3.872 \(\int \frac {e^{\tanh ^{-1}(a+b x)}}{1-a^2-2 a b x-b^2 x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac {\sqrt {a+b x+1}}{b \sqrt {-a-b x+1}} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {6164, 37} \[ \frac {\sqrt {a+b x+1}}{b \sqrt {-a-b x+1}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a + b*x]/(1 - a^2 - 2*a*b*x - b^2*x^2),x]

[Out]

Sqrt[1 + a + b*x]/(b*Sqrt[1 - a - b*x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 6164

Int[E^(ArcTanh[(a_) + (b_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(c/
(1 - a^2))^p, Int[u*(1 - a - b*x)^(p - n/2)*(1 + a + b*x)^(p + n/2), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]
&& EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e*(1 - a^2), 0] && (IntegerQ[p] || GtQ[c/(1 - a^2), 0])

Rubi steps

\begin {align*} \int \frac {e^{\tanh ^{-1}(a+b x)}}{1-a^2-2 a b x-b^2 x^2} \, dx &=\int \frac {1}{(1-a-b x)^{3/2} \sqrt {1+a+b x}} \, dx\\ &=\frac {\sqrt {1+a+b x}}{b \sqrt {1-a-b x}}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 12, normalized size = 0.44 \[ \frac {e^{\tanh ^{-1}(a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a + b*x]/(1 - a^2 - 2*a*b*x - b^2*x^2),x]

[Out]

E^ArcTanh[a + b*x]/b

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fricas [A]  time = 0.53, size = 37, normalized size = 1.37 \[ -\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{2} x + {\left (a - 1\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.40, size = 40, normalized size = 1.48 \[ \frac {2}{{\left (\frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} - 1\right )} {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 42, normalized size = 1.56 \[ -\frac {\left (b x +a -1\right ) \left (b x +a +1\right )^{2}}{b \left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x+a-1)*(b*x+a+1)^2/b/(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)

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maxima [B]  time = 0.43, size = 65, normalized size = 2.41 \[ -\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{2}}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}} {\left (b^{3} x + a b^{2} - b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^2/(sqrt(a^2*b^2 - (a^2 - 1)*b^2)*(b^3*x + a*b^2 - b^2))

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mupad [B]  time = 0.18, size = 26, normalized size = 0.96 \[ -\frac {\sqrt {1-{\left (a+b\,x\right )}^2}}{b\,\left (a+b\,x-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} - \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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