3.824 \(\int \frac{e^{\tanh ^{-1}(a+b x)}}{x^3} \, dx\)

Optimal. Leaf size=162 \[ -\frac{(2 a+1) b^2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(1-a)^2 (a+1) \sqrt{1-a^2}}-\frac{\sqrt{-a-b x+1} (a+b x+1)^{3/2}}{2 \left (1-a^2\right ) x^2}-\frac{(2 a+1) b \sqrt{-a-b x+1} \sqrt{a+b x+1}}{2 (1-a)^2 (a+1) x} \]

[Out]

-((1 + 2*a)*b*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*(1 - a)^2*(1 + a)*x) - (Sqrt[1 - a - b*x]*(1 + a + b*x)^
(3/2))/(2*(1 - a^2)*x^2) - ((1 + 2*a)*b^2*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*
x])])/((1 - a)^2*(1 + a)*Sqrt[1 - a^2])

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Rubi [A]  time = 0.108395, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6163, 96, 94, 93, 208} \[ -\frac{(2 a+1) b^2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(1-a)^2 (a+1) \sqrt{1-a^2}}-\frac{\sqrt{-a-b x+1} (a+b x+1)^{3/2}}{2 \left (1-a^2\right ) x^2}-\frac{(2 a+1) b \sqrt{-a-b x+1} \sqrt{a+b x+1}}{2 (1-a)^2 (a+1) x} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a + b*x]/x^3,x]

[Out]

-((1 + 2*a)*b*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*(1 - a)^2*(1 + a)*x) - (Sqrt[1 - a - b*x]*(1 + a + b*x)^
(3/2))/(2*(1 - a^2)*x^2) - ((1 + 2*a)*b^2*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*
x])])/((1 - a)^2*(1 + a)*Sqrt[1 - a^2])

Rule 6163

Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[((d + e*x)^m*(1
+ a*c + b*c*x)^(n/2))/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]

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 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[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^3} \, dx &=\int \frac{\sqrt{1+a+b x}}{x^3 \sqrt{1-a-b x}} \, dx\\ &=-\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 \left (1-a^2\right ) x^2}+\frac{((1+2 a) b) \int \frac{\sqrt{1+a+b x}}{x^2 \sqrt{1-a-b x}} \, dx}{2 \left (1-a^2\right )}\\ &=-\frac{(1+2 a) b \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 (1-a)^2 (1+a) x}-\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 \left (1-a^2\right ) x^2}+\frac{\left ((1+2 a) b^2\right ) \int \frac{1}{x \sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{2 (1-a)^2 (1+a)}\\ &=-\frac{(1+2 a) b \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 (1-a)^2 (1+a) x}-\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 \left (1-a^2\right ) x^2}+\frac{\left ((1+2 a) b^2\right ) \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)^2 (1+a)}\\ &=-\frac{(1+2 a) b \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 (1-a)^2 (1+a) x}-\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 \left (1-a^2\right ) x^2}-\frac{(1+2 a) b^2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{1+a+b x}}{\sqrt{1+a} \sqrt{1-a-b x}}\right )}{(1-a)^2 (1+a) \sqrt{1-a^2}}\\ \end{align*}

Mathematica [A]  time = 0.112476, size = 117, normalized size = 0.72 \[ \frac{\left (a^2-a b x-2 b x-1\right ) \sqrt{-a^2-2 a b x-b^2 x^2+1}}{2 (a-1)^2 (a+1) x^2}+\frac{(2 a+1) b^2 \tan ^{-1}\left (\frac{\sqrt{-a-b x+1}}{\sqrt{\frac{a-1}{a+1}} \sqrt{a+b x+1}}\right )}{(a-1)^{5/2} (a+1)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTanh[a + b*x]/x^3,x]

[Out]

((-1 + a^2 - 2*b*x - a*b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2])/(2*(-1 + a)^2*(1 + a)*x^2) + ((1 + 2*a)*b^2*Arc
Tan[Sqrt[1 - a - b*x]/(Sqrt[(-1 + a)/(1 + a)]*Sqrt[1 + a + b*x])])/((-1 + a)^(5/2)*(1 + a)^(3/2))

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Maple [B]  time = 0.072, size = 453, normalized size = 2.8 \begin{align*} -{\frac{b}{ \left ( -{a}^{2}+1 \right ) x}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}-{\frac{3\,a{b}^{2}}{2}\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{1}{ \left ( -2\,{a}^{2}+2 \right ){x}^{2}}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}-{\frac{3\,ab}{2\, \left ( -{a}^{2}+1 \right ) ^{2}x}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}-{\frac{3\,{a}^{2}{b}^{2}}{2}\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{5}{2}}}}-{\frac{{b}^{2}}{2}\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}{ \left ( -2\,{a}^{2}+2 \right ){x}^{2}}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}-{\frac{3\,{a}^{2}b}{2\, \left ( -{a}^{2}+1 \right ) ^{2}x}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}-{\frac{3\,{a}^{3}{b}^{2}}{2}\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{5}{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^3,x)

[Out]

-b/(-a^2+1)/x*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3/2*a*b^2/(-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)-1/2/(-a^2+1)/x^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3/2*a*b/(-a^2+1)^2/x*(-b^2*x^
2-2*a*b*x-a^2+1)^(1/2)-3/2*a^2*b^2/(-a^2+1)^(5/2)*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)-1/2*b^2/(-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
/2*a/(-a^2+1)/x^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3/2*a^2*b/(-a^2+1)^2/x*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3/2*a^3
*b^2/(-a^2+1)^(5/2)*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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.63096, size = 821, normalized size = 5.07 \begin{align*} \left [-\frac{\sqrt{-a^{2} + 1}{\left (2 \, a + 1\right )} b^{2} x^{2} \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 \,{\left (a^{4} -{\left (a^{3} + 2 \, a^{2} - a - 2\right )} b x - 2 \, a^{2} + 1\right )} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{4 \,{\left (a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right )} x^{2}}, \frac{\sqrt{a^{2} - 1}{\left (2 \, a + 1\right )} b^{2} x^{2} \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 ) +{\left (a^{4} -{\left (a^{3} + 2 \, a^{2} - a - 2\right )} b x - 2 \, a^{2} + 1\right )} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \,{\left (a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right )} x^{2}}\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^3,x, algorithm="fricas")

[Out]

[-1/4*(sqrt(-a^2 + 1)*(2*a + 1)*b^2*x^2*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*(a^4 - (a^3 + 2*a^2 - a - 2)*b*x -
2*a^2 + 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*x^2), 1/2*(sqrt(a^2 - 1)*(
2*a + 1)*b^2*x^2*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)) + (a^4 - (a^3 + 2*a^2 - a - 2)*b*x - 2*a^2 + 1)*sqrt(-b^2*x^2 - 2*a*b*x
- a^2 + 1))/((a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b x + 1}{x^{3} \sqrt{- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, 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**3,x)

[Out]

Integral((a + b*x + 1)/(x**3*sqrt(-(a + b*x - 1)*(a + b*x + 1))), x)

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Giac [B]  time = 1.30846, size = 840, normalized size = 5.19 \begin{align*} \text{result too large to display} \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^3,x, algorithm="giac")

[Out]

(2*a*b^3 + b^3)*arctan(((sqrt(-(b*x + a)^2 + 1)*abs(b) + b)*a/(b^2*x + a*b) - 1)/sqrt(a^2 - 1))/((a^3*abs(b) -
a^2*abs(b) - a*abs(b) + abs(b))*sqrt(a^2 - 1)) - (2*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^2*a^4*b^3/(b^2*x + a*
b)^2 + 2*a^4*b^3 - 5*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)*a^3*b^3/(b^2*x + a*b) + 2*(sqrt(-(b*x + a)^2 + 1)*abs
(b) + b)^2*a^3*b^3/(b^2*x + a*b)^2 - 3*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^3*a^3*b^3/(b^2*x + a*b)^3 + 2*a^3*b
^3 - 6*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)*a^2*b^3/(b^2*x + a*b) + 3*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^2*a^2
*b^3/(b^2*x + a*b)^2 - 2*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^3*a^2*b^3/(b^2*x + a*b)^3 - a^2*b^3 + 2*(sqrt(-(b
*x + a)^2 + 1)*abs(b) + b)*a*b^3/(b^2*x + a*b) + 4*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^2*a*b^3/(b^2*x + a*b)^2
+ 2*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^3*a*b^3/(b^2*x + a*b)^3 - 2*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^2*b^3
/(b^2*x + a*b)^2)/((a^5*abs(b) - a^4*abs(b) - a^3*abs(b) + a^2*abs(b))*((sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^2*
a/(b^2*x + a*b)^2 + a - 2*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)/(b^2*x + a*b))^2)