Integrand size = 14, antiderivative size = 143 \[ \int \frac {a+b \arctan \left (c x^2\right )}{x^2} \, dx=-\frac {a+b \arctan \left (c x^2\right )}{x}-\frac {b \sqrt {c} \arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2}}+\frac {b \sqrt {c} \arctan \left (1+\sqrt {2} \sqrt {c} x\right )}{\sqrt {2}}-\frac {b \sqrt {c} \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2}}+\frac {b \sqrt {c} \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2}} \]
(-a-b*arctan(c*x^2))/x+1/2*b*arctan(-1+x*2^(1/2)*c^(1/2))*c^(1/2)*2^(1/2)+ 1/2*b*arctan(1+x*2^(1/2)*c^(1/2))*c^(1/2)*2^(1/2)-1/4*b*ln(1+c*x^2-x*2^(1/ 2)*c^(1/2))*c^(1/2)*2^(1/2)+1/4*b*ln(1+c*x^2+x*2^(1/2)*c^(1/2))*c^(1/2)*2^ (1/2)
Time = 0.07 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \arctan \left (c x^2\right )}{x^2} \, dx=-\frac {a}{x}-\frac {b \arctan \left (c x^2\right )}{x}+\frac {b \sqrt {c} \arctan \left (\frac {-\sqrt {2}+2 \sqrt {c} x}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {b \sqrt {c} \arctan \left (\frac {\sqrt {2}+2 \sqrt {c} x}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {b \sqrt {c} \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2}}+\frac {b \sqrt {c} \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2}} \]
-(a/x) - (b*ArcTan[c*x^2])/x + (b*Sqrt[c]*ArcTan[(-Sqrt[2] + 2*Sqrt[c]*x)/ Sqrt[2]])/Sqrt[2] + (b*Sqrt[c]*ArcTan[(Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/Sq rt[2] - (b*Sqrt[c]*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]) + (b*Sq rt[c]*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2])
Time = 0.36 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5361, 755, 1476, 1082, 217, 1479, 25, 27, 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.
| \(\displaystyle \int \frac {a+b \arctan \left (c x^2\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle 2 b c \int \frac {1}{c^2 x^4+1}dx-\frac {a+b \arctan \left (c x^2\right )}{x}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 2 b c \left (\frac {1}{2} \int \frac {1-c x^2}{c^2 x^4+1}dx+\frac {1}{2} \int \frac {c x^2+1}{c^2 x^4+1}dx\right )-\frac {a+b \arctan \left (c x^2\right )}{x}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 2 b c \left (\frac {1}{2} \int \frac {1-c x^2}{c^2 x^4+1}dx+\frac {1}{2} \left (\frac {\int \frac {1}{x^2-\frac {\sqrt {2} x}{\sqrt {c}}+\frac {1}{c}}dx}{2 c}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} x}{\sqrt {c}}+\frac {1}{c}}dx}{2 c}\right )\right )-\frac {a+b \arctan \left (c x^2\right )}{x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 b c \left (\frac {1}{2} \int \frac {1-c x^2}{c^2 x^4+1}dx+\frac {1}{2} \left (\frac {\int \frac {1}{-\left (1-\sqrt {2} \sqrt {c} x\right )^2-1}d\left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{-\left (\sqrt {2} \sqrt {c} x+1\right )^2-1}d\left (\sqrt {2} \sqrt {c} x+1\right )}{\sqrt {2} \sqrt {c}}\right )\right )-\frac {a+b \arctan \left (c x^2\right )}{x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 b c \left (\frac {1}{2} \int \frac {1-c x^2}{c^2 x^4+1}dx+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} x+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}\right )\right )-\frac {a+b \arctan \left (c x^2\right )}{x}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 2 b c \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {c} x}{\sqrt {c} \left (x^2-\frac {\sqrt {2} x}{\sqrt {c}}+\frac {1}{c}\right )}dx}{2 \sqrt {2} \sqrt {c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {c} x+1\right )}{\sqrt {c} \left (x^2+\frac {\sqrt {2} x}{\sqrt {c}}+\frac {1}{c}\right )}dx}{2 \sqrt {2} \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} x+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}\right )\right )-\frac {a+b \arctan \left (c x^2\right )}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 b c \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {c} x}{\sqrt {c} \left (x^2-\frac {\sqrt {2} x}{\sqrt {c}}+\frac {1}{c}\right )}dx}{2 \sqrt {2} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {c} x+1\right )}{\sqrt {c} \left (x^2+\frac {\sqrt {2} x}{\sqrt {c}}+\frac {1}{c}\right )}dx}{2 \sqrt {2} \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} x+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}\right )\right )-\frac {a+b \arctan \left (c x^2\right )}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 b c \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {c} x}{x^2-\frac {\sqrt {2} x}{\sqrt {c}}+\frac {1}{c}}dx}{2 \sqrt {2} c}+\frac {\int \frac {\sqrt {2} \sqrt {c} x+1}{x^2+\frac {\sqrt {2} x}{\sqrt {c}}+\frac {1}{c}}dx}{2 c}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} x+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}\right )\right )-\frac {a+b \arctan \left (c x^2\right )}{x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 b c \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} x+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \sqrt {c}}-\frac {\log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \sqrt {c}}\right )\right )-\frac {a+b \arctan \left (c x^2\right )}{x}\) |
-((a + b*ArcTan[c*x^2])/x) + 2*b*c*((-(ArcTan[1 - Sqrt[2]*Sqrt[c]*x]/(Sqrt [2]*Sqrt[c])) + ArcTan[1 + Sqrt[2]*Sqrt[c]*x]/(Sqrt[2]*Sqrt[c]))/2 + (-1/2 *Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2]/(Sqrt[2]*Sqrt[c]) + Log[1 + Sqrt[2]*Sq rt[c]*x + c*x^2]/(2*Sqrt[2]*Sqrt[c]))/2)
3.1.71.3.1 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_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & & IntegerQ[m])) && NeQ[m, -1]
Time = 0.35 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.75
| method | result | size |
| default | \(-\frac {a}{x}+b \left (-\frac {\arctan \left (c \,x^{2}\right )}{x}+\frac {c \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4}\right )\) | \(107\) |
| parts | \(-\frac {a}{x}+b \left (-\frac {\arctan \left (c \,x^{2}\right )}{x}+\frac {c \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4}\right )\) | \(107\) |
-a/x+b*(-1/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*arctan(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 = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.97 \[ \int \frac {a+b \arctan \left (c x^2\right )}{x^2} \, dx=\frac {\left (-b^{4} c^{2}\right )^{\frac {1}{4}} x \log \left (b c x + \left (-b^{4} c^{2}\right )^{\frac {1}{4}}\right ) + i \, \left (-b^{4} c^{2}\right )^{\frac {1}{4}} x \log \left (b c x + i \, \left (-b^{4} c^{2}\right )^{\frac {1}{4}}\right ) - i \, \left (-b^{4} c^{2}\right )^{\frac {1}{4}} x \log \left (b c x - i \, \left (-b^{4} c^{2}\right )^{\frac {1}{4}}\right ) - \left (-b^{4} c^{2}\right )^{\frac {1}{4}} x \log \left (b c x - \left (-b^{4} c^{2}\right )^{\frac {1}{4}}\right ) - 2 \, b \arctan \left (c x^{2}\right ) - 2 \, a}{2 \, x} \]
1/2*((-b^4*c^2)^(1/4)*x*log(b*c*x + (-b^4*c^2)^(1/4)) + I*(-b^4*c^2)^(1/4) *x*log(b*c*x + I*(-b^4*c^2)^(1/4)) - I*(-b^4*c^2)^(1/4)*x*log(b*c*x - I*(- b^4*c^2)^(1/4)) - (-b^4*c^2)^(1/4)*x*log(b*c*x - (-b^4*c^2)^(1/4)) - 2*b*a rctan(c*x^2) - 2*a)/x
Time = 8.53 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \arctan \left (c x^2\right )}{x^2} \, dx=\begin {cases} - \frac {a}{x} - b c \sqrt [4]{- \frac {1}{c^{2}}} \log {\left (x - \sqrt [4]{- \frac {1}{c^{2}}} \right )} + \frac {b c \sqrt [4]{- \frac {1}{c^{2}}} \log {\left (x^{2} + \sqrt {- \frac {1}{c^{2}}} \right )}}{2} + b c \sqrt [4]{- \frac {1}{c^{2}}} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{c^{2}}}} \right )} - \frac {b \operatorname {atan}{\left (c x^{2} \right )}}{\sqrt [4]{- \frac {1}{c^{2}}}} - \frac {b \operatorname {atan}{\left (c x^{2} \right )}}{x} & \text {for}\: c \neq 0 \\- \frac {a}{x} & \text {otherwise} \end {cases} \]
Piecewise((-a/x - b*c*(-1/c**2)**(1/4)*log(x - (-1/c**2)**(1/4)) + b*c*(-1 /c**2)**(1/4)*log(x**2 + sqrt(-1/c**2))/2 + b*c*(-1/c**2)**(1/4)*atan(x/(- 1/c**2)**(1/4)) - b*atan(c*x**2)/(-1/c**2)**(1/4) - b*atan(c*x**2)/x, Ne(c , 0)), (-a/x, True))
Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.92 \[ \int \frac {a+b \arctan \left (c x^2\right )}{x^2} \, dx=\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {\sqrt {2} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right )}{\sqrt {c}} - \frac {\sqrt {2} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right )}{\sqrt {c}}\right )} - \frac {4 \, \arctan \left (c x^{2}\right )}{x}\right )} b - \frac {a}{x} \]
1/4*(c*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*c*x + sqrt(2)*sqrt(c))/sqrt(c))/sq rt(c) + 2*sqrt(2)*arctan(1/2*sqrt(2)*(2*c*x - sqrt(2)*sqrt(c))/sqrt(c))/sq rt(c) + sqrt(2)*log(c*x^2 + sqrt(2)*sqrt(c)*x + 1)/sqrt(c) - sqrt(2)*log(c *x^2 - sqrt(2)*sqrt(c)*x + 1)/sqrt(c)) - 4*arctan(c*x^2)/x)*b - a/x
Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.97 \[ \int \frac {a+b \arctan \left (c x^2\right )}{x^2} \, dx=\frac {1}{4} \, b c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{\sqrt {{\left | c \right |}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{\sqrt {{\left | c \right |}}} + \frac {\sqrt {2} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{\sqrt {{\left | c \right |}}} - \frac {\sqrt {2} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{\sqrt {{\left | c \right |}}}\right )} - \frac {b \arctan \left (c x^{2}\right ) + a}{x} \]
1/4*b*c*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(abs(c)))*sqrt(ab s(c)))/sqrt(abs(c)) + 2*sqrt(2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)/sqrt(abs (c)))*sqrt(abs(c)))/sqrt(abs(c)) + sqrt(2)*log(x^2 + sqrt(2)*x/sqrt(abs(c) ) + 1/abs(c))/sqrt(abs(c)) - sqrt(2)*log(x^2 - sqrt(2)*x/sqrt(abs(c)) + 1/ abs(c))/sqrt(abs(c))) - (b*arctan(c*x^2) + a)/x
Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.38 \[ \int \frac {a+b \arctan \left (c x^2\right )}{x^2} \, dx=-\frac {a}{x}-\frac {b\,\mathrm {atan}\left (c\,x^2\right )}{x}-{\left (-1\right )}^{1/4}\,b\,\sqrt {c}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\right )\,1{}\mathrm {i}-{\left (-1\right )}^{1/4}\,b\,\sqrt {c}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\,1{}\mathrm {i}\right ) \]