\(\int e^{2 \coth ^{-1}(a x)} (c-\frac {c}{a x})^2 \, dx\) [390]

Optimal result

Integrand size = 22, antiderivative size = 16 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2}{a^2 x}+c^2 x \]

[Out]

c^2/a^2/x+c^2*x

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6302, 6266, 6264, 74, 14} \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2}{a^2 x}+c^2 x \]

[In]

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

[Out]

c^2/(a^2*x) + c^2*x

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 74

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

Rule 6264

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

Rule 6266

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

Rule 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2}{a^2 x}+c^2 x \]

[In]

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

[Out]

c^2/(a^2*x) + c^2*x

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {c^{2} \left (a^{2} x +\frac {1}{x}\right )}{a^{2}}\)	\(17\)
risch	\(\frac {c^{2}}{a^{2} x}+c^{2} x\)	\(17\)
gosper	\(\frac {c^{2} \left (a^{2} x^{2}+1\right )}{x \,a^{2}}\)	\(20\)
parallelrisch	\(\frac {a^{2} c^{2} x^{2}+c^{2}}{a^{2} x}\)	\(22\)
norman	\(\frac {\frac {c^{2}}{a}+a \,c^{2} x^{2}}{a x}\)	\(24\)
meijerg	\(-\frac {c^{2} \left (-a x -\ln \left (-a x +1\right )\right )}{a}-\frac {c^{2} \ln \left (-a x +1\right )}{a}+\frac {c^{2} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )\right )}{a}+\frac {c^{2} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{a x}\right )}{a}\)	\(94\)

[In]

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

[Out]

c^2/a^2*(a^2*x+1/x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {a^{2} c^{2} x^{2} + c^{2}}{a^{2} x} \]

[In]

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

[Out]

(a^2*c^2*x^2 + c^2)/(a^2*x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {a^{2} c^{2} x + \frac {c^{2}}{x}}{a^{2}} \]

[In]

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

[Out]

(a**2*c**2*x + c**2/x)/a**2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=c^{2} x + \frac {c^{2}}{a^{2} x} \]

[In]

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

[Out]

c^2*x + c^2/(a^2*x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=c^{2} x + \frac {c^{2}}{a^{2} x} \]

[In]

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

[Out]

c^2*x + c^2/(a^2*x)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2\,\left (a^2\,x^2+1\right )}{a^2\,x} \]

[In]

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

[Out]

(c^2*(a^2*x^2 + 1))/(a^2*x)