\(\int e^{-\coth ^{-1}(a x)} (c-\frac {c}{a x})^2 \, dx\) [415]

Optimal result

Integrand size = 22, antiderivative size = 77 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {3 c^2 \csc ^{-1}(a x)}{a}-\frac {3 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6312, 1821, 1823, 858, 222, 272, 65, 214} \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {3 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+c^2 x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a}-\frac {3 c^2 \csc ^{-1}(a x)}{a} \]

[In]

Int[(c - c/(a*x))^2/E^ArcCoth[a*x],x]

[Out]

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

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 222

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 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 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rule 1821

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 6312

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

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.71 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2 \left (\sqrt {1-\frac {1}{a^2 x^2}} (-1+a x)-3 \arcsin \left (\frac {1}{a x}\right )-3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.74

method	result	size
risch	\(-\frac {\left (a x +1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}}{x \,a^{2}}+\frac {\left (-\frac {3 a \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\sqrt {\left (a x -1\right ) \left (a x +1\right )}-3 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a \left (a x -1\right )}\)	\(134\)
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{2} \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -3 a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+4 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -4 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x \right )}{\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x \sqrt {a^{2}}}\)	\(227\)

[In]

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

[Out]

-(a*x+1)/x*c^2/a^2*((a*x-1)/(a*x+1))^(1/2)+1/a*(-3*a*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)+((a*x
-1)*(a*x+1))^(1/2)-3*arctan(1/(a^2*x^2-1)^(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 = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.47 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {6 \, a c^{2} x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 3 \, a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 3 \, a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} c^{2} x^{2} - c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.64 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-{\left (\frac {4 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} - \frac {6 \, c^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} a \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.69 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {6 \, c^{2} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {3 \, c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c^{2} \mathrm {sgn}\left (a x + 1\right )}{a} - \frac {2 \, c^{2} \mathrm {sgn}\left (a x + 1\right )}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} {\left | a \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.17 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {4\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}+\frac {6\,c^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {6\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]

[In]

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

[Out]

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