3.457 \(\int e^{2 \tanh ^{-1}(a x)} (c-\frac{c}{a x})^5 \, dx\)

Optimal. Leaf size=62 \[ \frac{c^5}{a^3 x^2}-\frac{c^5}{a^4 x^3}+\frac{c^5}{4 a^5 x^4}+\frac{2 c^5}{a^2 x}+\frac{3 c^5 \log (x)}{a}+c^5 (-x) \]

[Out]

c^5/(4*a^5*x^4) - c^5/(a^4*x^3) + c^5/(a^3*x^2) + (2*c^5)/(a^2*x) - c^5*x + (3*c^5*Log[x])/a

________________________________________________________________________________________

Rubi [A]  time = 0.113199, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6131, 6129, 75} \[ \frac{c^5}{a^3 x^2}-\frac{c^5}{a^4 x^3}+\frac{c^5}{4 a^5 x^4}+\frac{2 c^5}{a^2 x}+\frac{3 c^5 \log (x)}{a}+c^5 (-x) \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*(c - c/(a*x))^5,x]

[Out]

c^5/(4*a^5*x^4) - c^5/(a^4*x^3) + c^5/(a^3*x^2) + (2*c^5)/(a^2*x) - c^5*x + (3*c^5*Log[x])/a

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 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

\begin{align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^5 \, dx &=-\frac{c^5 \int \frac{e^{2 \tanh ^{-1}(a x)} (1-a x)^5}{x^5} \, dx}{a^5}\\ &=-\frac{c^5 \int \frac{(1-a x)^4 (1+a x)}{x^5} \, dx}{a^5}\\ &=-\frac{c^5 \int \left (a^5+\frac{1}{x^5}-\frac{3 a}{x^4}+\frac{2 a^2}{x^3}+\frac{2 a^3}{x^2}-\frac{3 a^4}{x}\right ) \, dx}{a^5}\\ &=\frac{c^5}{4 a^5 x^4}-\frac{c^5}{a^4 x^3}+\frac{c^5}{a^3 x^2}+\frac{2 c^5}{a^2 x}-c^5 x+\frac{3 c^5 \log (x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.216819, size = 64, normalized size = 1.03 \[ \frac{c^5}{a^3 x^2}-\frac{c^5}{a^4 x^3}+\frac{c^5}{4 a^5 x^4}+\frac{2 c^5}{a^2 x}+\frac{3 c^5 \log (a x)}{a}+c^5 (-x) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcTanh[a*x])*(c - c/(a*x))^5,x]

[Out]

c^5/(4*a^5*x^4) - c^5/(a^4*x^3) + c^5/(a^3*x^2) + (2*c^5)/(a^2*x) - c^5*x + (3*c^5*Log[a*x])/a

________________________________________________________________________________________

Maple [A]  time = 0.036, size = 61, normalized size = 1. \begin{align*}{\frac{{c}^{5}}{4\,{a}^{5}{x}^{4}}}-{\frac{{c}^{5}}{{a}^{4}{x}^{3}}}+{\frac{{c}^{5}}{{x}^{2}{a}^{3}}}+2\,{\frac{{c}^{5}}{{a}^{2}x}}-{c}^{5}x+3\,{\frac{{c}^{5}\ln \left ( x \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^5,x)

[Out]

1/4*c^5/a^5/x^4-c^5/a^4/x^3+c^5/x^2/a^3+2*c^5/a^2/x-c^5*x+3*c^5*ln(x)/a

________________________________________________________________________________________

Maxima [A]  time = 0.947721, size = 78, normalized size = 1.26 \begin{align*} -c^{5} x + \frac{3 \, c^{5} \log \left (x\right )}{a} + \frac{8 \, a^{3} c^{5} x^{3} + 4 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + c^{5}}{4 \, a^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^5,x, algorithm="maxima")

[Out]

-c^5*x + 3*c^5*log(x)/a + 1/4*(8*a^3*c^5*x^3 + 4*a^2*c^5*x^2 - 4*a*c^5*x + c^5)/(a^5*x^4)

________________________________________________________________________________________

Fricas [A]  time = 2.07021, size = 143, normalized size = 2.31 \begin{align*} -\frac{4 \, a^{5} c^{5} x^{5} - 12 \, a^{4} c^{5} x^{4} \log \left (x\right ) - 8 \, a^{3} c^{5} x^{3} - 4 \, a^{2} c^{5} x^{2} + 4 \, a c^{5} x - c^{5}}{4 \, a^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^5,x, algorithm="fricas")

[Out]

-1/4*(4*a^5*c^5*x^5 - 12*a^4*c^5*x^4*log(x) - 8*a^3*c^5*x^3 - 4*a^2*c^5*x^2 + 4*a*c^5*x - c^5)/(a^5*x^4)

________________________________________________________________________________________

Sympy [A]  time = 0.461193, size = 63, normalized size = 1.02 \begin{align*} \frac{- a^{5} c^{5} x + 3 a^{4} c^{5} \log{\left (x \right )} + \frac{8 a^{3} c^{5} x^{3} + 4 a^{2} c^{5} x^{2} - 4 a c^{5} x + c^{5}}{4 x^{4}}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**5,x)

[Out]

(-a**5*c**5*x + 3*a**4*c**5*log(x) + (8*a**3*c**5*x**3 + 4*a**2*c**5*x**2 - 4*a*c**5*x + c**5)/(4*x**4))/a**5

________________________________________________________________________________________

Giac [A]  time = 1.19038, size = 80, normalized size = 1.29 \begin{align*} -c^{5} x + \frac{3 \, c^{5} \log \left ({\left | x \right |}\right )}{a} + \frac{8 \, a^{3} c^{5} x^{3} + 4 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + c^{5}}{4 \, a^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^5,x, algorithm="giac")

[Out]

-c^5*x + 3*c^5*log(abs(x))/a + 1/4*(8*a^3*c^5*x^3 + 4*a^2*c^5*x^2 - 4*a*c^5*x + c^5)/(a^5*x^4)