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

Optimal. Leaf size=93 \[ \frac{\sqrt{a+b x+1}}{(1-a) \sqrt{-a-b x+1}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(1-a) \sqrt{1-a^2}} \]

[Out]

Sqrt[1 + a + b*x]/((1 - a)*Sqrt[1 - a - b*x]) - (2*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1
- a - b*x])])/((1 - a)*Sqrt[1 - a^2])

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Rubi [A]  time = 0.131745, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {6164, 96, 93, 208} \[ \frac{\sqrt{a+b x+1}}{(1-a) \sqrt{-a-b x+1}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(1-a) \sqrt{1-a^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 + a + b*x]/((1 - a)*Sqrt[1 - a - b*x]) - (2*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1
- a - b*x])])/((1 - a)*Sqrt[1 - a^2])

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])

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.242331, size = 118, normalized size = 1.27 \[ -\frac{-\frac{\sqrt{-a^2-2 a b x-b^2 x^2+1}}{a+b x-1}-\frac{\log \left (\sqrt{1-a^2} \sqrt{-a^2-2 a b x-b^2 x^2+1}-a^2-a b x+1\right )}{\sqrt{1-a^2}}+\frac{\log (x)}{\sqrt{1-a^2}}}{a-1} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((-(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]/(-1 + a + b*x)) + Log[x]/Sqrt[1 - a^2] - Log[1 - a^2 - a*b*x + Sqrt[1 -
a^2]*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]]/Sqrt[1 - a^2])/(-1 + a))

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Maple [B]  time = 0.037, size = 391, normalized size = 4.2 \begin{align*} 2\,{\frac{b \left ( -2\,{b}^{2}x-2\,ab \right ) }{ \left ( -4\,{b}^{2} \left ( -{a}^{2}+1 \right ) -4\,{a}^{2}{b}^{2} \right ) \sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}+{\frac{1}{-{a}^{2}+1}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}+{\frac{xab}{-{a}^{2}+1}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}+{\frac{{a}^{2}}{-{a}^{2}+1}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}-{\ln \left ({\frac{1}{x} \left ( -2\,{a}^{2}+2-2\,xab+2\,\sqrt{-{a}^{2}+1}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \right ) } \right ) \left ( -{a}^{2}+1 \right ) ^{-{\frac{3}{2}}}}+{\frac{a}{-{a}^{2}+1}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}+{\frac{{a}^{2}bx}{-{a}^{2}+1}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}+{\frac{{a}^{3}}{-{a}^{2}+1}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}-{a\ln \left ({\frac{1}{x} \left ( -2\,{a}^{2}+2-2\,xab+2\,\sqrt{-{a}^{2}+1}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \right ) } \right ) \left ( -{a}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*(-2*b^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*a^2*b^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+1/(-a^2+1)/(-b^2*x^2-2*a*b*x-a
^2+1)^(1/2)+a*b/(-a^2+1)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+a^2/(-a^2+1)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/(-a^2+
1)^(3/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x)+1/(-a^2+1)/(-b^2*x^2-2*a*b*x
-a^2+1)^(1/2)*a+a^2*b/(-a^2+1)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+1/(-a^2+1)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a^3-
1/(-a^2+1)^(3/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x)*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{b x + a + 1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \sqrt{-{\left (b x + a\right )}^{2} + 1} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.76028, size = 730, normalized size = 7.85 \begin{align*} \left [-\frac{\sqrt{-a^{2} + 1}{\left (b x + a - 1\right )} \log \left (\frac{{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \,{\left (a^{3} - a\right )} b x - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (a b x + a^{2} - 1\right )} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (a^{2} - 1\right )}}{2 \,{\left (a^{4} - 2 \, a^{3} +{\left (a^{3} - a^{2} - a + 1\right )} b x + 2 \, a - 1\right )}}, -\frac{\sqrt{a^{2} - 1}{\left (b x + a - 1\right )} \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (a b x + a^{2} - 1\right )} \sqrt{a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \,{\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) - \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (a^{2} - 1\right )}}{a^{4} - 2 \, a^{3} +{\left (a^{3} - a^{2} - a + 1\right )} b x + 2 \, a - 1}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(sqrt(-a^2 + 1)*(b*x + a - 1)*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 2*sqrt(-b^2*x^2 - 2*a
*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2
- 1))/(a^4 - 2*a^3 + (a^3 - a^2 - a + 1)*b*x + 2*a - 1), -(sqrt(a^2 - 1)*(b*x + a - 1)*arctan(sqrt(-b^2*x^2 -
2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) -
sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1))/(a^4 - 2*a^3 + (a^3 - a^2 - a + 1)*b*x + 2*a - 1)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{a x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x^{2} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} - x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21274, size = 151, normalized size = 1.62 \begin{align*} -\frac{2 \, b \arctan \left (\frac{\frac{{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt{a^{2} - 1}}\right )}{\sqrt{a^{2} - 1}{\left (a{\left | b \right |} -{\left | b \right |}\right )}} - \frac{2 \, b}{{\left (a{\left | b \right |} -{\left | b \right |}\right )}{\left (\frac{\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b}{b^{2} x + a b} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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