Integrand size = 15, antiderivative size = 59 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=-\frac {a \arctan \left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c} \left (a^2+c\right )}+\frac {\log (x)}{a^2+c}-\frac {\log \left (c+(a+b x)^2\right )}{2 \left (a^2+c\right )} \]
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=-\frac {\frac {2 a \arctan \left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c}}-2 \log (b x)+\log \left (c+(a+b x)^2\right )}{2 \left (a^2+c\right )} \]
Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {896, 25, 479, 452, 216, 240}
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.
\(\displaystyle \int \frac {1}{x \left ((a+b x)^2+c\right )} \, dx\) |
\(\Big \downarrow \) 896 |
\(\displaystyle \int \frac {1}{b x \left ((a+b x)^2+c\right )}d(a+b x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {1}{b x \left ((a+b x)^2+c\right )}d(a+b x)\) |
\(\Big \downarrow \) 479 |
\(\displaystyle \frac {\log (-b x)}{a^2+c}-\frac {\int \frac {2 a+b x}{(a+b x)^2+c}d(a+b x)}{a^2+c}\) |
\(\Big \downarrow \) 452 |
\(\displaystyle \frac {\log (-b x)}{a^2+c}-\frac {a \int \frac {1}{(a+b x)^2+c}d(a+b x)+\int \frac {a+b x}{(a+b x)^2+c}d(a+b x)}{a^2+c}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\log (-b x)}{a^2+c}-\frac {\int \frac {a+b x}{(a+b x)^2+c}d(a+b x)+\frac {a \arctan \left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c}}}{a^2+c}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {\log (-b x)}{a^2+c}-\frac {\frac {a \arctan \left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {1}{2} \log \left ((a+b x)^2+c\right )}{a^2+c}\) |
Log[-(b*x)]/(a^2 + c) - ((a*ArcTan[(a + b*x)/Sqrt[c]])/Sqrt[c] + Log[c + ( a + b*x)^2]/2)/(a^2 + c)
3.1.82.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c Int[1/ (a + b*x^2), x], x] + Simp[d Int[x/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c^2 + a*d^2, 0]
Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[d*(Log [RemoveContent[c + d*x, x]]/(b*c^2 + a*d^2)), x] + Simp[b/(b*c^2 + a*d^2) Int[(c - d*x)/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1) Subst[Int[Si mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
Time = 0.84 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.25
method | result | size |
default | \(\frac {\ln \left (x \right )}{a^{2}+c}-\frac {b \left (\frac {\ln \left (b^{2} x^{2}+2 a b x +a^{2}+c \right )}{2 b}+\frac {a \arctan \left (\frac {2 b^{2} x +2 a b}{2 \sqrt {c}\, b}\right )}{\sqrt {c}\, b}\right )}{a^{2}+c}\) | \(74\) |
risch | \(\frac {\ln \left (x \right )}{a^{2}+c}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (c \,a^{2}+c^{2}\right ) \textit {\_Z}^{2}+2 c \textit {\_Z} \right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a^{2} b +3 b c \right ) \textit {\_R} +3 b \right ) x +\left (-a^{3}-a c \right ) \textit {\_R} +2 a \right )\right )}{2}\) | \(75\) |
ln(x)/(a^2+c)-b/(a^2+c)*(1/2/b*ln(b^2*x^2+2*a*b*x+a^2+c)+a/c^(1/2)/b*arcta n(1/2*(2*b^2*x+2*a*b)/c^(1/2)/b))
Time = 0.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.61 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=\left [-\frac {a \sqrt {-c} \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 2 \, {\left (b x + a\right )} \sqrt {-c} - c}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right ) + c \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) - 2 \, c \log \left (x\right )}{2 \, {\left (a^{2} c + c^{2}\right )}}, -\frac {2 \, a \sqrt {c} \arctan \left (\frac {b x + a}{\sqrt {c}}\right ) + c \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) - 2 \, c \log \left (x\right )}{2 \, {\left (a^{2} c + c^{2}\right )}}\right ] \]
[-1/2*(a*sqrt(-c)*log((b^2*x^2 + 2*a*b*x + a^2 + 2*(b*x + a)*sqrt(-c) - c) /(b^2*x^2 + 2*a*b*x + a^2 + c)) + c*log(b^2*x^2 + 2*a*b*x + a^2 + c) - 2*c *log(x))/(a^2*c + c^2), -1/2*(2*a*sqrt(c)*arctan((b*x + a)/sqrt(c)) + c*lo g(b^2*x^2 + 2*a*b*x + a^2 + c) - 2*c*log(x))/(a^2*c + c^2)]
Leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (49) = 98\).
Time = 1.85 (sec) , antiderivative size = 738, normalized size of antiderivative = 12.51 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=\left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) \log {\left (x + \frac {- 4 a^{6} c \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} + 4 a^{4} c^{2} \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} - 6 a^{4} c \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) + 20 a^{2} c^{3} \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} - 12 a^{2} c^{2} \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) + 10 a^{2} c + 12 c^{4} \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} - 6 c^{3} \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) - 6 c^{2}}{a^{3} b + 9 a b c} \right )} + \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) \log {\left (x + \frac {- 4 a^{6} c \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} + 4 a^{4} c^{2} \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} - 6 a^{4} c \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) + 20 a^{2} c^{3} \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} - 12 a^{2} c^{2} \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) + 10 a^{2} c + 12 c^{4} \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} - 6 c^{3} \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) - 6 c^{2}}{a^{3} b + 9 a b c} \right )} + \frac {\log {\left (x + \frac {- \frac {4 a^{6} c}{\left (a^{2} + c\right )^{2}} + \frac {4 a^{4} c^{2}}{\left (a^{2} + c\right )^{2}} - \frac {6 a^{4} c}{a^{2} + c} + \frac {20 a^{2} c^{3}}{\left (a^{2} + c\right )^{2}} - \frac {12 a^{2} c^{2}}{a^{2} + c} + 10 a^{2} c + \frac {12 c^{4}}{\left (a^{2} + c\right )^{2}} - \frac {6 c^{3}}{a^{2} + c} - 6 c^{2}}{a^{3} b + 9 a b c} \right )}}{a^{2} + c} \]
(-a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))*log(x + (-4*a**6*c*(-a*s qrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))**2 + 4*a**4*c**2*(-a*sqrt(-c) /(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))**2 - 6*a**4*c*(-a*sqrt(-c)/(2*c*(a** 2 + c)) - 1/(2*(a**2 + c))) + 20*a**2*c**3*(-a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))**2 - 12*a**2*c**2*(-a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2* (a**2 + c))) + 10*a**2*c + 12*c**4*(-a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a **2 + c)))**2 - 6*c**3*(-a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c))) - 6*c**2)/(a**3*b + 9*a*b*c)) + (a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))*log(x + (-4*a**6*c*(a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))* *2 + 4*a**4*c**2*(a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))**2 - 6*a **4*c*(a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c))) + 20*a**2*c**3*(a*s qrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))**2 - 12*a**2*c**2*(a*sqrt(-c) /(2*c*(a**2 + c)) - 1/(2*(a**2 + c))) + 10*a**2*c + 12*c**4*(a*sqrt(-c)/(2 *c*(a**2 + c)) - 1/(2*(a**2 + c)))**2 - 6*c**3*(a*sqrt(-c)/(2*c*(a**2 + c) ) - 1/(2*(a**2 + c))) - 6*c**2)/(a**3*b + 9*a*b*c)) + log(x + (-4*a**6*c/( a**2 + c)**2 + 4*a**4*c**2/(a**2 + c)**2 - 6*a**4*c/(a**2 + c) + 20*a**2*c **3/(a**2 + c)**2 - 12*a**2*c**2/(a**2 + c) + 10*a**2*c + 12*c**4/(a**2 + c)**2 - 6*c**3/(a**2 + c) - 6*c**2)/(a**3*b + 9*a*b*c))/(a**2 + c)
Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=-\frac {a \arctan \left (\frac {b^{2} x + a b}{b \sqrt {c}}\right )}{{\left (a^{2} + c\right )} \sqrt {c}} - \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, {\left (a^{2} + c\right )}} + \frac {\log \left (x\right )}{a^{2} + c} \]
-a*arctan((b^2*x + a*b)/(b*sqrt(c)))/((a^2 + c)*sqrt(c)) - 1/2*log(b^2*x^2 + 2*a*b*x + a^2 + c)/(a^2 + c) + log(x)/(a^2 + c)
Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=-\frac {a \arctan \left (\frac {b x + a}{\sqrt {c}}\right )}{{\left (a^{2} + c\right )} \sqrt {c}} - \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, {\left (a^{2} + c\right )}} + \frac {\log \left ({\left | x \right |}\right )}{a^{2} + c} \]
-a*arctan((b*x + a)/sqrt(c))/((a^2 + c)*sqrt(c)) - 1/2*log(b^2*x^2 + 2*a*b *x + a^2 + c)/(a^2 + c) + log(abs(x))/(a^2 + c)
Time = 10.58 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.93 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=\frac {\ln \left (x\right )}{a^2+c}-\frac {\ln \left (2\,a\,b^3+3\,b^4\,x+\frac {b^3\,\left (c+a\,\sqrt {-c}\right )\,\left (a^3+b\,x\,a^2+c\,a-3\,b\,c\,x\right )}{c\,\left (a^2+c\right )}\right )\,\left (c+a\,\sqrt {-c}\right )}{2\,\left (a^2\,c+c^2\right )}-\frac {\ln \left (2\,a\,b^3+3\,b^4\,x+\frac {b^3\,\left (c-a\,\sqrt {-c}\right )\,\left (a^3+b\,x\,a^2+c\,a-3\,b\,c\,x\right )}{c\,\left (a^2+c\right )}\right )\,\left (c-a\,\sqrt {-c}\right )}{2\,\left (a^2\,c+c^2\right )} \]