Integrand size = 14, antiderivative size = 69 \[ \int \frac {1}{\left (b+2 a x-b x^2\right )^2} \, dx=-\frac {a-b x}{2 \left (a^2+b^2\right ) \left (b+2 a x-b x^2\right )}-\frac {b \text {arctanh}\left (\frac {a-b x}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}} \] Output:
-1/2*(-b*x+a)/(a^2+b^2)/(-b*x^2+2*a*x+b)-1/2*b*arctanh((-b*x+a)/(a^2+b^2)^ (1/2))/(a^2+b^2)^(3/2)
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\left (b+2 a x-b x^2\right )^2} \, dx=\frac {\frac {-a+b x}{b+2 a x-b x^2}-\frac {b \arctan \left (\frac {-a+b x}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}}{2 \left (a^2+b^2\right )} \] Input:
Integrate[(b + 2*a*x - b*x^2)^(-2),x]
Output:
((-a + b*x)/(b + 2*a*x - b*x^2) - (b*ArcTan[(-a + b*x)/Sqrt[-a^2 - b^2]])/ Sqrt[-a^2 - b^2])/(2*(a^2 + b^2))
Time = 0.34 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1086, 1083, 219}
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.
\(\displaystyle \int \frac {1}{\left (2 a x-b x^2+b\right )^2} \, dx\) |
\(\Big \downarrow \) 1086 |
\(\displaystyle \frac {b \int \frac {1}{-b x^2+2 a x+b}dx}{2 \left (a^2+b^2\right )}-\frac {a-b x}{2 \left (a^2+b^2\right ) \left (2 a x-b x^2+b\right )}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle -\frac {b \int \frac {1}{4 \left (a^2+b^2\right )-(2 a-2 b x)^2}d(2 a-2 b x)}{a^2+b^2}-\frac {a-b x}{2 \left (a^2+b^2\right ) \left (2 a x-b x^2+b\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {b \text {arctanh}\left (\frac {2 a-2 b x}{2 \sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {a-b x}{2 \left (a^2+b^2\right ) \left (2 a x-b x^2+b\right )}\) |
Input:
Int[(b + 2*a*x - b*x^2)^(-2),x]
Output:
-1/2*(a - b*x)/((a^2 + b^2)*(b + 2*a*x - b*x^2)) - (b*ArcTanh[(2*a - 2*b*x )/(2*Sqrt[a^2 + b^2])])/(2*(a^2 + b^2)^(3/2))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
Time = 0.72 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.20
method | result | size |
default | \(\frac {-2 b x +2 a}{\left (-4 a^{2}-4 b^{2}\right ) \left (-b \,x^{2}+2 a x +b \right )}+\frac {2 b \,\operatorname {arctanh}\left (\frac {-2 b x +2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (-4 a^{2}-4 b^{2}\right ) \sqrt {a^{2}+b^{2}}}\) | \(83\) |
risch | \(\frac {\frac {b x}{4 a^{2}+4 b^{2}}-\frac {a}{4 \left (a^{2}+b^{2}\right )}}{-\frac {1}{2} b \,x^{2}+a x +\frac {1}{2} b}+\frac {b \ln \left (\left (a^{2} b +b^{3}\right ) x +\left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{3}-a \,b^{2}\right )}{4 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {b \ln \left (\left (-a^{2} b -b^{3}\right ) x +\left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{3}+a \,b^{2}\right )}{4 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\) | \(134\) |
Input:
int(1/(-b*x^2+2*a*x+b)^2,x,method=_RETURNVERBOSE)
Output:
(-2*b*x+2*a)/(-4*a^2-4*b^2)/(-b*x^2+2*a*x+b)+2*b/(-4*a^2-4*b^2)/(a^2+b^2)^ (1/2)*arctanh(1/2*(-2*b*x+2*a)/(a^2+b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (65) = 130\).
Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.48 \[ \int \frac {1}{\left (b+2 a x-b x^2\right )^2} \, dx=-\frac {2 \, a^{3} + 2 \, a b^{2} + {\left (b^{2} x^{2} - 2 \, a b x - b^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} + b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b x - a\right )}}{b x^{2} - 2 \, a x - b}\right ) - 2 \, {\left (a^{2} b + b^{3}\right )} x}{4 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} x^{2} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} x\right )}} \] Input:
integrate(1/(-b*x^2+2*a*x+b)^2,x, algorithm="fricas")
Output:
-1/4*(2*a^3 + 2*a*b^2 + (b^2*x^2 - 2*a*b*x - b^2)*sqrt(a^2 + b^2)*log((b^2 *x^2 - 2*a*b*x + 2*a^2 + b^2 + 2*sqrt(a^2 + b^2)*(b*x - a))/(b*x^2 - 2*a*x - b)) - 2*(a^2*b + b^3)*x)/(a^4*b + 2*a^2*b^3 + b^5 - (a^4*b + 2*a^2*b^3 + b^5)*x^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*x)
Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (56) = 112\).
Time = 0.29 (sec) , antiderivative size = 218, normalized size of antiderivative = 3.16 \[ \int \frac {1}{\left (b+2 a x-b x^2\right )^2} \, dx=- \frac {b \sqrt {\frac {1}{\left (a^{2} + b^{2}\right )^{3}}} \log {\left (x + \frac {- a^{4} b \sqrt {\frac {1}{\left (a^{2} + b^{2}\right )^{3}}} - 2 a^{2} b^{3} \sqrt {\frac {1}{\left (a^{2} + b^{2}\right )^{3}}} - a b - b^{5} \sqrt {\frac {1}{\left (a^{2} + b^{2}\right )^{3}}}}{b^{2}} \right )}}{4} + \frac {b \sqrt {\frac {1}{\left (a^{2} + b^{2}\right )^{3}}} \log {\left (x + \frac {a^{4} b \sqrt {\frac {1}{\left (a^{2} + b^{2}\right )^{3}}} + 2 a^{2} b^{3} \sqrt {\frac {1}{\left (a^{2} + b^{2}\right )^{3}}} - a b + b^{5} \sqrt {\frac {1}{\left (a^{2} + b^{2}\right )^{3}}}}{b^{2}} \right )}}{4} + \frac {a - b x}{- 2 a^{2} b - 2 b^{3} + x^{2} \cdot \left (2 a^{2} b + 2 b^{3}\right ) + x \left (- 4 a^{3} - 4 a b^{2}\right )} \] Input:
integrate(1/(-b*x**2+2*a*x+b)**2,x)
Output:
-b*sqrt((a**2 + b**2)**(-3))*log(x + (-a**4*b*sqrt((a**2 + b**2)**(-3)) - 2*a**2*b**3*sqrt((a**2 + b**2)**(-3)) - a*b - b**5*sqrt((a**2 + b**2)**(-3 )))/b**2)/4 + b*sqrt((a**2 + b**2)**(-3))*log(x + (a**4*b*sqrt((a**2 + b** 2)**(-3)) + 2*a**2*b**3*sqrt((a**2 + b**2)**(-3)) - a*b + b**5*sqrt((a**2 + b**2)**(-3)))/b**2)/4 + (a - b*x)/(-2*a**2*b - 2*b**3 + x**2*(2*a**2*b + 2*b**3) + x*(-4*a**3 - 4*a*b**2))
Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\left (b+2 a x-b x^2\right )^2} \, dx=-\frac {b \log \left (\frac {b x - a - \sqrt {a^{2} + b^{2}}}{b x - a + \sqrt {a^{2} + b^{2}}}\right )}{4 \, {\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} + \frac {b x - a}{2 \, {\left (a^{2} b + b^{3} - {\left (a^{2} b + b^{3}\right )} x^{2} + 2 \, {\left (a^{3} + a b^{2}\right )} x\right )}} \] Input:
integrate(1/(-b*x^2+2*a*x+b)^2,x, algorithm="maxima")
Output:
-1/4*b*log((b*x - a - sqrt(a^2 + b^2))/(b*x - a + sqrt(a^2 + b^2)))/(a^2 + b^2)^(3/2) + 1/2*(b*x - a)/(a^2*b + b^3 - (a^2*b + b^3)*x^2 + 2*(a^3 + a* b^2)*x)
Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\left (b+2 a x-b x^2\right )^2} \, dx=-\frac {b \log \left (\frac {{\left | 2 \, b x - 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b x - 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{4 \, {\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {b x - a}{2 \, {\left (b x^{2} - 2 \, a x - b\right )} {\left (a^{2} + b^{2}\right )}} \] Input:
integrate(1/(-b*x^2+2*a*x+b)^2,x, algorithm="giac")
Output:
-1/4*b*log(abs(2*b*x - 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*x - 2*a + 2*sqrt(a ^2 + b^2)))/(a^2 + b^2)^(3/2) - 1/2*(b*x - a)/((b*x^2 - 2*a*x - b)*(a^2 + b^2))
Time = 9.39 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\left (b+2 a x-b x^2\right )^2} \, dx=-\frac {\frac {a}{2\,\left (a^2+b^2\right )}-\frac {b\,x}{2\,\left (a^2+b^2\right )}}{-b\,x^2+2\,a\,x+b}+\frac {b\,\mathrm {atan}\left (\frac {a\,b^2\,1{}\mathrm {i}+a^3\,1{}\mathrm {i}-b\,x\,\left (a^2+b^2\right )\,1{}\mathrm {i}}{{\left (a^2+b^2\right )}^{3/2}}\right )\,1{}\mathrm {i}}{2\,{\left (a^2+b^2\right )}^{3/2}} \] Input:
int(1/(b + 2*a*x - b*x^2)^2,x)
Output:
(b*atan((a*b^2*1i + a^3*1i - b*x*(a^2 + b^2)*1i)/(a^2 + b^2)^(3/2))*1i)/(2 *(a^2 + b^2)^(3/2)) - (a/(2*(a^2 + b^2)) - (b*x)/(2*(a^2 + b^2)))/(b + 2*a *x - b*x^2)
Time = 0.21 (sec) , antiderivative size = 222, normalized size of antiderivative = 3.22 \[ \int \frac {1}{\left (b+2 a x-b x^2\right )^2} \, dx=\frac {4 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {-b i x +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b i x -2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {-b i x +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{2} i \,x^{2}+2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {-b i x +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{2} i -2 a^{4}+a^{2} b^{2} x^{2}-3 a^{2} b^{2}+b^{4} x^{2}-b^{4}}{4 a \left (-a^{4} b \,x^{2}-2 a^{2} b^{3} x^{2}-b^{5} x^{2}+2 a^{5} x +4 a^{3} b^{2} x +2 a \,b^{4} x +a^{4} b +2 a^{2} b^{3}+b^{5}\right )} \] Input:
int(1/(-b*x^2+2*a*x+b)^2,x)
Output:
(4*sqrt(a**2 + b**2)*atan((a*i - b*i*x)/sqrt(a**2 + b**2))*a**2*b*i*x - 2* sqrt(a**2 + b**2)*atan((a*i - b*i*x)/sqrt(a**2 + b**2))*a*b**2*i*x**2 + 2* sqrt(a**2 + b**2)*atan((a*i - b*i*x)/sqrt(a**2 + b**2))*a*b**2*i - 2*a**4 + a**2*b**2*x**2 - 3*a**2*b**2 + b**4*x**2 - b**4)/(4*a*(2*a**5*x - a**4*b *x**2 + a**4*b + 4*a**3*b**2*x - 2*a**2*b**3*x**2 + 2*a**2*b**3 + 2*a*b**4 *x - b**5*x**2 + b**5))