Integrand size = 18, antiderivative size = 55 \[ \int \arctan \left (\frac {-\sqrt {2}+2 x}{\sqrt {2}}\right ) \, dx=\frac {\arctan \left (1-\sqrt {2} x\right )}{\sqrt {2}}-x \arctan \left (1-\sqrt {2} x\right )-\frac {\log \left (1-\sqrt {2} x+x^2\right )}{2 \sqrt {2}} \] Output:
x*arctan(-1+x*2^(1/2))-1/2*arctan(-1+x*2^(1/2))*2^(1/2)-1/4*ln(1+x^2-x*2^( 1/2))*2^(1/2)
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \arctan \left (\frac {-\sqrt {2}+2 x}{\sqrt {2}}\right ) \, dx=\frac {1}{4} \left (2 \left (\sqrt {2}-2 x\right ) \arctan \left (1-\sqrt {2} x\right )-\sqrt {2} \log \left (1-\sqrt {2} x+x^2\right )\right ) \] Input:
Integrate[ArcTan[(-Sqrt[2] + 2*x)/Sqrt[2]],x]
Output:
(2*(Sqrt[2] - 2*x)*ArcTan[1 - Sqrt[2]*x] - Sqrt[2]*Log[1 - Sqrt[2]*x + x^2 ])/4
Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5726, 27, 1142, 25, 27, 1082, 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.
\(\displaystyle \int \arctan \left (\frac {2 x-\sqrt {2}}{\sqrt {2}}\right ) \, dx\) |
\(\Big \downarrow \) 5726 |
\(\displaystyle -\int \frac {x}{\sqrt {2} \left (x^2-\sqrt {2} x+1\right )}dx-x \arctan \left (1-\sqrt {2} x\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {x}{x^2-\sqrt {2} x+1}dx}{\sqrt {2}}-x \arctan \left (1-\sqrt {2} x\right )\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle -\frac {\frac {\int \frac {1}{x^2-\sqrt {2} x+1}dx}{\sqrt {2}}+\frac {1}{2} \int -\frac {\sqrt {2} \left (1-\sqrt {2} x\right )}{x^2-\sqrt {2} x+1}dx}{\sqrt {2}}-x \arctan \left (1-\sqrt {2} x\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {\int \frac {1}{x^2-\sqrt {2} x+1}dx}{\sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \left (1-\sqrt {2} x\right )}{x^2-\sqrt {2} x+1}dx}{\sqrt {2}}-x \arctan \left (1-\sqrt {2} x\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\int \frac {1}{x^2-\sqrt {2} x+1}dx}{\sqrt {2}}-\frac {\int \frac {1-\sqrt {2} x}{x^2-\sqrt {2} x+1}dx}{\sqrt {2}}}{\sqrt {2}}-x \arctan \left (1-\sqrt {2} x\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle -\frac {\int \frac {1}{-\left (1-\sqrt {2} x\right )^2-1}d\left (1-\sqrt {2} x\right )-\frac {\int \frac {1-\sqrt {2} x}{x^2-\sqrt {2} x+1}dx}{\sqrt {2}}}{\sqrt {2}}-x \arctan \left (1-\sqrt {2} x\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {-\frac {\int \frac {1-\sqrt {2} x}{x^2-\sqrt {2} x+1}dx}{\sqrt {2}}-\arctan \left (1-\sqrt {2} x\right )}{\sqrt {2}}-x \arctan \left (1-\sqrt {2} x\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle -\frac {\frac {1}{2} \log \left (x^2-\sqrt {2} x+1\right )-\arctan \left (1-\sqrt {2} x\right )}{\sqrt {2}}-x \arctan \left (1-\sqrt {2} x\right )\) |
Input:
Int[ArcTan[(-Sqrt[2] + 2*x)/Sqrt[2]],x]
Output:
-(x*ArcTan[1 - Sqrt[2]*x]) - (-ArcTan[1 - Sqrt[2]*x] + Log[1 - Sqrt[2]*x + x^2]/2)/Sqrt[2]
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] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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[ArcTan[u_], x_Symbol] :> Simp[x*ArcTan[u], x] - Int[SimplifyIntegrand[x *(D[u, x]/(1 + u^2)), x], x] /; InverseFunctionFreeQ[u, x]
Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\frac {\sqrt {2}\, \left (\left (-1+x \sqrt {2}\right ) \arctan \left (-1+x \sqrt {2}\right )-\frac {\ln \left (\left (-1+x \sqrt {2}\right )^{2}+1\right )}{2}\right )}{2}\) | \(37\) |
default | \(\frac {\sqrt {2}\, \left (\left (-1+x \sqrt {2}\right ) \arctan \left (-1+x \sqrt {2}\right )-\frac {\ln \left (\left (-1+x \sqrt {2}\right )^{2}+1\right )}{2}\right )}{2}\) | \(37\) |
parts | \(\arctan \left (\frac {\left (2 x -\sqrt {2}\right ) \sqrt {2}}{2}\right ) x -2 \sqrt {2}\, \left (\frac {\ln \left (1+x^{2}-x \sqrt {2}\right )}{8}+\frac {\arctan \left (\frac {\left (2 x -\sqrt {2}\right ) \sqrt {2}}{2}\right )}{4}\right )\) | \(56\) |
risch | \(\frac {i x \ln \left (1+\frac {i \left (-2 x +\sqrt {2}\right ) \sqrt {2}}{2}\right )}{2}-\frac {i x \ln \left (1-\frac {i \left (-2 x +\sqrt {2}\right ) \sqrt {2}}{2}\right )}{2}-\frac {\sqrt {2}\, \ln \left (4-4 x \sqrt {2}+4 x^{2}\right )}{4}-\frac {\sqrt {2}\, \arctan \left (\frac {\left (2 x -\sqrt {2}\right ) \sqrt {2}}{2}\right )}{2}\) | \(81\) |
parallelrisch | \(\frac {\sqrt {2}\, \ln \left (1+x^{2}-x \sqrt {2}\right ) x^{2}+6 \sqrt {2}\, \arctan \left (\frac {\left (2 x -\sqrt {2}\right ) \sqrt {2}}{2}\right ) x^{2}-4 x^{3} \arctan \left (\frac {\left (2 x -\sqrt {2}\right ) \sqrt {2}}{2}\right )+\ln \left (1+x^{2}-x \sqrt {2}\right ) \sqrt {2}+2 \sqrt {2}\, \arctan \left (\frac {\left (2 x -\sqrt {2}\right ) \sqrt {2}}{2}\right )-2 \ln \left (1+x^{2}-x \sqrt {2}\right ) x -8 \arctan \left (\frac {\left (2 x -\sqrt {2}\right ) \sqrt {2}}{2}\right ) x}{4 x \sqrt {2}-4 x^{2}-4}\) | \(149\) |
Input:
int(arctan(1/2*(2*x-2^(1/2))*2^(1/2)),x,method=_RETURNVERBOSE)
Output:
1/2*2^(1/2)*((-1+x*2^(1/2))*arctan(-1+x*2^(1/2))-1/2*ln((-1+x*2^(1/2))^2+1 ))
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.67 \[ \int \arctan \left (\frac {-\sqrt {2}+2 x}{\sqrt {2}}\right ) \, dx=\frac {1}{2} \, {\left (2 \, x - \sqrt {2}\right )} \arctan \left (\sqrt {2} x - 1\right ) - \frac {1}{4} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) \] Input:
integrate(arctan(1/2*(2*x-2^(1/2))*2^(1/2)),x, algorithm="fricas")
Output:
1/2*(2*x - sqrt(2))*arctan(sqrt(2)*x - 1) - 1/4*sqrt(2)*log(x^2 - sqrt(2)* x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (48) = 96\).
Time = 0.44 (sec) , antiderivative size = 230, normalized size of antiderivative = 4.18 \[ \int \arctan \left (\frac {-\sqrt {2}+2 x}{\sqrt {2}}\right ) \, dx=\frac {4 x^{3} \operatorname {atan}{\left (\sqrt {2} x - 1 \right )}}{4 x^{2} - 4 \sqrt {2} x + 4} - \frac {\sqrt {2} x^{2} \log {\left (x^{2} - \sqrt {2} x + 1 \right )}}{4 x^{2} - 4 \sqrt {2} x + 4} - \frac {6 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} x - 1 \right )}}{4 x^{2} - 4 \sqrt {2} x + 4} + \frac {2 x \log {\left (x^{2} - \sqrt {2} x + 1 \right )}}{4 x^{2} - 4 \sqrt {2} x + 4} + \frac {8 x \operatorname {atan}{\left (\sqrt {2} x - 1 \right )}}{4 x^{2} - 4 \sqrt {2} x + 4} - \frac {\sqrt {2} \log {\left (x^{2} - \sqrt {2} x + 1 \right )}}{4 x^{2} - 4 \sqrt {2} x + 4} - \frac {2 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x - 1 \right )}}{4 x^{2} - 4 \sqrt {2} x + 4} \] Input:
integrate(atan(1/2*(2*x-2**(1/2))*2**(1/2)),x)
Output:
4*x**3*atan(sqrt(2)*x - 1)/(4*x**2 - 4*sqrt(2)*x + 4) - sqrt(2)*x**2*log(x **2 - sqrt(2)*x + 1)/(4*x**2 - 4*sqrt(2)*x + 4) - 6*sqrt(2)*x**2*atan(sqrt (2)*x - 1)/(4*x**2 - 4*sqrt(2)*x + 4) + 2*x*log(x**2 - sqrt(2)*x + 1)/(4*x **2 - 4*sqrt(2)*x + 4) + 8*x*atan(sqrt(2)*x - 1)/(4*x**2 - 4*sqrt(2)*x + 4 ) - sqrt(2)*log(x**2 - sqrt(2)*x + 1)/(4*x**2 - 4*sqrt(2)*x + 4) - 2*sqrt( 2)*atan(sqrt(2)*x - 1)/(4*x**2 - 4*sqrt(2)*x + 4)
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95 \[ \int \arctan \left (\frac {-\sqrt {2}+2 x}{\sqrt {2}}\right ) \, dx=\frac {1}{4} \, \sqrt {2} {\left (\sqrt {2} {\left (2 \, x - \sqrt {2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \log \left (\frac {1}{2} \, {\left (2 \, x - \sqrt {2}\right )}^{2} + 1\right )\right )} \] Input:
integrate(arctan(1/2*(2*x-2^(1/2))*2^(1/2)),x, algorithm="maxima")
Output:
1/4*sqrt(2)*(sqrt(2)*(2*x - sqrt(2))*arctan(1/2*sqrt(2)*(2*x - sqrt(2))) - log(1/2*(2*x - sqrt(2))^2 + 1))
Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95 \[ \int \arctan \left (\frac {-\sqrt {2}+2 x}{\sqrt {2}}\right ) \, dx=\frac {1}{4} \, \sqrt {2} {\left (\sqrt {2} {\left (2 \, x - \sqrt {2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \log \left (\frac {1}{2} \, {\left (2 \, x - \sqrt {2}\right )}^{2} + 1\right )\right )} \] Input:
integrate(arctan(1/2*(2*x-2^(1/2))*2^(1/2)),x, algorithm="giac")
Output:
1/4*sqrt(2)*(sqrt(2)*(2*x - sqrt(2))*arctan(1/2*sqrt(2)*(2*x - sqrt(2))) - log(1/2*(2*x - sqrt(2))^2 + 1))
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \arctan \left (\frac {-\sqrt {2}+2 x}{\sqrt {2}}\right ) \, dx=\mathrm {atan}\left (\frac {\sqrt {2}\,\left (2\,x-\sqrt {2}\right )}{2}\right )\,\left (x-\frac {\sqrt {2}}{2}\right )-\frac {\sqrt {2}\,\ln \left ({\left (2\,x-\sqrt {2}\right )}^2+2\right )}{4} \] Input:
int(atan((2^(1/2)*(2*x - 2^(1/2)))/2),x)
Output:
atan((2^(1/2)*(2*x - 2^(1/2)))/2)*(x - 2^(1/2)/2) - (2^(1/2)*log((2*x - 2^ (1/2))^2 + 2))/4
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65 \[ \int \arctan \left (\frac {-\sqrt {2}+2 x}{\sqrt {2}}\right ) \, dx=-\frac {\sqrt {2}\, \mathit {atan} \left (\sqrt {2}\, x -1\right )}{2}+\mathit {atan} \left (\sqrt {2}\, x -1\right ) x -\frac {\sqrt {2}\, \mathrm {log}\left (-\sqrt {2}\, x +x^{2}+1\right )}{4} \] Input:
int(atan(1/2*(2*x-2^(1/2))*2^(1/2)),x)
Output:
( - 2*sqrt(2)*atan(sqrt(2)*x - 1) + 4*atan(sqrt(2)*x - 1)*x - sqrt(2)*log( - sqrt(2)*x + x**2 + 1))/4