\(\int e^{-\coth ^{-1}(a x)} (c-a^2 c x^2) \, dx\) [593]

Optimal result

Integrand size = 20, antiderivative size = 145 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {1}{2} c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {1}{2} a c \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {1}{3} a^2 c \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2} x^3-\frac {c \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{2 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6326, 6330, 96, 94, 214} \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {1}{3} a^2 c x^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {c \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{2 a}+\frac {1}{2} a c x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {1}{2} c x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} \]

[In]

Int[(c - a^2*c*x^2)/E^ArcCoth[a*x],x]

[Out]

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

Rule 94

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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

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 6326

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 6330

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]

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.42 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {c \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (2+3 a x-2 a^2 x^2\right )-3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{6 a} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.74

method	result	size
risch	\(-\frac {\left (2 a^{2} x^{2}-3 a x -2\right ) \left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{6 a}-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 \sqrt {a^{2}}\, \left (a x -1\right )}\)	\(108\)
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c \left (3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a}\)	\(119\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.63 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (2 \, a^{3} c x^{3} - a^{2} c x^{2} - 5 \, a c x - 2 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.18 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {1}{6} \, a {\left (\frac {2 \, {\left (3 \, c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 8 \, c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 3 \, c \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - a^{2}} - \frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.57 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{2 \, {\left | a \right |}} - \frac {1}{6} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (2 \, a c x \mathrm {sgn}\left (a x + 1\right ) - 3 \, c \mathrm {sgn}\left (a x + 1\right )\right )} x - \frac {2 \, c \mathrm {sgn}\left (a x + 1\right )}{a}\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 3.89 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.91 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {\frac {8\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}-c\,\sqrt {\frac {a\,x-1}{a\,x+1}}+c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a-\frac {3\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {3\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}}-\frac {c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]

[In]

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

[Out]

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