3.89 \(\int \frac {1}{b+2 a x+b x^2} \, dx\)

Optimal. Leaf size=35 \[ -\frac {\tanh ^{-1}\left (\frac {a+b x}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \]

[Out]

-arctanh((b*x+a)/(a^2-b^2)^(1/2))/(a^2-b^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {618, 206} \[ -\frac {\tanh ^{-1}\left (\frac {a+b x}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*a*x + b*x^2)^(-1),x]

[Out]

-(ArcTanh[(a + b*x)/Sqrt[a^2 - b^2]]/Sqrt[a^2 - b^2])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {1}{b+2 a x+b x^2} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{4 \left (a^2-b^2\right )-x^2} \, dx,x,2 a+2 b x\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {a+b x}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 34, normalized size = 0.97 \[ \frac {\tan ^{-1}\left (\frac {a+b x}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*a*x + b*x^2)^(-1),x]

[Out]

ArcTan[(a + b*x)/Sqrt[-a^2 + b^2]]/Sqrt[-a^2 + b^2]

________________________________________________________________________________________

fricas [A]  time = 1.11, size = 124, normalized size = 3.54 \[ \left [\frac {\log \left (\frac {b^{2} x^{2} + 2 \, a b x + 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b x + a\right )}}{b x^{2} + 2 \, a x + b}\right )}{2 \, \sqrt {a^{2} - b^{2}}}, -\frac {\sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b x + a\right )}}{a^{2} - b^{2}}\right )}{a^{2} - b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+2*a*x+b),x, algorithm="fricas")

[Out]

[1/2*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))/sqrt(a^2 - b^2),
-sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*x + a)/(a^2 - b^2))/(a^2 - b^2)]

________________________________________________________________________________________

giac [A]  time = 0.37, size = 30, normalized size = 0.86 \[ \frac {\arctan \left (\frac {b x + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+2*a*x+b),x, algorithm="giac")

[Out]

arctan((b*x + a)/sqrt(-a^2 + b^2))/sqrt(-a^2 + b^2)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 35, normalized size = 1.00 \[ \frac {\arctan \left (\frac {2 b x +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+2*a*x+b),x)

[Out]

1/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*x+2*a)/(-a^2+b^2)^(1/2))

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+2*a*x+b),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more details)Is 4*a^2-4*b^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 0.27, size = 33, normalized size = 0.94 \[ -\frac {\mathrm {atanh}\left (\frac {a+b\,x}{\sqrt {a+b}\,\sqrt {a-b}}\right )}{\sqrt {a+b}\,\sqrt {a-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b + 2*a*x + b*x^2),x)

[Out]

-atanh((a + b*x)/((a + b)^(1/2)*(a - b)^(1/2)))/((a + b)^(1/2)*(a - b)^(1/2))

________________________________________________________________________________________

sympy [B]  time = 0.23, size = 100, normalized size = 2.86 \[ \frac {\sqrt {\frac {1}{\left (a - b\right ) \left (a + b\right )}} \log {\left (x + \frac {- a^{2} \sqrt {\frac {1}{\left (a - b\right ) \left (a + b\right )}} + a + b^{2} \sqrt {\frac {1}{\left (a - b\right ) \left (a + b\right )}}}{b} \right )}}{2} - \frac {\sqrt {\frac {1}{\left (a - b\right ) \left (a + b\right )}} \log {\left (x + \frac {a^{2} \sqrt {\frac {1}{\left (a - b\right ) \left (a + b\right )}} + a - b^{2} \sqrt {\frac {1}{\left (a - b\right ) \left (a + b\right )}}}{b} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+2*a*x+b),x)

[Out]

sqrt(1/((a - b)*(a + b)))*log(x + (-a**2*sqrt(1/((a - b)*(a + b))) + a + b**2*sqrt(1/((a - b)*(a + b))))/b)/2
- sqrt(1/((a - b)*(a + b)))*log(x + (a**2*sqrt(1/((a - b)*(a + b))) + a - b**2*sqrt(1/((a - b)*(a + b))))/b)/2

________________________________________________________________________________________