### 3.592 $$\int e^{-\coth ^{-1}(a x)} (c-a^2 c x^2)^2 \, dx$$

Optimal. Leaf size=233 $\frac{1}{5} a^4 c^2 x^5 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{1}{4} a^3 c^2 x^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{5/2}+\frac{1}{4} a^2 c^2 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{1}{8} a c^2 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{3}{8} c^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{8 a}$

[Out]

(-3*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/8 - (a*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2)/8 + (a^2*
c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3)/4 - (a^3*c^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)*x^4)/4 + (
a^4*c^2*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)*x^5)/5 - (3*c^2*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/
(8*a)

________________________________________________________________________________________

Rubi [A]  time = 0.199331, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.227, Rules used = {6191, 6195, 94, 92, 208} $\frac{1}{5} a^4 c^2 x^5 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{1}{4} a^3 c^2 x^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{5/2}+\frac{1}{4} a^2 c^2 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{1}{8} a c^2 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{3}{8} c^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{8 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-3*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/8 - (a*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2)/8 + (a^2*
c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3)/4 - (a^3*c^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)*x^4)/4 + (
a^4*c^2*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)*x^5)/5 - (3*c^2*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/
(8*a)

Rule 6191

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

Rule 6195

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

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 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-a^2 c x^2\right )^2 \, dx &=\left (a^4 c^2\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^2 x^4 \, dx\\ &=-\left (\left (a^4 c^2\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{3/2}}{x^6} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5+\left (a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{3/2}}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{4} a^3 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x^4+\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5-\frac{1}{4} \left (3 a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{4} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{4} a^3 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x^4+\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5+\frac{1}{4} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^3 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{8} a c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{1}{4} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{4} a^3 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x^4+\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5+\frac{1}{8} \left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^2 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3}{8} c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{1}{8} a c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{1}{4} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{4} a^3 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x^4+\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5+\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a}\\ &=-\frac{3}{8} c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{1}{8} a c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{1}{4} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{4} a^3 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x^4+\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5-\frac{\left (3 c^2\right ) \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 )}{8 a^2}\\ &=-\frac{3}{8} c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{1}{8} a c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{1}{4} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{4} a^3 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x^4+\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{8 a}\\ \end{align*}

Mathematica [A]  time = 0.10375, size = 79, normalized size = 0.34 $\frac{c^2 \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^4 x^4-10 a^3 x^3-16 a^2 x^2+25 a x+8\right )-15 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{40 a}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(8 + 25*a*x - 16*a^2*x^2 - 10*a^3*x^3 + 8*a^4*x^4) - 15*Log[(1 + Sqrt[1 - 1/(a
^2*x^2)])*x]))/(40*a)

________________________________________________________________________________________

Maple [A]  time = 0.138, size = 183, normalized size = 0.8 \begin{align*} -{\frac{ \left ( ax+1 \right ){c}^{2}}{120\,a}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -24\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+30\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+40\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-16\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}-45\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+45\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a \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((-a^2*c*x^2+c)^2*((a*x-1)/(a*x+1))^(1/2),x)

[Out]

-1/120*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*c^2/a*(-24*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^2*a^2+30*(a^2)^(1/2)*(a^2*x^
2-1)^(3/2)*x*a+40*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)-16*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)-45*(a^2)^(1/2)*(a^2*x^2
-1)^(1/2)*x*a+45*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a)/((a*x-1)*(a*x+1))^(1/2)/(a^2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.13966, size = 350, normalized size = 1.5 \begin{align*} -\frac{1}{40} \, a{\left (\frac{15 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{15 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{2 \,{\left (15 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} - 70 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 128 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 70 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 15 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{5 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{10 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{10 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac{5 \,{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac{{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - a^{2}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*a*(15*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 15*c^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 2*(15
*c^2*((a*x - 1)/(a*x + 1))^(9/2) - 70*c^2*((a*x - 1)/(a*x + 1))^(7/2) - 128*c^2*((a*x - 1)/(a*x + 1))^(5/2) +
70*c^2*((a*x - 1)/(a*x + 1))^(3/2) - 15*c^2*sqrt((a*x - 1)/(a*x + 1)))/(5*(a*x - 1)*a^2/(a*x + 1) - 10*(a*x -
1)^2*a^2/(a*x + 1)^2 + 10*(a*x - 1)^3*a^2/(a*x + 1)^3 - 5*(a*x - 1)^4*a^2/(a*x + 1)^4 + (a*x - 1)^5*a^2/(a*x +
1)^5 - a^2))

________________________________________________________________________________________

Fricas [A]  time = 2.00816, size = 285, normalized size = 1.22 \begin{align*} -\frac{15 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 15 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (8 \, a^{5} c^{2} x^{5} - 2 \, a^{4} c^{2} x^{4} - 26 \, a^{3} c^{2} x^{3} + 9 \, a^{2} c^{2} x^{2} + 33 \, a c^{2} x + 8 \, c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{40 \, a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*(15*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*c^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (8*a^5*c^2*x^5
- 2*a^4*c^2*x^4 - 26*a^3*c^2*x^3 + 9*a^2*c^2*x^2 + 33*a*c^2*x + 8*c^2)*sqrt((a*x - 1)/(a*x + 1)))/a

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{2} \left (\int - 2 a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int a^{4} x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*(Integral(-2*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(a**4*x**4*sqrt(a*x/(a*x + 1) - 1/
(a*x + 1)), x) + Integral(sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x))

________________________________________________________________________________________

Giac [A]  time = 1.16409, size = 170, normalized size = 0.73 \begin{align*} \frac{3 \, c^{2} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{8 \,{\left | a \right |}} + \frac{1}{40} \, \sqrt{a^{2} x^{2} - 1}{\left ({\left (25 \, c^{2} \mathrm{sgn}\left (a x + 1\right ) - 2 \,{\left (8 \, a c^{2} \mathrm{sgn}\left (a x + 1\right ) -{\left (4 \, a^{3} c^{2} x \mathrm{sgn}\left (a x + 1\right ) - 5 \, a^{2} c^{2} \mathrm{sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x + \frac{8 \, c^{2} \mathrm{sgn}\left (a x + 1\right )}{a}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*c^2*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + 1/40*sqrt(a^2*x^2 - 1)*((25*c^2*sgn(a*x
+ 1) - 2*(8*a*c^2*sgn(a*x + 1) - (4*a^3*c^2*x*sgn(a*x + 1) - 5*a^2*c^2*sgn(a*x + 1))*x)*x)*x + 8*c^2*sgn(a*x +
1)/a)