Integrand size = 10, antiderivative size = 103 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=a x+b x \arctan \left (c x^2\right )+\frac {b \arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}-\frac {b \arctan \left (1+\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}+\frac {b \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} x}{1+c x^2}\right )}{\sqrt {2} \sqrt {c}} \] Output:
a*x+b*x*arctan(c*x^2)-1/2*b*arctan(-1+2^(1/2)*c^(1/2)*x)*2^(1/2)/c^(1/2)-1 /2*b*arctan(1+2^(1/2)*c^(1/2)*x)*2^(1/2)/c^(1/2)+1/2*b*arctanh(2^(1/2)*c^( 1/2)*x/(c*x^2+1))*2^(1/2)/c^(1/2)
Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.04 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=a x+b x \arctan \left (c x^2\right )-\frac {b \left (-2 \arctan \left (1-\sqrt {2} \sqrt {c} x\right )+2 \arctan \left (1+\sqrt {2} \sqrt {c} x\right )+\log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )-\log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )\right )}{2 \sqrt {2} \sqrt {c}} \] Input:
Integrate[a + b*ArcTan[c*x^2],x]
Output:
a*x + b*x*ArcTan[c*x^2] - (b*(-2*ArcTan[1 - Sqrt[2]*Sqrt[c]*x] + 2*ArcTan[ 1 + Sqrt[2]*Sqrt[c]*x] + Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2] - Log[1 + Sqrt [2]*Sqrt[c]*x + c*x^2]))/(2*Sqrt[2]*Sqrt[c])
Time = 0.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.36, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2009}
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.
\(\displaystyle \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a x+b x \arctan \left (c x^2\right )+\frac {b \arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}-\frac {b \arctan \left (\sqrt {2} \sqrt {c} x+1\right )}{\sqrt {2} \sqrt {c}}-\frac {b \log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \sqrt {c}}+\frac {b \log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \sqrt {c}}\) |
Input:
Int[a + b*ArcTan[c*x^2],x]
Output:
a*x + b*x*ArcTan[c*x^2] + (b*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/(Sqrt[2]*Sqrt[ c]) - (b*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/(Sqrt[2]*Sqrt[c]) - (b*Log[1 - Sqr t[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]*Sqrt[c]) + (b*Log[1 + Sqrt[2]*Sqrt[c]* x + c*x^2])/(2*Sqrt[2]*Sqrt[c])
Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00
method | result | size |
default | \(a x +b \left (x \arctan \left (c \,x^{2}\right )-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}\right )\) | \(103\) |
parts | \(a x +b \left (x \arctan \left (c \,x^{2}\right )-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}\right )\) | \(103\) |
Input:
int(a+b*arctan(c*x^2),x,method=_RETURNVERBOSE)
Output:
a*x+b*(x*arctan(c*x^2)-1/4/c/(1/c^2)^(1/4)*2^(1/2)*(ln((x^2-(1/c^2)^(1/4)* x*2^(1/2)+(1/c^2)^(1/2))/(x^2+(1/c^2)^(1/4)*x*2^(1/2)+(1/c^2)^(1/2)))+2*ar ctan(2^(1/2)/(1/c^2)^(1/4)*x+1)+2*arctan(2^(1/2)/(1/c^2)^(1/4)*x-1)))
Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=b x \arctan \left (c x^{2}\right ) + a x - \frac {\sqrt {2} b \arctan \left (\sqrt {2} \sqrt {c} x + 1\right )}{2 \, \sqrt {c}} - \frac {\sqrt {2} b \arctan \left (\sqrt {2} \sqrt {c} x - 1\right )}{2 \, \sqrt {c}} + \frac {\sqrt {2} b \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right )}{4 \, \sqrt {c}} - \frac {\sqrt {2} b \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right )}{4 \, \sqrt {c}} \] Input:
integrate(a+b*arctan(c*x^2),x, algorithm="fricas")
Output:
b*x*arctan(c*x^2) + a*x - 1/2*sqrt(2)*b*arctan(sqrt(2)*sqrt(c)*x + 1)/sqrt (c) - 1/2*sqrt(2)*b*arctan(sqrt(2)*sqrt(c)*x - 1)/sqrt(c) + 1/4*sqrt(2)*b* log(c*x^2 + sqrt(2)*sqrt(c)*x + 1)/sqrt(c) - 1/4*sqrt(2)*b*log(c*x^2 - sqr t(2)*sqrt(c)*x + 1)/sqrt(c)
Time = 3.64 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.07 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=a x + b \left (\begin {cases} x \operatorname {atan}{\left (c x^{2} \right )} + \sqrt [4]{- \frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{2} \right )} - \frac {\log {\left (x - \sqrt [4]{- \frac {1}{c^{2}}} \right )}}{c \sqrt [4]{- \frac {1}{c^{2}}}} + \frac {\log {\left (x^{2} + \sqrt {- \frac {1}{c^{2}}} \right )}}{2 c \sqrt [4]{- \frac {1}{c^{2}}}} - \frac {\operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{c^{2}}}} \right )}}{c \sqrt [4]{- \frac {1}{c^{2}}}} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right ) \] Input:
integrate(a+b*atan(c*x**2),x)
Output:
a*x + b*Piecewise((x*atan(c*x**2) + (-1/c**2)**(1/4)*atan(c*x**2) - log(x - (-1/c**2)**(1/4))/(c*(-1/c**2)**(1/4)) + log(x**2 + sqrt(-1/c**2))/(2*c* (-1/c**2)**(1/4)) - atan(x/(-1/c**2)**(1/4))/(c*(-1/c**2)**(1/4)), Ne(c, 0 )), (0, True))
Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.23 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=-\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}}\right )} - 4 \, x \arctan \left (c x^{2}\right )\right )} b + a x \] Input:
integrate(a+b*arctan(c*x^2),x, algorithm="maxima")
Output:
-1/4*(c*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*c*x + sqrt(2)*sqrt(c))/sqrt(c))/c ^(3/2) + 2*sqrt(2)*arctan(1/2*sqrt(2)*(2*c*x - sqrt(2)*sqrt(c))/sqrt(c))/c ^(3/2) - sqrt(2)*log(c*x^2 + sqrt(2)*sqrt(c)*x + 1)/c^(3/2) + sqrt(2)*log( c*x^2 - sqrt(2)*sqrt(c)*x + 1)/c^(3/2)) - 4*x*arctan(c*x^2))*b + a*x
Time = 0.13 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.45 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=-\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {2} \sqrt {{\left | c \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{c^{2}} + \frac {2 \, \sqrt {2} \sqrt {{\left | c \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{c^{2}} - \frac {\sqrt {2} \sqrt {{\left | c \right |}} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{c^{2}} + \frac {\sqrt {2} \sqrt {{\left | c \right |}} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{c^{2}}\right )} - 4 \, x \arctan \left (c x^{2}\right )\right )} b + a x \] Input:
integrate(a+b*arctan(c*x^2),x, algorithm="giac")
Output:
-1/4*(c*(2*sqrt(2)*sqrt(abs(c))*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(abs (c)))*sqrt(abs(c)))/c^2 + 2*sqrt(2)*sqrt(abs(c))*arctan(1/2*sqrt(2)*(2*x - sqrt(2)/sqrt(abs(c)))*sqrt(abs(c)))/c^2 - sqrt(2)*sqrt(abs(c))*log(x^2 + sqrt(2)*x/sqrt(abs(c)) + 1/abs(c))/c^2 + sqrt(2)*sqrt(abs(c))*log(x^2 - sq rt(2)*x/sqrt(abs(c)) + 1/abs(c))/c^2) - 4*x*arctan(c*x^2))*b + a*x
Time = 0.69 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.48 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=a\,x+b\,x\,\mathrm {atan}\left (c\,x^2\right )-\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\right )}{\sqrt {c}}-\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{\sqrt {c}} \] Input:
int(a + b*atan(c*x^2),x)
Output:
a*x + b*x*atan(c*x^2) - ((-1)^(1/4)*b*atan((-1)^(1/4)*c^(1/2)*x))/c^(1/2) - ((-1)^(1/4)*b*atan((-1)^(1/4)*c^(1/2)*x*1i)*1i)/c^(1/2)
Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\frac {4 \sqrt {c}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {c}\, \sqrt {2}-2 c x}{\sqrt {c}\, \sqrt {2}}\right ) b +2 \sqrt {c}\, \sqrt {2}\, \mathit {atan} \left (c \,x^{2}\right ) b +4 \mathit {atan} \left (c \,x^{2}\right ) b c x -\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {c}\, \sqrt {2}\, x +c \,x^{2}+1\right ) b +\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {c}\, \sqrt {2}\, x +c \,x^{2}+1\right ) b +4 a c x}{4 c} \] Input:
int(a+b*atan(c*x^2),x)
Output:
(4*sqrt(c)*sqrt(2)*atan((sqrt(c)*sqrt(2) - 2*c*x)/(sqrt(c)*sqrt(2)))*b + 2 *sqrt(c)*sqrt(2)*atan(c*x**2)*b + 4*atan(c*x**2)*b*c*x - sqrt(c)*sqrt(2)*l og( - sqrt(c)*sqrt(2)*x + c*x**2 + 1)*b + sqrt(c)*sqrt(2)*log(sqrt(c)*sqrt (2)*x + c*x**2 + 1)*b + 4*a*c*x)/(4*c)