Integrand size = 14, antiderivative size = 106 \[ \int \arctan \left (x \sqrt {1-x^2}\right ) \, dx=-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {1-x^2}\right )+x \arctan \left (x \sqrt {1-x^2}\right )+\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {1-x^2}\right ) \]
x*arctan(x*(-x^2+1)^(1/2))+1/2*arctanh(1/2*(-x^2+1)^(1/2)*(-2+2*5^(1/2))^( 1/2))*(-2+2*5^(1/2))^(1/2)-1/2*arctan(1/2*(-x^2+1)^(1/2)*(2+2*5^(1/2))^(1/ 2))*(2+2*5^(1/2))^(1/2)
Time = 0.18 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \arctan \left (x \sqrt {1-x^2}\right ) \, dx=x \arctan \left (x \sqrt {1-x^2}\right )-\frac {\sqrt {1+\sqrt {5}} \arctan \left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {1-x^2}\right )-\sqrt {-1+\sqrt {5}} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {1-x^2}\right )}{\sqrt {2}} \]
x*ArcTan[x*Sqrt[1 - x^2]] - (Sqrt[1 + Sqrt[5]]*ArcTan[Sqrt[(1 + Sqrt[5])/2 ]*Sqrt[1 - x^2]] - Sqrt[-1 + Sqrt[5]]*ArcTanh[Sqrt[(-1 + Sqrt[5])/2]*Sqrt[ 1 - x^2]])/Sqrt[2]
Time = 0.32 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5726, 2238, 1197, 1480, 217, 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.
| \(\displaystyle \int \arctan \left (x \sqrt {1-x^2}\right ) \, dx\) |
| \(\Big \downarrow \) 5726 |
| \(\displaystyle x \arctan \left (x \sqrt {1-x^2}\right )-\int \frac {x \left (1-2 x^2\right )}{\sqrt {1-x^2} \left (-x^4+x^2+1\right )}dx\) |
| \(\Big \downarrow \) 2238 |
| \(\displaystyle x \arctan \left (x \sqrt {1-x^2}\right )-\frac {1}{2} \int \frac {1-2 x^2}{\sqrt {1-x^2} \left (-x^4+x^2+1\right )}dx^2\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle x \arctan \left (x \sqrt {1-x^2}\right )-\int \frac {1-2 x^4}{-x^8+x^4+1}d\sqrt {1-x^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \int \frac {1}{\frac {1}{2} \left (1-\sqrt {5}\right )-x^4}d\sqrt {1-x^2}+\int \frac {1}{\frac {1}{2} \left (1+\sqrt {5}\right )-x^4}d\sqrt {1-x^2}+x \arctan \left (x \sqrt {1-x^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \int \frac {1}{\frac {1}{2} \left (1+\sqrt {5}\right )-x^4}d\sqrt {1-x^2}-\sqrt {\frac {2}{\sqrt {5}-1}} \arctan \left (\sqrt {\frac {2}{\sqrt {5}-1}} \sqrt {1-x^2}\right )+x \arctan \left (x \sqrt {1-x^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\sqrt {\frac {2}{\sqrt {5}-1}} \arctan \left (\sqrt {\frac {2}{\sqrt {5}-1}} \sqrt {1-x^2}\right )+x \arctan \left (x \sqrt {1-x^2}\right )+\sqrt {\frac {2}{1+\sqrt {5}}} \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {1-x^2}\right )\) |
-(Sqrt[2/(-1 + Sqrt[5])]*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*Sqrt[1 - x^2]]) + x *ArcTan[x*Sqrt[1 - x^2]] + Sqrt[2/(1 + Sqrt[5])]*ArcTanh[Sqrt[2/(1 + Sqrt[ 5])]*Sqrt[1 - x^2]]
3.1.50.3.1 Defintions of rubi rules used
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_)^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[((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[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[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. \(209\) vs. \(2(79)=158\).
Time = 0.10 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.98
| method | result | size |
| default | \(x \arctan \left (x \sqrt {-x^{2}+1}\right )+\frac {\sqrt {5}\, \arctan \left (\frac {\frac {2 \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}-2 \sqrt {5}+4}{4 \sqrt {-2+\sqrt {5}}}\right )}{5 \sqrt {-2+\sqrt {5}}}+\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\frac {2 \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+4+2 \sqrt {5}}{4 \sqrt {2+\sqrt {5}}}\right )}{5 \sqrt {2+\sqrt {5}}}+\frac {\left (-\frac {1}{2}+\frac {3 \sqrt {5}}{10}\right ) \arctan \left (\frac {\frac {2 \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}-2 \sqrt {5}+4}{4 \sqrt {-2+\sqrt {5}}}\right )}{\sqrt {-2+\sqrt {5}}}+\frac {\left (\frac {1}{2}+\frac {3 \sqrt {5}}{10}\right ) \operatorname {arctanh}\left (\frac {\frac {2 \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+4+2 \sqrt {5}}{4 \sqrt {2+\sqrt {5}}}\right )}{\sqrt {2+\sqrt {5}}}\) | \(210\) |
| parts | \(x \arctan \left (x \sqrt {-x^{2}+1}\right )+\frac {\sqrt {5}\, \arctan \left (\frac {\frac {2 \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}-2 \sqrt {5}+4}{4 \sqrt {-2+\sqrt {5}}}\right )}{5 \sqrt {-2+\sqrt {5}}}+\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\frac {2 \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+4+2 \sqrt {5}}{4 \sqrt {2+\sqrt {5}}}\right )}{5 \sqrt {2+\sqrt {5}}}+\frac {\left (-\frac {1}{2}+\frac {3 \sqrt {5}}{10}\right ) \arctan \left (\frac {\frac {2 \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}-2 \sqrt {5}+4}{4 \sqrt {-2+\sqrt {5}}}\right )}{\sqrt {-2+\sqrt {5}}}+\frac {\left (\frac {1}{2}+\frac {3 \sqrt {5}}{10}\right ) \operatorname {arctanh}\left (\frac {\frac {2 \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+4+2 \sqrt {5}}{4 \sqrt {2+\sqrt {5}}}\right )}{\sqrt {2+\sqrt {5}}}\) | \(210\) |
x*arctan(x*(-x^2+1)^(1/2))+1/5*5^(1/2)/(-2+5^(1/2))^(1/2)*arctan(1/4*(2*(( -x^2+1)^(1/2)-1)^2/x^2-2*5^(1/2)+4)/(-2+5^(1/2))^(1/2))+1/5*5^(1/2)/(2+5^( 1/2))^(1/2)*arctanh(1/4*(2*((-x^2+1)^(1/2)-1)^2/x^2+4+2*5^(1/2))/(2+5^(1/2 ))^(1/2))+(-1/2+3/10*5^(1/2))/(-2+5^(1/2))^(1/2)*arctan(1/4*(2*((-x^2+1)^( 1/2)-1)^2/x^2-2*5^(1/2)+4)/(-2+5^(1/2))^(1/2))+(1/2+3/10*5^(1/2))/(2+5^(1/ 2))^(1/2)*arctanh(1/4*(2*((-x^2+1)^(1/2)-1)^2/x^2+4+2*5^(1/2))/(2+5^(1/2)) ^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (79) = 158\).
Time = 0.27 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.93 \[ \int \arctan \left (x \sqrt {1-x^2}\right ) \, dx=x \arctan \left (\sqrt {-x^{2} + 1} x\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left ({\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {\sqrt {5} - 1} + 4 \, \sqrt {-x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (-{\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {\sqrt {5} - 1} + 4 \, \sqrt {-x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {5} - 1} \log \left ({\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {-\sqrt {5} - 1} + 4 \, \sqrt {-x^{2} + 1}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {5} - 1} \log \left (-{\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {-\sqrt {5} - 1} + 4 \, \sqrt {-x^{2} + 1}\right ) \]
x*arctan(sqrt(-x^2 + 1)*x) + 1/4*sqrt(2)*sqrt(sqrt(5) - 1)*log((sqrt(5)*sq rt(2) + sqrt(2))*sqrt(sqrt(5) - 1) + 4*sqrt(-x^2 + 1)) - 1/4*sqrt(2)*sqrt( sqrt(5) - 1)*log(-(sqrt(5)*sqrt(2) + sqrt(2))*sqrt(sqrt(5) - 1) + 4*sqrt(- x^2 + 1)) - 1/4*sqrt(2)*sqrt(-sqrt(5) - 1)*log((sqrt(5)*sqrt(2) - sqrt(2)) *sqrt(-sqrt(5) - 1) + 4*sqrt(-x^2 + 1)) + 1/4*sqrt(2)*sqrt(-sqrt(5) - 1)*l og(-(sqrt(5)*sqrt(2) - sqrt(2))*sqrt(-sqrt(5) - 1) + 4*sqrt(-x^2 + 1))
Timed out. \[ \int \arctan \left (x \sqrt {1-x^2}\right ) \, dx=\text {Timed out} \]
\[ \int \arctan \left (x \sqrt {1-x^2}\right ) \, dx=\int { \arctan \left (\sqrt {-x^{2} + 1} x\right ) \,d x } \]
x*arctan(sqrt(x + 1)*x*sqrt(-x + 1)) - integrate((2*x^3 - x)*e^(1/2*log(x + 1) + 1/2*log(-x + 1))/(x^2 + (x^4 - x^2)*e^(log(x + 1) + log(-x + 1)) - 1), x)
Time = 0.34 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.05 \[ \int \arctan \left (x \sqrt {1-x^2}\right ) \, dx=x \arctan \left (\sqrt {-x^{2} + 1} x\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {\sqrt {-x^{2} + 1}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (\sqrt {-x^{2} + 1} + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | \sqrt {-x^{2} + 1} - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) \]
x*arctan(sqrt(-x^2 + 1)*x) - 1/2*sqrt(2*sqrt(5) + 2)*arctan(sqrt(-x^2 + 1) /sqrt(1/2*sqrt(5) - 1/2)) + 1/4*sqrt(2*sqrt(5) - 2)*log(sqrt(-x^2 + 1) + s qrt(1/2*sqrt(5) + 1/2)) - 1/4*sqrt(2*sqrt(5) - 2)*log(abs(sqrt(-x^2 + 1) - sqrt(1/2*sqrt(5) + 1/2)))
Time = 1.15 (sec) , antiderivative size = 455, normalized size of antiderivative = 4.29 \[ \int \arctan \left (x \sqrt {1-x^2}\right ) \, dx=x\,\mathrm {atan}\left (x\,\sqrt {1-x^2}\right )+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\,\left (\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}-2\,{\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )}^{3/2}\right )}{\left (2\,\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}-4\,{\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )\,\left (\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}-2\,{\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}^{3/2}\right )}{\left (2\,\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}-4\,{\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}^{3/2}\right )\,\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}}+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\,\left (\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}-2\,{\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )}^{3/2}\right )}{\left (2\,\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}-4\,{\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )\,\left (\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}-2\,{\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}^{3/2}\right )}{\left (2\,\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}-4\,{\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}^{3/2}\right )\,\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}} \]
x*atan(x*(1 - x^2)^(1/2)) + (log((((x*(5^(1/2)/2 + 1/2)^(1/2) - 1)*1i)/(1/ 2 - 5^(1/2)/2)^(1/2) - (1 - x^2)^(1/2)*1i)/(x - (5^(1/2)/2 + 1/2)^(1/2)))* ((5^(1/2)/2 + 1/2)^(1/2) - 2*(5^(1/2)/2 + 1/2)^(3/2)))/((2*(5^(1/2)/2 + 1/ 2)^(1/2) - 4*(5^(1/2)/2 + 1/2)^(3/2))*(1/2 - 5^(1/2)/2)^(1/2)) + (log((((x *(1/2 - 5^(1/2)/2)^(1/2) - 1)*1i)/(5^(1/2)/2 + 1/2)^(1/2) - (1 - x^2)^(1/2 )*1i)/(x - (1/2 - 5^(1/2)/2)^(1/2)))*((1/2 - 5^(1/2)/2)^(1/2) - 2*(1/2 - 5 ^(1/2)/2)^(3/2)))/((2*(1/2 - 5^(1/2)/2)^(1/2) - 4*(1/2 - 5^(1/2)/2)^(3/2)) *(5^(1/2)/2 + 1/2)^(1/2)) + (log((((x*(5^(1/2)/2 + 1/2)^(1/2) + 1)*1i)/(1/ 2 - 5^(1/2)/2)^(1/2) + (1 - x^2)^(1/2)*1i)/(x + (5^(1/2)/2 + 1/2)^(1/2)))* ((5^(1/2)/2 + 1/2)^(1/2) - 2*(5^(1/2)/2 + 1/2)^(3/2)))/((2*(5^(1/2)/2 + 1/ 2)^(1/2) - 4*(5^(1/2)/2 + 1/2)^(3/2))*(1/2 - 5^(1/2)/2)^(1/2)) + (log((((x *(1/2 - 5^(1/2)/2)^(1/2) + 1)*1i)/(5^(1/2)/2 + 1/2)^(1/2) + (1 - x^2)^(1/2 )*1i)/(x + (1/2 - 5^(1/2)/2)^(1/2)))*((1/2 - 5^(1/2)/2)^(1/2) - 2*(1/2 - 5 ^(1/2)/2)^(3/2)))/((2*(1/2 - 5^(1/2)/2)^(1/2) - 4*(1/2 - 5^(1/2)/2)^(3/2)) *(5^(1/2)/2 + 1/2)^(1/2))