \left (x \sqrt {1+x^2}\right )+\frac {1}{2} \arctan \left (\sqrt {3}-2 \sqrt {1+x^2}\right )-\frac {1}{2} \arctan \left (\sqrt {3}+2 \sqrt {1+x^2}\right )-\frac {1}{4} \sqrt {3} \log \left (2+x^2-\sqrt {3} \sqrt {1+x^2}\right )+\frac {1}{4} \sqrt {3} \log \left (2+x^2+\sqrt {3} \sqrt {1+x^2}\right ) \]
-1/2*arctan(-3^(1/2)+2*(x^2+1)^(1/2))+x*arctan(x*(x^2+1)^(1/2))-1/2*arctan (3^(1/2)+2*(x^2+1)^(1/2))-1/4*ln(2+x^2-3^(1/2)*(x^2+1)^(1/2))*3^(1/2)+1/4* ln(2+x^2+3^(1/2)*(x^2+1)^(1/2))*3^(1/2)
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.79 \[ \int \arctan \left (x \sqrt {1+x^2}\right ) \, dx=-\frac {1}{2} \left (1-i \sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {1+x^2}\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt {1+x^2}\right )+x \arctan \left (x \sqrt {1+x^2}\right ) \]
-1/2*((1 - I*Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*Sqrt[1 + x^2])/2]) - ((1 + I *Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*Sqrt[1 + x^2])/2])/2 + x*ArcTan[x*Sqrt[1 + x^2]]
Time = 0.39 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5726, 2238, 1197, 25, 1483, 27, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \arctan \left (x \sqrt {x^2+1}\right ) \, dx\) |
| \(\Big \downarrow \) 5726 |
| \(\displaystyle x \arctan \left (x \sqrt {x^2+1}\right )-\int \frac {x \left (2 x^2+1\right )}{\sqrt {x^2+1} \left (x^4+x^2+1\right )}dx\) |
| \(\Big \downarrow \) 2238 |
| \(\displaystyle x \arctan \left (x \sqrt {x^2+1}\right )-\frac {1}{2} \int \frac {2 x^2+1}{\sqrt {x^2+1} \left (x^4+x^2+1\right )}dx^2\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle x \arctan \left (x \sqrt {x^2+1}\right )-\int -\frac {1-2 x^4}{x^8-x^4+1}d\sqrt {x^2+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {1-2 x^4}{x^8-x^4+1}d\sqrt {x^2+1}+x \arctan \left (x \sqrt {x^2+1}\right )\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\int \frac {\sqrt {3}-3 \sqrt {x^2+1}}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}}{2 \sqrt {3}}+\frac {\int \frac {\sqrt {3} \left (\sqrt {3} \sqrt {x^2+1}+1\right )}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}}{2 \sqrt {3}}+x \arctan \left (x \sqrt {x^2+1}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {3}-3 \sqrt {x^2+1}}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}}{2 \sqrt {3}}+\frac {1}{2} \int \frac {\sqrt {3} \sqrt {x^2+1}+1}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}+x \arctan \left (x \sqrt {x^2+1}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {-\frac {1}{2} \sqrt {3} \int \frac {1}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}-\frac {3}{2} \int -\frac {\sqrt {3}-2 \sqrt {x^2+1}}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}}{2 \sqrt {3}}+\frac {1}{2} \left (\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt {x^2+1}+\sqrt {3}}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}-\frac {1}{2} \int \frac {1}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}\right )+x \arctan \left (x \sqrt {x^2+1}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3}{2} \int \frac {\sqrt {3}-2 \sqrt {x^2+1}}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}-\frac {1}{2} \sqrt {3} \int \frac {1}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}}{2 \sqrt {3}}+\frac {1}{2} \left (\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt {x^2+1}+\sqrt {3}}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}-\frac {1}{2} \int \frac {1}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}\right )+x \arctan \left (x \sqrt {x^2+1}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\sqrt {3} \int \frac {1}{-x^4-1}d\left (2 \sqrt {x^2+1}-\sqrt {3}\right )+\frac {3}{2} \int \frac {\sqrt {3}-2 \sqrt {x^2+1}}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}}{2 \sqrt {3}}+\frac {1}{2} \left (\int \frac {1}{-x^4-1}d\left (2 \sqrt {x^2+1}+\sqrt {3}\right )+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt {x^2+1}+\sqrt {3}}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}\right )+x \arctan \left (x \sqrt {x^2+1}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {3}{2} \int \frac {\sqrt {3}-2 \sqrt {x^2+1}}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}+\sqrt {3} \arctan \left (\sqrt {3}-2 \sqrt {x^2+1}\right )}{2 \sqrt {3}}+\frac {1}{2} \left (\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt {x^2+1}+\sqrt {3}}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}-\arctan \left (2 \sqrt {x^2+1}+\sqrt {3}\right )\right )+x \arctan \left (x \sqrt {x^2+1}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle x \arctan \left (x \sqrt {x^2+1}\right )+\frac {\sqrt {3} \arctan \left (\sqrt {3}-2 \sqrt {x^2+1}\right )-\frac {3}{2} \log \left (x^4-\sqrt {3} \sqrt {x^2+1}+1\right )}{2 \sqrt {3}}+\frac {1}{2} \left (\frac {1}{2} \sqrt {3} \log \left (x^4+\sqrt {3} \sqrt {x^2+1}+1\right )-\arctan \left (2 \sqrt {x^2+1}+\sqrt {3}\right )\right )\) |
x*ArcTan[x*Sqrt[1 + x^2]] + (Sqrt[3]*ArcTan[Sqrt[3] - 2*Sqrt[1 + x^2]] - ( 3*Log[1 + x^4 - Sqrt[3]*Sqrt[1 + x^2]])/2)/(2*Sqrt[3]) + (-ArcTan[Sqrt[3] + 2*Sqrt[1 + x^2]] + (Sqrt[3]*Log[1 + x^4 + Sqrt[3]*Sqrt[1 + x^2]])/2)/2
3.1.47.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Int[(Px_)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[(Px /. x -> Sqrt[x])*(d + e*x )^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x^2]
Int[ArcTan[u_], x_Symbol] :> Simp[x*ArcTan[u], x] - Int[SimplifyIntegrand[x *(D[u, x]/(1 + u^2)), x], x] /; InverseFunctionFreeQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(507\) vs. \(2(92)=184\).
Time = 0.17 (sec) , antiderivative size = 508, normalized size of antiderivative = 4.23
| method | result | size |
| default | \(x \arctan \left (x \sqrt {x^{2}+1}\right )+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+2}\, \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )}{3 \sqrt {\frac {\frac {\left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+1}{\left (\frac {-1+x}{-1-x}+1\right )^{2}}}\, \left (\frac {-1+x}{-1-x}+1\right )}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )}{3 \sqrt {\frac {\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {1+x}{1-x}+1\right )^{2}}}\, \left (\frac {1+x}{1-x}+1\right )}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )-3 \arctan \left (\frac {\sqrt {\frac {2 \left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+2}\, \left (-1+x \right )}{\left (\frac {\left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+1\right ) \left (-1-x \right )}\right )\right )}{12 \sqrt {\frac {\frac {\left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+1}{\left (\frac {-1+x}{-1-x}+1\right )^{2}}}\, \left (\frac {-1+x}{-1-x}+1\right )}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )-3 \arctan \left (\frac {\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (1+x \right )}{\left (\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{12 \sqrt {\frac {\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {1+x}{1-x}+1\right )^{2}}}\, \left (\frac {1+x}{1-x}+1\right )}\) | \(508\) |
| parts | \(x \arctan \left (x \sqrt {x^{2}+1}\right )+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+2}\, \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )}{3 \sqrt {\frac {\frac {\left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+1}{\left (\frac {-1+x}{-1-x}+1\right )^{2}}}\, \left (\frac {-1+x}{-1-x}+1\right )}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )}{3 \sqrt {\frac {\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {1+x}{1-x}+1\right )^{2}}}\, \left (\frac {1+x}{1-x}+1\right )}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )-3 \arctan \left (\frac {\sqrt {\frac {2 \left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+2}\, \left (-1+x \right )}{\left (\frac {\left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+1\right ) \left (-1-x \right )}\right )\right )}{12 \sqrt {\frac {\frac {\left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+1}{\left (\frac {-1+x}{-1-x}+1\right )^{2}}}\, \left (\frac {-1+x}{-1-x}+1\right )}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )-3 \arctan \left (\frac {\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (1+x \right )}{\left (\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{12 \sqrt {\frac {\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {1+x}{1-x}+1\right )^{2}}}\, \left (\frac {1+x}{1-x}+1\right )}\) | \(508\) |
x*arctan(x*(x^2+1)^(1/2))+1/3*2^(1/2)/(((-1+x)^2/(-1-x)^2+1)/((-1+x)/(-1-x )+1)^2)^(1/2)/((-1+x)/(-1-x)+1)*(2*(-1+x)^2/(-1-x)^2+2)^(1/2)*3^(1/2)*arct anh(1/2*(2*(-1+x)^2/(-1-x)^2+2)^(1/2)*3^(1/2))+1/3*2^(1/2)/(((1+x)^2/(1-x) ^2+1)/((1+x)/(1-x)+1)^2)^(1/2)/((1+x)/(1-x)+1)*(2*(1+x)^2/(1-x)^2+2)^(1/2) *3^(1/2)*arctanh(1/2*(2*(1+x)^2/(1-x)^2+2)^(1/2)*3^(1/2))-1/12*2^(1/2)*(2* (-1+x)^2/(-1-x)^2+2)^(1/2)*(3^(1/2)*arctanh(1/2*(2*(-1+x)^2/(-1-x)^2+2)^(1 /2)*3^(1/2))-3*arctan(1/((-1+x)^2/(-1-x)^2+1)*(2*(-1+x)^2/(-1-x)^2+2)^(1/2 )*(-1+x)/(-1-x)))/(((-1+x)^2/(-1-x)^2+1)/((-1+x)/(-1-x)+1)^2)^(1/2)/((-1+x )/(-1-x)+1)-1/12*2^(1/2)*(2*(1+x)^2/(1-x)^2+2)^(1/2)*(3^(1/2)*arctanh(1/2* (2*(1+x)^2/(1-x)^2+2)^(1/2)*3^(1/2))-3*arctan(1/((1+x)^2/(1-x)^2+1)*(2*(1+ x)^2/(1-x)^2+2)^(1/2)*(1+x)/(1-x)))/(((1+x)^2/(1-x)^2+1)/((1+x)/(1-x)+1)^2 )^(1/2)/((1+x)/(1-x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (92) = 184\).
Time = 0.26 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.48 \[ \int \arctan \left (x \sqrt {1+x^2}\right ) \, dx=x \arctan \left (\sqrt {x^{2} + 1} x\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {-3} + 1} \log \left (4 \, x^{2} + \sqrt {2} {\left (\sqrt {-3} x + x\right )} \sqrt {-\sqrt {-3} + 1} - \sqrt {x^{2} + 1} {\left (\sqrt {2} {\left (\sqrt {-3} + 1\right )} \sqrt {-\sqrt {-3} + 1} + 4 \, x\right )} + 4\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {-3} + 1} \log \left (4 \, x^{2} - \sqrt {2} {\left (\sqrt {-3} x + x\right )} \sqrt {-\sqrt {-3} + 1} + \sqrt {x^{2} + 1} {\left (\sqrt {2} {\left (\sqrt {-3} + 1\right )} \sqrt {-\sqrt {-3} + 1} - 4 \, x\right )} + 4\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {-3} + 1} \log \left (4 \, x^{2} - 4 \, \sqrt {x^{2} + 1} x + {\left (\sqrt {2} \sqrt {x^{2} + 1} {\left (\sqrt {-3} - 1\right )} - \sqrt {2} {\left (\sqrt {-3} x - x\right )}\right )} \sqrt {\sqrt {-3} + 1} + 4\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {-3} + 1} \log \left (4 \, x^{2} - 4 \, \sqrt {x^{2} + 1} x - {\left (\sqrt {2} \sqrt {x^{2} + 1} {\left (\sqrt {-3} - 1\right )} - \sqrt {2} {\left (\sqrt {-3} x - x\right )}\right )} \sqrt {\sqrt {-3} + 1} + 4\right ) \]
x*arctan(sqrt(x^2 + 1)*x) - 1/4*sqrt(2)*sqrt(-sqrt(-3) + 1)*log(4*x^2 + sq rt(2)*(sqrt(-3)*x + x)*sqrt(-sqrt(-3) + 1) - sqrt(x^2 + 1)*(sqrt(2)*(sqrt( -3) + 1)*sqrt(-sqrt(-3) + 1) + 4*x) + 4) + 1/4*sqrt(2)*sqrt(-sqrt(-3) + 1) *log(4*x^2 - sqrt(2)*(sqrt(-3)*x + x)*sqrt(-sqrt(-3) + 1) + sqrt(x^2 + 1)* (sqrt(2)*(sqrt(-3) + 1)*sqrt(-sqrt(-3) + 1) - 4*x) + 4) - 1/4*sqrt(2)*sqrt (sqrt(-3) + 1)*log(4*x^2 - 4*sqrt(x^2 + 1)*x + (sqrt(2)*sqrt(x^2 + 1)*(sqr t(-3) - 1) - sqrt(2)*(sqrt(-3)*x - x))*sqrt(sqrt(-3) + 1) + 4) + 1/4*sqrt( 2)*sqrt(sqrt(-3) + 1)*log(4*x^2 - 4*sqrt(x^2 + 1)*x - (sqrt(2)*sqrt(x^2 + 1)*(sqrt(-3) - 1) - sqrt(2)*(sqrt(-3)*x - x))*sqrt(sqrt(-3) + 1) + 4)
\[ \int \arctan \left (x \sqrt {1+x^2}\right ) \, dx=\int \operatorname {atan}{\left (x \sqrt {x^{2} + 1} \right )}\, dx \]
\[ \int \arctan \left (x \sqrt {1+x^2}\right ) \, dx=\int { \arctan \left (\sqrt {x^{2} + 1} x\right ) \,d x } \]
x*arctan(sqrt(x^2 + 1)*x) - integrate((2*x^3 + x)*sqrt(x^2 + 1)/((x^4 + x^ 2)*(x^2 + 1) + x^2 + 1), x)
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.77 \[ \int \arctan \left (x \sqrt {1+x^2}\right ) \, dx=x \arctan \left (\sqrt {x^{2} + 1} x\right ) + \frac {1}{4} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} \sqrt {x^{2} + 1} + 2\right ) - \frac {1}{4} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} \sqrt {x^{2} + 1} + 2\right ) - \frac {1}{2} \, \arctan \left (\sqrt {3} + 2 \, \sqrt {x^{2} + 1}\right ) - \frac {1}{2} \, \arctan \left (-\sqrt {3} + 2 \, \sqrt {x^{2} + 1}\right ) \]
x*arctan(sqrt(x^2 + 1)*x) + 1/4*sqrt(3)*log(x^2 + sqrt(3)*sqrt(x^2 + 1) + 2) - 1/4*sqrt(3)*log(x^2 - sqrt(3)*sqrt(x^2 + 1) + 2) - 1/2*arctan(sqrt(3) + 2*sqrt(x^2 + 1)) - 1/2*arctan(-sqrt(3) + 2*sqrt(x^2 + 1))
Time = 1.10 (sec) , antiderivative size = 413, normalized size of antiderivative = 3.44 \[ \int \arctan \left (x \sqrt {1+x^2}\right ) \, dx=x\,\mathrm {atan}\left (x\,\sqrt {x^2+1}\right )-\frac {\left (\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (\frac {x}{2}+\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}+1+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\left (2\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{\sqrt {{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}\,\left (4\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3+1+\sqrt {3}\,1{}\mathrm {i}\right )}-\frac {\left (\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\left (2\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{\sqrt {{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}\,\left (4\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3-1+\sqrt {3}\,1{}\mathrm {i}\right )}-\frac {\left (\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (\frac {x}{2}+\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}+1-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\left (2\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{\sqrt {{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}\,\left (4\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3-1+\sqrt {3}\,1{}\mathrm {i}\right )}-\frac {\left (\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\left (2\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{\sqrt {{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}\,\left (4\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3+1+\sqrt {3}\,1{}\mathrm {i}\right )} \]
x*atan(x*(x^2 + 1)^(1/2)) - ((log(x - (3^(1/2)*1i)/2 - 1/2) - log(x/2 + (3 ^(1/2)/2 + 1i/2)*(x^2 + 1)^(1/2) + (3^(1/2)*x*1i)/2 + 1))*((3^(1/2)*1i)/2 + 2*((3^(1/2)*1i)/2 + 1/2)^3 + 1/2))/((((3^(1/2)*1i)/2 + 1/2)^2 + 1)^(1/2) *(3^(1/2)*1i + 4*((3^(1/2)*1i)/2 + 1/2)^3 + 1)) - ((log(x - (3^(1/2)*1i)/2 + 1/2) - log((3^(1/2)/2 - 1i/2)*(x^2 + 1)^(1/2) - x/2 + (3^(1/2)*x*1i)/2 + 1))*((3^(1/2)*1i)/2 + 2*((3^(1/2)*1i)/2 - 1/2)^3 - 1/2))/((((3^(1/2)*1i) /2 - 1/2)^2 + 1)^(1/2)*(3^(1/2)*1i + 4*((3^(1/2)*1i)/2 - 1/2)^3 - 1)) - (( log(x + (3^(1/2)*1i)/2 - 1/2) - log(x/2 + (3^(1/2)/2 - 1i/2)*(x^2 + 1)^(1/ 2) - (3^(1/2)*x*1i)/2 + 1))*((3^(1/2)*1i)/2 + 2*((3^(1/2)*1i)/2 - 1/2)^3 - 1/2))/((((3^(1/2)*1i)/2 - 1/2)^2 + 1)^(1/2)*(3^(1/2)*1i + 4*((3^(1/2)*1i) /2 - 1/2)^3 - 1)) - ((log(x + (3^(1/2)*1i)/2 + 1/2) - log((3^(1/2)/2 + 1i/ 2)*(x^2 + 1)^(1/2) - x/2 - (3^(1/2)*x*1i)/2 + 1))*((3^(1/2)*1i)/2 + 2*((3^ (1/2)*1i)/2 + 1/2)^3 + 1/2))/((((3^(1/2)*1i)/2 + 1/2)^2 + 1)^(1/2)*(3^(1/2 )*1i + 4*((3^(1/2)*1i)/2 + 1/2)^3 + 1))