Integrand size = 10, antiderivative size = 140 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=a x+b x \arctan \left (c x^2\right )+\frac {b \arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}-\frac {b \arctan \left (1+\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}-\frac {b \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \sqrt {c}}+\frac {b \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \sqrt {c}} \]
a*x+b*x*arctan(c*x^2)-1/2*b*arctan(-1+x*2^(1/2)*c^(1/2))*2^(1/2)/c^(1/2)-1 /2*b*arctan(1+x*2^(1/2)*c^(1/2))*2^(1/2)/c^(1/2)-1/4*b*ln(1+c*x^2-x*2^(1/2 )*c^(1/2))*2^(1/2)/c^(1/2)+1/4*b*ln(1+c*x^2+x*2^(1/2)*c^(1/2))*2^(1/2)/c^( 1/2)
Time = 0.06 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.76 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=a x+b x \arctan \left (c x^2\right )-\frac {b \left (-2 \arctan \left (1-\sqrt {2} \sqrt {c} x\right )+2 \arctan \left (1+\sqrt {2} \sqrt {c} x\right )+\log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )-\log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )\right )}{2 \sqrt {2} \sqrt {c}} \]
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 = 0.26 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, 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 |
| \(\displaystyle a x+b x \arctan \left (c x^2\right )+\frac {b \arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}-\frac {b \arctan \left (\sqrt {2} \sqrt {c} x+1\right )}{\sqrt {2} \sqrt {c}}-\frac {b \log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \sqrt {c}}+\frac {b \log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \sqrt {c}}\) |
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])
3.1.70.3.1 Defintions of rubi rules used
Time = 0.36 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.74
| 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 | \(a x +b \left (x \arctan \left (c \,x^{2}\right )-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}\right )\) | \(103\) |
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)))
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=b x \arctan \left (c x^{2}\right ) + a x - \frac {1}{2} \, \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {1}{4}} \log \left (b^{3} x + \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {3}{4}} c\right ) + \frac {1}{2} i \, \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {1}{4}} \log \left (b^{3} x + i \, \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {3}{4}} c\right ) - \frac {1}{2} i \, \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {1}{4}} \log \left (b^{3} x - i \, \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {3}{4}} c\right ) + \frac {1}{2} \, \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {1}{4}} \log \left (b^{3} x - \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {3}{4}} c\right ) \]
b*x*arctan(c*x^2) + a*x - 1/2*(-b^4/c^2)^(1/4)*log(b^3*x + (-b^4/c^2)^(3/4 )*c) + 1/2*I*(-b^4/c^2)^(1/4)*log(b^3*x + I*(-b^4/c^2)^(3/4)*c) - 1/2*I*(- b^4/c^2)^(1/4)*log(b^3*x - I*(-b^4/c^2)^(3/4)*c) + 1/2*(-b^4/c^2)^(1/4)*lo g(b^3*x - (-b^4/c^2)^(3/4)*c)
Time = 4.34 (sec) , antiderivative size = 617, normalized size of antiderivative = 4.41 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=a x + b \left (\begin {cases} 0 & \text {for}\: c = 0 \\- \infty i x & \text {for}\: c = - \frac {i}{x^{2}} \\\infty i x & \text {for}\: c = \frac {i}{x^{2}} \\\frac {2 c^{5} x^{5} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} \operatorname {atan}{\left (c x^{2} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} - \frac {2 c^{4} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{2}} \log {\left (x - \sqrt [4]{- \frac {1}{c^{2}}} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} + \frac {c^{4} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{2}} \log {\left (x^{2} + \sqrt {- \frac {1}{c^{2}}} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} - \frac {2 c^{4} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{c^{2}}}} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} + \frac {2 c^{3} x \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} \operatorname {atan}{\left (c x^{2} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} - \frac {2 c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{2}} \log {\left (x - \sqrt [4]{- \frac {1}{c^{2}}} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} + \frac {c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{2}} \log {\left (x^{2} + \sqrt {- \frac {1}{c^{2}}} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} - \frac {2 c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{c^{2}}}} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} + \frac {2 c x^{4} \operatorname {atan}{\left (c x^{2} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} + \frac {2 \operatorname {atan}{\left (c x^{2} \right )}}{2 c^{6} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} & \text {otherwise} \end {cases}\right ) \]
a*x + b*Piecewise((0, Eq(c, 0)), (-oo*I*x, Eq(c, -I/x**2)), (oo*I*x, Eq(c, I/x**2)), (2*c**5*x**5*(-1/c**2)**(7/4)*atan(c*x**2)/(2*c**5*x**4*(-1/c** 2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) - 2*c**4*x**4*(-1/c**2)**(3/2)*log(x - (-1/c**2)**(1/4))/(2*c**5*x**4*(-1/c**2)**(7/4) + 2*c**3*(-1/c**2)**(7/4 )) + c**4*x**4*(-1/c**2)**(3/2)*log(x**2 + sqrt(-1/c**2))/(2*c**5*x**4*(-1 /c**2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) - 2*c**4*x**4*(-1/c**2)**(3/2)*at an(x/(-1/c**2)**(1/4))/(2*c**5*x**4*(-1/c**2)**(7/4) + 2*c**3*(-1/c**2)**( 7/4)) + 2*c**3*x*(-1/c**2)**(7/4)*atan(c*x**2)/(2*c**5*x**4*(-1/c**2)**(7/ 4) + 2*c**3*(-1/c**2)**(7/4)) - 2*c**2*(-1/c**2)**(3/2)*log(x - (-1/c**2)* *(1/4))/(2*c**5*x**4*(-1/c**2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) + c**2*(- 1/c**2)**(3/2)*log(x**2 + sqrt(-1/c**2))/(2*c**5*x**4*(-1/c**2)**(7/4) + 2 *c**3*(-1/c**2)**(7/4)) - 2*c**2*(-1/c**2)**(3/2)*atan(x/(-1/c**2)**(1/4)) /(2*c**5*x**4*(-1/c**2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) + 2*c*x**4*atan( c*x**2)/(2*c**5*x**4*(-1/c**2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) + 2*atan( c*x**2)/(2*c**6*x**4*(-1/c**2)**(7/4) + 2*c**4*(-1/c**2)**(7/4)), True))
Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=-\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}}\right )} - 4 \, x \arctan \left (c x^{2}\right )\right )} b + a x \]
-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 = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.06 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=-\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {2} \sqrt {{\left | c \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{c^{2}} + \frac {2 \, \sqrt {2} \sqrt {{\left | c \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{c^{2}} - \frac {\sqrt {2} \sqrt {{\left | c \right |}} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{c^{2}} + \frac {\sqrt {2} \sqrt {{\left | c \right |}} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{c^{2}}\right )} - 4 \, x \arctan \left (c x^{2}\right )\right )} b + a x \]
-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 = 0.45 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.35 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=a\,x+b\,x\,\mathrm {atan}\left (c\,x^2\right )-\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\right )}{\sqrt {c}}-\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{\sqrt {c}} \]