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

Optimal. Leaf size=16 \[ \frac{c^2}{a^2 x}+c^2 x \]

[Out]

c^2/(a^2*x) + c^2*x

________________________________________________________________________________________

Rubi [A]  time = 0.121349, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6167, 6131, 6129, 73, 14} \[ \frac{c^2}{a^2 x}+c^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^2/(a^2*x) + c^2*x

Rule 6167

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]

Rule 6131

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 6129

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 73

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]

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]]

Rubi steps

\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^2 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^2 \, dx\\ &=-\frac{c^2 \int \frac{e^{2 \tanh ^{-1}(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]  time = 0.0886441, size = 16, normalized size = 1. \[ \frac{c^2}{a^2 x}+c^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^2/(a^2*x) + c^2*x

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 17, normalized size = 1.1 \begin{align*}{\frac{{c}^{2} \left ({a}^{2}x+{x}^{-1} \right ) }{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.00241, size = 22, normalized size = 1.38 \begin{align*} c^{2} x + \frac{c^{2}}{a^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 1.42953, size = 39, normalized size = 2.44 \begin{align*} \frac{a^{2} c^{2} x^{2} + c^{2}}{a^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0.259576, size = 15, normalized size = 0.94 \begin{align*} \frac{a^{2} c^{2} x + \frac{c^{2}}{x}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Giac [A]  time = 1.14569, size = 22, normalized size = 1.38 \begin{align*} c^{2} x + \frac{c^{2}}{a^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)