3.458 \(\int e^{2 \tanh ^{-1}(a x)} (c-\frac{c}{a x})^4 \, dx\)

Optimal. Leaf size=40 \[ \frac{c^4}{a^3 x^2}-\frac{c^4}{3 a^4 x^3}+\frac{2 c^4 \log (x)}{a}+c^4 (-x) \]

[Out]

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

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Rubi [A]  time = 0.102118, antiderivative size = 40, 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} \[ \frac{c^4}{a^3 x^2}-\frac{c^4}{3 a^4 x^3}+\frac{2 c^4 \log (x)}{a}+c^4 (-x) \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*(c - c/(a*x))^4,x]

[Out]

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

Mathematica [A]  time = 0.148076, size = 42, normalized size = 1.05 \[ \frac{c^4}{a^3 x^2}-\frac{c^4}{3 a^4 x^3}+\frac{2 c^4 \log (a x)}{a}+c^4 (-x) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcTanh[a*x])*(c - c/(a*x))^4,x]

[Out]

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

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Maple [A]  time = 0.034, size = 39, normalized size = 1. \begin{align*} -{\frac{{c}^{4}}{3\,{a}^{4}{x}^{3}}}+{\frac{{c}^{4}}{{x}^{2}{a}^{3}}}-{c}^{4}x+2\,{\frac{{c}^{4}\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)^4,x)

[Out]

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

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Maxima [A]  time = 0.949206, size = 51, normalized size = 1.27 \begin{align*} -c^{4} x + \frac{2 \, c^{4} \log \left (x\right )}{a} + \frac{3 \, a c^{4} x - c^{4}}{3 \, a^{4} x^{3}} \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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.06078, size = 99, normalized size = 2.48 \begin{align*} -\frac{3 \, a^{4} c^{4} x^{4} - 6 \, a^{3} c^{4} x^{3} \log \left (x\right ) - 3 \, a c^{4} x + c^{4}}{3 \, a^{4} x^{3}} \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)^4,x, algorithm="fricas")

[Out]

-1/3*(3*a^4*c^4*x^4 - 6*a^3*c^4*x^3*log(x) - 3*a*c^4*x + c^4)/(a^4*x^3)

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Sympy [A]  time = 0.362913, size = 39, normalized size = 0.98 \begin{align*} \frac{- a^{4} c^{4} x + 2 a^{3} c^{4} \log{\left (x \right )} + \frac{3 a c^{4} x - c^{4}}{3 x^{3}}}{a^{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)**4,x)

[Out]

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

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Giac [A]  time = 1.20082, size = 53, normalized size = 1.32 \begin{align*} -c^{4} x + \frac{2 \, c^{4} \log \left ({\left | x \right |}\right )}{a} + \frac{3 \, a c^{4} x - c^{4}}{3 \, a^{4} x^{3}} \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)^4,x, algorithm="giac")

[Out]

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