\(\int e^{3 \coth ^{-1}(a x)} (c-\frac {c}{a x})^2 \, dx\) [398]

Optimal result

Integrand size = 22, antiderivative size = 63 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right ) x}{a}+\frac {c^2 \csc ^{-1}(a x)}{a}+\frac {c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 63, 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, 864, 827, 858, 222, 272, 65, 214} \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+\frac {c^2 x \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}{a}+\frac {c^2 \csc ^{-1}(a x)}{a} \]

[In]

Int[E^(3*ArcCoth[a*x])*(c - c/(a*x))^2,x]

[Out]

(c^2*Sqrt[1 - 1/(a^2*x^2)]*(a - x^(-1))*x)/a + (c^2*ArcCsc[a*x])/a + (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 827

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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 864

Int[((x_)^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + c*(x/e))*(a + c*x
^2)^(p - 1), x] /; FreeQ[{a, c, d, e, n, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||
!IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2]))

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(154\) vs. \(2(63)=126\).

Time = 0.17 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.44 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {c^2 \left (-1+a x+a^2 x^2-a^3 x^3+4 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )+a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 \arcsin \left (\frac {1}{a x}\right )-a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2} \]

[In]

Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^2,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(130\) vs. \(2(59)=118\).

Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.08

method	result	size
risch	\(-\frac {\left (a x -1\right ) c^{2}}{x \,a^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {a \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\sqrt {\left (a x -1\right ) \left (a x +1\right )}+\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(131\)
default	\(\frac {\left (a x -1\right )^{2} c^{2} \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right )}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x \sqrt {a^{2}}}\)	\(174\)

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^2,x,method=_RETURNVERBOSE)

[Out]

-(a*x-1)/x*c^2/a^2/((a*x-1)/(a*x+1))^(1/2)+1/a*(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)+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 = 0.25 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.81 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {2 \, a c^{2} x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (a^{2} c^{2} x^{2} - c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^2,x, algorithm="fricas")

[Out]

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

\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^{2} \left (\int \left (- \frac {2 a}{\frac {a x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{2}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \frac {1}{\frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx\right )}{a^{2}} \]

[In]

integrate(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a/x)**2,x)

[Out]

c**2*(Integral(-2*a/(a*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/
(a*x + 1)), x) + Integral(a**2/(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - sqrt(a*x/(a*x + 1) - 1/(a*x
+ 1))/(a*x + 1)), x) + Integral(1/(a*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x**2*sqrt(a*x/(a*x + 1
) - 1/(a*x + 1))/(a*x + 1)), x))/a**2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (59) = 118\).

Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.98 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-{\left (\frac {4 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac {2 \, c^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} a \]

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (59) = 118\).

Time = 0.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.19 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {2 \, c^{2} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{a \mathrm {sgn}\left (a x + 1\right )} - \frac {c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c^{2}}{a \mathrm {sgn}\left (a x + 1\right )} - \frac {2 \, c^{2}}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 3.81 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.43 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {4\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {2\,c^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {2\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]

[In]

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

[Out]

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