3.475 \(\int e^{4 \tanh ^{-1}(a x)} (c-\frac {c}{a x})^5 \, dx\)

Optimal. Leaf size=64 \[ \frac {c^5}{4 a^5 x^4}-\frac {c^5}{3 a^4 x^3}-\frac {c^5}{a^3 x^2}+\frac {2 c^5}{a^2 x}-\frac {c^5 \log (x)}{a}+c^5 x \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 64, normalized size of antiderivative = 1.00, 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, 88} \[ -\frac {c^5}{a^3 x^2}-\frac {c^5}{3 a^4 x^3}+\frac {c^5}{4 a^5 x^4}+\frac {2 c^5}{a^2 x}-\frac {c^5 \log (x)}{a}+c^5 x \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcTanh[a*x])*(c - c/(a*x))^5,x]

[Out]

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

Rule 88

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

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.28, size = 66, normalized size = 1.03 \[ \frac {c^5}{4 a^5 x^4}-\frac {c^5}{3 a^4 x^3}-\frac {c^5}{a^3 x^2}+\frac {2 c^5}{a^2 x}-\frac {c^5 \log (a x)}{a}+c^5 x \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(4*ArcTanh[a*x])*(c - c/(a*x))^5,x]

[Out]

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

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fricas [A]  time = 0.44, size = 67, normalized size = 1.05 \[ \frac {12 \, a^{5} c^{5} x^{5} - 12 \, a^{4} c^{5} x^{4} \log \relax (x) + 24 \, a^{3} c^{5} x^{3} - 12 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + 3 \, c^{5}}{12 \, a^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^5,x, algorithm="fricas")

[Out]

1/12*(12*a^5*c^5*x^5 - 12*a^4*c^5*x^4*log(x) + 24*a^3*c^5*x^3 - 12*a^2*c^5*x^2 - 4*a*c^5*x + 3*c^5)/(a^5*x^4)

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giac [A]  time = 0.20, size = 60, normalized size = 0.94 \[ c^{5} x - \frac {c^{5} \log \left ({\left | x \right |}\right )}{a} + \frac {24 \, a^{3} c^{5} x^{3} - 12 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + 3 \, c^{5}}{12 \, a^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^5,x, algorithm="giac")

[Out]

c^5*x - c^5*log(abs(x))/a + 1/12*(24*a^3*c^5*x^3 - 12*a^2*c^5*x^2 - 4*a*c^5*x + 3*c^5)/(a^5*x^4)

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maple [A]  time = 0.03, size = 61, normalized size = 0.95 \[ \frac {c^{5}}{4 a^{5} x^{4}}-\frac {c^{5}}{3 a^{4} x^{3}}-\frac {c^{5}}{x^{2} a^{3}}+\frac {2 c^{5}}{a^{2} x}+c^{5} x -\frac {c^{5} \ln \relax (x )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^5,x)

[Out]

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

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maxima [A]  time = 0.35, size = 59, normalized size = 0.92 \[ c^{5} x - \frac {c^{5} \log \relax (x)}{a} + \frac {24 \, a^{3} c^{5} x^{3} - 12 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + 3 \, c^{5}}{12 \, a^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^5,x, algorithm="maxima")

[Out]

c^5*x - c^5*log(x)/a + 1/12*(24*a^3*c^5*x^3 - 12*a^2*c^5*x^2 - 4*a*c^5*x + 3*c^5)/(a^5*x^4)

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mupad [B]  time = 0.07, size = 51, normalized size = 0.80 \[ -\frac {c^5\,\left (4\,a\,x+12\,a^2\,x^2-24\,a^3\,x^3-12\,a^5\,x^5+12\,a^4\,x^4\,\ln \relax (x)-3\right )}{12\,a^5\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - c/(a*x))^5*(a*x + 1)^4)/(a^2*x^2 - 1)^2,x)

[Out]

-(c^5*(4*a*x + 12*a^2*x^2 - 24*a^3*x^3 - 12*a^5*x^5 + 12*a^4*x^4*log(x) - 3))/(12*a^5*x^4)

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sympy [A]  time = 0.28, size = 63, normalized size = 0.98 \[ \frac {a^{5} c^{5} x - a^{4} c^{5} \log {\relax (x )} + \frac {24 a^{3} c^{5} x^{3} - 12 a^{2} c^{5} x^{2} - 4 a c^{5} x + 3 c^{5}}{12 x^{4}}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a/x)**5,x)

[Out]

(a**5*c**5*x - a**4*c**5*log(x) + (24*a**3*c**5*x**3 - 12*a**2*c**5*x**2 - 4*a*c**5*x + 3*c**5)/(12*x**4))/a**
5

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