### 3.809 $$\int e^{-\coth ^{-1}(a x)} (c-\frac{c}{a^2 x^2}) \, dx$$

Optimal. Leaf size=108 $c x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}+\frac{2 c \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{a}+\frac{c \csc ^{-1}(a x)}{a}-\frac{c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a}$

[Out]

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

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Rubi [A]  time = 0.0763682, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.45, Rules used = {6194, 97, 154, 21, 105, 41, 216, 92, 208} $c x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}+\frac{2 c \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{a}+\frac{c \csc ^{-1}(a x)}{a}-\frac{c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c - c/(a^2*x^2))/E^ArcCoth[a*x],x]

[Out]

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

Rule 6194

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> -Dist[c^p, Subst[Int[((1 - x/a)^(p
- n/2)*(1 + x/a)^(p + n/2))/x^2, x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !Integ
erQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegersQ[2*p, p + n/2]

Rule 97

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)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 216

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

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

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

Mathematica [A]  time = 0.083748, size = 55, normalized size = 0.51 $\frac{c \left (\sqrt{1-\frac{1}{a^2 x^2}} (a x+1)-\log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )+\sin ^{-1}\left (\frac{1}{a x}\right )\right )}{a}$

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - c/(a^2*x^2))/E^ArcCoth[a*x],x]

[Out]

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

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Maple [A]  time = 0.13, size = 166, normalized size = 1.5 \begin{align*} -{\frac{c \left ( ax+1 \right ) }{{a}^{2}x}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+ \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}}-\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) x{a}^{2}-ax\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*c*(-(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^2*a^2+(a^2*x^2-1)^(3/2)*(a^2)^(1/2)-(a^2)
^(1/2)*(a^2*x^2-1)^(1/2)*x*a+ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x*a^2-a*x*(a^2)^(1/2)*arcta
n(1/(a^2*x^2-1)^(1/2)))/((a*x-1)*(a*x+1))^(1/2)/a^2/x/(a^2)^(1/2)

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Maxima [A]  time = 1.51355, size = 158, normalized size = 1.46 \begin{align*} -a{\left (\frac{4 \, c \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac{2 \, c \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*(4*c*sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)^2*a^2/(a*x + 1)^2 - a^2) + 2*c*arctan(sqrt((a*x - 1)/(a*x + 1)))/
a^2 + c*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - c*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2)

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Fricas [A]  time = 1.42899, size = 262, normalized size = 2.43 \begin{align*} -\frac{2 \, a c x \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + a c x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - a c x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (a^{2} c x^{2} + 2 \, a c x + c\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*c*x*arctan(sqrt((a*x - 1)/(a*x + 1))) + a*c*x*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - a*c*x*log(sqrt((a*x -
1)/(a*x + 1)) - 1) - (a^2*c*x^2 + 2*a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^2*x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14917, size = 163, normalized size = 1.51 \begin{align*} -\frac{2 \, c \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right )}{a} + \frac{c \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac{\sqrt{a^{2} x^{2} - 1} c \mathrm{sgn}\left (a x + 1\right )}{a} + \frac{2 \, c \mathrm{sgn}\left (a x + 1\right )}{{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1\right )}{\left | a \right |}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1)/a + c*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x
+ 1)/abs(a) + sqrt(a^2*x^2 - 1)*c*sgn(a*x + 1)/a + 2*c*sgn(a*x + 1)/(((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)*ab
s(a))