\(\int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx\) [160]

Optimal result

Integrand size = 16, antiderivative size = 78 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=-\frac {1}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3+\frac {c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6310, 6313, 821, 272, 43, 65, 214} \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a}-\frac {1}{2} a c^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {1}{3} a^2 c^2 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \]

[In]

Int[E^ArcCoth[a*x]*(c - a*c*x)^2,x]

[Out]

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

Rule 43

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 6310

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

Rule 6313

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[-c^n, Subst[Int[(c +
d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^2 x^2 \, dx \\ & = -\left (\left (a^2 c^2\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}}{x^4} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {1}{3} a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3+\left (a c^2\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^3} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3+\frac {1}{2} \left (a c^2\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a^2}}}{x^2} \, dx,x,\frac {1}{x^2}\right ) \\ & = -\frac {1}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {c^2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a} \\ & = -\frac {1}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3+\frac {1}{2} \left (a c^2\right ) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right ) \\ & = -\frac {1}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3+\frac {c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.82 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {c^2 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-2-3 a x+2 a^2 x^2\right )+3 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{6 a} \]

[In]

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

[Out]

(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-2 - 3*a*x + 2*a^2*x^2) + 3*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.44

method	result	size
risch	\(\frac {\left (2 a^{2} x^{2}-3 a x -2\right ) \left (a x -1\right ) c^{2}}{6 a \sqrt {\frac {a x -1}{a x +1}}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 \sqrt {a^{2}}\, \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(112\)
default	\(-\frac {\left (a x -1\right ) c^{2} \left (3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{6 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a}\)	\(121\)

[In]

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

[Out]

1/6*(2*a^2*x^2-3*a*x-2)*(a*x-1)/a*c^2/((a*x-1)/(a*x+1))^(1/2)+1/2*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2
)^(1/2)*c^2/(a*x+1)/((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.32 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (2 \, a^{3} c^{2} x^{3} - a^{2} c^{2} x^{2} - 5 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a} \]

[In]

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

[Out]

1/6*(3*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 3*c^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (2*a^3*c^2*x^3 - a^
2*c^2*x^2 - 5*a*c^2*x - 2*c^2)*sqrt((a*x - 1)/(a*x + 1)))/a

Sympy [F]

\[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=c^{2} \left (\int \left (- \frac {2 a x}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {a^{2} x^{2}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \frac {1}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (66) = 132\).

Time = 0.21 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.32 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {1}{6} \, a {\left (\frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (3 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 8 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 3 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - a^{2}}\right )} \]

[In]

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

[Out]

1/6*a*(3*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 3*c^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 2*(3*c^2*
((a*x - 1)/(a*x + 1))^(5/2) + 8*c^2*((a*x - 1)/(a*x + 1))^(3/2) - 3*c^2*sqrt((a*x - 1)/(a*x + 1)))/(3*(a*x - 1
)*a^2/(a*x + 1) - 3*(a*x - 1)^2*a^2/(a*x + 1)^2 + (a*x - 1)^3*a^2/(a*x + 1)^3 - a^2))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.26 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {1}{6} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (\frac {2 \, a c^{2} x}{\mathrm {sgn}\left (a x + 1\right )} - \frac {3 \, c^{2}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x - \frac {2 \, c^{2}}{a \mathrm {sgn}\left (a x + 1\right )}\right )} - \frac {c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{2 \, {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \]

[In]

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

[Out]

1/6*sqrt(a^2*x^2 - 1)*((2*a*c^2*x/sgn(a*x + 1) - 3*c^2/sgn(a*x + 1))*x - 2*c^2/(a*sgn(a*x + 1))) - 1/2*c^2*log
(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(abs(a)*sgn(a*x + 1))

Mupad [B] (verification not implemented)

Time = 4.23 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.77 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {\frac {8\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}-c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}+c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a-\frac {3\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {3\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}}+\frac {c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]

[In]

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

[Out]

((8*c^2*((a*x - 1)/(a*x + 1))^(3/2))/3 - c^2*((a*x - 1)/(a*x + 1))^(1/2) + c^2*((a*x - 1)/(a*x + 1))^(5/2))/(a
- (3*a*(a*x - 1))/(a*x + 1) + (3*a*(a*x - 1)^2)/(a*x + 1)^2 - (a*(a*x - 1)^3)/(a*x + 1)^3) + (c^2*atanh(((a*x
- 1)/(a*x + 1))^(1/2)))/a