Integrand size = 19, antiderivative size = 90 \[ \int \frac {1}{x \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=\frac {\sqrt {b} c \arctan \left (\frac {\sqrt {a} \left (c+d x^2\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \left (b+a c^2\right )}+\frac {c^2 \log (x)}{b+a c^2}+\frac {b \log \left (b+a \left (c+d x^2\right )^2\right )}{4 a \left (b+a c^2\right )} \] Output:
1/2*b^(1/2)*c*arctan(a^(1/2)*(d*x^2+c)/b^(1/2))/a^(1/2)/(a*c^2+b)+c^2*ln(x )/(a*c^2+b)+1/4*b*ln(b+a*(d*x^2+c)^2)/a/(a*c^2+b)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=\frac {4 a c^2 \log (x)+\left (b-i \sqrt {a} \sqrt {b} c\right ) \log \left (-i \sqrt {b}+\sqrt {a} \left (c+d x^2\right )\right )+\sqrt {b} \left (\sqrt {b}+i \sqrt {a} c\right ) \log \left (i \sqrt {b}+\sqrt {a} \left (c+d x^2\right )\right )}{4 a \left (b+a c^2\right )} \] Input:
Integrate[1/(x*(a + b/(c + d*x^2)^2)),x]
Output:
(4*a*c^2*Log[x] + (b - I*Sqrt[a]*Sqrt[b]*c)*Log[(-I)*Sqrt[b] + Sqrt[a]*(c + d*x^2)] + Sqrt[b]*(Sqrt[b] + I*Sqrt[a]*c)*Log[I*Sqrt[b] + Sqrt[a]*(c + d *x^2)])/(4*a*(b + a*c^2))
Time = 0.72 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {7293, 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.
\(\displaystyle \int \frac {1}{x \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {b d x \left (2 c+d x^2\right )}{\left (a c^2+b\right ) \left (a c^2+2 a c d x^2+a d^2 x^4+b\right )}+\frac {c^2}{x \left (a c^2+b\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {b} c \arctan \left (\frac {\sqrt {a} \left (c+d x^2\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \left (a c^2+b\right )}+\frac {b \log \left (a c^2+2 a c d x^2+a d^2 x^4+b\right )}{4 a \left (a c^2+b\right )}+\frac {c^2 \log (x)}{a c^2+b}\) |
Input:
Int[1/(x*(a + b/(c + d*x^2)^2)),x]
Output:
(Sqrt[b]*c*ArcTan[(Sqrt[a]*(c + d*x^2))/Sqrt[b]])/(2*Sqrt[a]*(b + a*c^2)) + (c^2*Log[x])/(b + a*c^2) + (b*Log[b + a*c^2 + 2*a*c*d*x^2 + a*d^2*x^4])/ (4*a*(b + a*c^2))
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {b d \left (\frac {\ln \left (a \,d^{2} x^{4}+2 a d \,x^{2} c +a \,c^{2}+b \right )}{2 d a}+\frac {c \arctan \left (\frac {2 a \,d^{2} x^{2}+2 a c d}{2 d \sqrt {a b}}\right )}{d \sqrt {a b}}\right )}{2 a \,c^{2}+2 b}+\frac {c^{2} \ln \left (x \right )}{a \,c^{2}+b}\) | \(99\) |
risch | \(\frac {c^{2} \ln \left (x \right )}{a \,c^{2}+b}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (c^{2} a^{3}+a^{2} b \right ) \textit {\_Z}^{2}-2 a \textit {\_Z} b +b \right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a^{3} c^{2} d +5 a^{2} b d \right ) \textit {\_R}^{2}+\left (4 a^{2} c^{2} d -9 d a b \right ) \textit {\_R} +4 b d \right ) x^{2}+\left (-a^{3} c^{3}-c \,a^{2} b \right ) \textit {\_R}^{2}+4 a^{2} c^{3} \textit {\_R} \right )\right )}{4}\) | \(127\) |
Input:
int(1/x/(a+b/(d*x^2+c)^2),x,method=_RETURNVERBOSE)
Output:
1/2*b*d/(a*c^2+b)*(1/2/d/a*ln(a*d^2*x^4+2*a*c*d*x^2+a*c^2+b)+c/d/(a*b)^(1/ 2)*arctan(1/2*(2*a*d^2*x^2+2*a*c*d)/d/(a*b)^(1/2)))+c^2*ln(x)/(a*c^2+b)
Time = 0.09 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.41 \[ \int \frac {1}{x \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=\left [\frac {4 \, a c^{2} \log \left (x\right ) + a c \sqrt {-\frac {b}{a}} \log \left (\frac {a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2} + 2 \, {\left (a d x^{2} + a c\right )} \sqrt {-\frac {b}{a}} - b}{a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2} + b}\right ) + b \log \left (a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2} + b\right )}{4 \, {\left (a^{2} c^{2} + a b\right )}}, \frac {4 \, a c^{2} \log \left (x\right ) + 2 \, a c \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left (a d x^{2} + a c\right )} \sqrt {\frac {b}{a}}}{b}\right ) + b \log \left (a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2} + b\right )}{4 \, {\left (a^{2} c^{2} + a b\right )}}\right ] \] Input:
integrate(1/x/(a+b/(d*x^2+c)^2),x, algorithm="fricas")
Output:
[1/4*(4*a*c^2*log(x) + a*c*sqrt(-b/a)*log((a*d^2*x^4 + 2*a*c*d*x^2 + a*c^2 + 2*(a*d*x^2 + a*c)*sqrt(-b/a) - b)/(a*d^2*x^4 + 2*a*c*d*x^2 + a*c^2 + b) ) + b*log(a*d^2*x^4 + 2*a*c*d*x^2 + a*c^2 + b))/(a^2*c^2 + a*b), 1/4*(4*a* c^2*log(x) + 2*a*c*sqrt(b/a)*arctan((a*d*x^2 + a*c)*sqrt(b/a)/b) + b*log(a *d^2*x^4 + 2*a*c*d*x^2 + a*c^2 + b))/(a^2*c^2 + a*b)]
Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (76) = 152\).
Time = 48.94 (sec) , antiderivative size = 287, normalized size of antiderivative = 3.19 \[ \int \frac {1}{x \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=\frac {c^{2} \log {\left (x \right )}}{a c^{2} + b} + \left (\frac {b}{4 a \left (a c^{2} + b\right )} - \frac {c \sqrt {- a^{3} b}}{4 a^{2} \left (a c^{2} + b\right )}\right ) \log {\left (x^{2} + \frac {4 a^{2} c^{2} \left (\frac {b}{4 a \left (a c^{2} + b\right )} - \frac {c \sqrt {- a^{3} b}}{4 a^{2} \left (a c^{2} + b\right )}\right ) + 4 a b \left (\frac {b}{4 a \left (a c^{2} + b\right )} - \frac {c \sqrt {- a^{3} b}}{4 a^{2} \left (a c^{2} + b\right )}\right ) + a c^{2} - b}{a c d} \right )} + \left (\frac {b}{4 a \left (a c^{2} + b\right )} + \frac {c \sqrt {- a^{3} b}}{4 a^{2} \left (a c^{2} + b\right )}\right ) \log {\left (x^{2} + \frac {4 a^{2} c^{2} \left (\frac {b}{4 a \left (a c^{2} + b\right )} + \frac {c \sqrt {- a^{3} b}}{4 a^{2} \left (a c^{2} + b\right )}\right ) + 4 a b \left (\frac {b}{4 a \left (a c^{2} + b\right )} + \frac {c \sqrt {- a^{3} b}}{4 a^{2} \left (a c^{2} + b\right )}\right ) + a c^{2} - b}{a c d} \right )} \] Input:
integrate(1/x/(a+b/(d*x**2+c)**2),x)
Output:
c**2*log(x)/(a*c**2 + b) + (b/(4*a*(a*c**2 + b)) - c*sqrt(-a**3*b)/(4*a**2 *(a*c**2 + b)))*log(x**2 + (4*a**2*c**2*(b/(4*a*(a*c**2 + b)) - c*sqrt(-a* *3*b)/(4*a**2*(a*c**2 + b))) + 4*a*b*(b/(4*a*(a*c**2 + b)) - c*sqrt(-a**3* b)/(4*a**2*(a*c**2 + b))) + a*c**2 - b)/(a*c*d)) + (b/(4*a*(a*c**2 + b)) + c*sqrt(-a**3*b)/(4*a**2*(a*c**2 + b)))*log(x**2 + (4*a**2*c**2*(b/(4*a*(a *c**2 + b)) + c*sqrt(-a**3*b)/(4*a**2*(a*c**2 + b))) + 4*a*b*(b/(4*a*(a*c* *2 + b)) + c*sqrt(-a**3*b)/(4*a**2*(a*c**2 + b))) + a*c**2 - b)/(a*c*d))
Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=\frac {c^{2} \log \left (x^{2}\right )}{2 \, {\left (a c^{2} + b\right )}} + \frac {b c \arctan \left (\frac {a d^{2} x^{2} + a c d}{\sqrt {a b} d}\right )}{2 \, {\left (a c^{2} + b\right )} \sqrt {a b}} + \frac {b \log \left (a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2} + b\right )}{4 \, {\left (a^{2} c^{2} + a b\right )}} \] Input:
integrate(1/x/(a+b/(d*x^2+c)^2),x, algorithm="maxima")
Output:
1/2*c^2*log(x^2)/(a*c^2 + b) + 1/2*b*c*arctan((a*d^2*x^2 + a*c*d)/(sqrt(a* b)*d))/((a*c^2 + b)*sqrt(a*b)) + 1/4*b*log(a*d^2*x^4 + 2*a*c*d*x^2 + a*c^2 + b)/(a^2*c^2 + a*b)
Exception generated. \[ \int \frac {1}{x \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/x/(a+b/(d*x^2+c)^2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%{[1,0]:[1,0,%%%{1,[1,1]%%%}]%%},[0,1]%%%}+%%%{%%%{1,[0,1 ]%%%},[0,
Time = 11.81 (sec) , antiderivative size = 2098, normalized size of antiderivative = 23.31 \[ \int \frac {1}{x \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=\text {Too large to display} \] Input:
int(1/(x*(a + b/(c + d*x^2)^2)),x)
Output:
(c^2*log(x))/(b + a*c^2) + (4*a*b*log(b + a*c^2 + a*d^2*x^4 + 2*a*c*d*x^2) )/(16*a^2*b + 16*a^3*c^2) + (b^(1/2)*c*atan(((x^2*((a*(b + a*c^2)^(3/2)*(( b^(1/2)*c*(4*a^2*b*c*d^7 - (4*a*b*(36*a^3*b*c*d^7 - 16*a^4*c^3*d^7 + (4*a* b*(16*a^5*c^3*d^7 - 80*a^4*b*c*d^7))/(16*a^2*b + 16*a^3*c^2)))/(16*a^2*b + 16*a^3*c^2)))/(4*(b + a*c^2)^(1/2)*(a*b + a^2*c^2)^(1/2)) - (4*a*b*((b^(1 /2)*c*(36*a^3*b*c*d^7 - 16*a^4*c^3*d^7 + (4*a*b*(16*a^5*c^3*d^7 - 80*a^4*b *c*d^7))/(16*a^2*b + 16*a^3*c^2)))/(4*(b + a*c^2)^(1/2)*(a*b + a^2*c^2)^(1 /2)) + (a*b^(3/2)*c*(16*a^5*c^3*d^7 - 80*a^4*b*c*d^7))/((b + a*c^2)^(1/2)* (a*b + a^2*c^2)^(1/2)*(16*a^2*b + 16*a^3*c^2))))/(16*a^2*b + 16*a^3*c^2) + (b^(3/2)*c^3*(16*a^5*c^3*d^7 - 80*a^4*b*c*d^7))/(64*(b + a*c^2)^(3/2)*(a* b + a^2*c^2)^(3/2)))*(b^2 + 13*a^2*c^4 - 10*a*b*c^2))/((a*(b + a*c^2))^(5/ 2)*(b + a*c^2)*(b^6 + 16*a^6*c^12 - 4*a*b^5*c^2 + 48*a^5*b*c^10 - a^2*b^4* c^4 + 22*a^3*b^3*c^6 + 51*a^4*b^2*c^8) + b*c^2*(a*(b + a*c^2))^(7/2)*(b^4 + 9*a^4*c^8 - 5*a*b^3*c^2 + 18*a^3*b*c^6 + 4*a^2*b^2*c^4)) - (4*c*((4*a*b* (4*a^2*b*c*d^7 - (4*a*b*(36*a^3*b*c*d^7 - 16*a^4*c^3*d^7 + (4*a*b*(16*a^5* c^3*d^7 - 80*a^4*b*c*d^7))/(16*a^2*b + 16*a^3*c^2)))/(16*a^2*b + 16*a^3*c^ 2)))/(16*a^2*b + 16*a^3*c^2) + (b^(1/2)*c*((b^(1/2)*c*(36*a^3*b*c*d^7 - 16 *a^4*c^3*d^7 + (4*a*b*(16*a^5*c^3*d^7 - 80*a^4*b*c*d^7))/(16*a^2*b + 16*a^ 3*c^2)))/(4*(b + a*c^2)^(1/2)*(a*b + a^2*c^2)^(1/2)) + (a*b^(3/2)*c*(16*a^ 5*c^3*d^7 - 80*a^4*b*c*d^7))/((b + a*c^2)^(1/2)*(a*b + a^2*c^2)^(1/2)*(...
Time = 0.21 (sec) , antiderivative size = 294, normalized size of antiderivative = 3.27 \[ \int \frac {1}{x \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=\frac {-2 \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}-2 \sqrt {a}\, d x}{\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {2}}\right ) c -2 \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}+2 \sqrt {a}\, d x}{\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {2}}\right ) c +\mathrm {log}\left (-\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}\, x +\sqrt {a \,c^{2}+b}+\sqrt {a}\, d \,x^{2}\right ) b +\mathrm {log}\left (\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}\, x +\sqrt {a \,c^{2}+b}+\sqrt {a}\, d \,x^{2}\right ) b +4 \,\mathrm {log}\left (x \right ) a \,c^{2}}{4 a \left (a \,c^{2}+b \right )} \] Input:
int(1/x/(a+b/(d*x^2+c)^2),x)
Output:
( - 2*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*atan((sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2) - 2*sqrt( a)*d*x)/(sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)))*c - 2*sqrt (sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*atan ((sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2) + 2*sqrt(a)*d*x)/(s qrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)))*c + log( - sqrt(d)*s qrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2)*x + sqrt(a*c**2 + b) + sqrt(a) *d*x**2)*b + log(sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2)*x + sqrt(a*c**2 + b) + sqrt(a)*d*x**2)*b + 4*log(x)*a*c**2)/(4*a*(a*c**2 + b))