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

Optimal. Leaf size=90 \[ \frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}+\frac {2 c^4 \log (x)}{a}+c^4 x \]

[Out]

1/7*c^4/a^8/x^7+1/3*c^4/a^7/x^6-2/5*c^4/a^6/x^5-3/2*c^4/a^5/x^4+3*c^4/a^3/x^2+2*c^4/a^2/x+c^4*x+2*c^4*ln(x)/a

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Rubi [A]  time = 0.16, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6167, 6157, 6150, 88} \[ \frac {3 c^4}{a^3 x^2}-\frac {3 c^4}{2 a^5 x^4}-\frac {2 c^4}{5 a^6 x^5}+\frac {c^4}{3 a^7 x^6}+\frac {c^4}{7 a^8 x^7}+\frac {2 c^4}{a^2 x}+\frac {2 c^4 \log (x)}{a}+c^4 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^4/(7*a^8*x^7) + c^4/(3*a^7*x^6) - (2*c^4)/(5*a^6*x^5) - (3*c^4)/(2*a^5*x^4) + (3*c^4)/(a^3*x^2) + (2*c^4)/(a
^2*x) + c^4*x + (2*c^4*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 6150

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

Rule 6157

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

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]

Rubi steps

\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx\\ &=-\frac {c^4 \int \frac {e^{2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^4}{x^8} \, dx}{a^8}\\ &=-\frac {c^4 \int \frac {(1-a x)^3 (1+a x)^5}{x^8} \, dx}{a^8}\\ &=-\frac {c^4 \int \left (-a^8+\frac {1}{x^8}+\frac {2 a}{x^7}-\frac {2 a^2}{x^6}-\frac {6 a^3}{x^5}+\frac {6 a^5}{x^3}+\frac {2 a^6}{x^2}-\frac {2 a^7}{x}\right ) \, dx}{a^8}\\ &=\frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}+c^4 x+\frac {2 c^4 \log (x)}{a}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 90, normalized size = 1.00 \[ \frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}+\frac {2 c^4 \log (x)}{a}+c^4 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^4/(7*a^8*x^7) + c^4/(3*a^7*x^6) - (2*c^4)/(5*a^6*x^5) - (3*c^4)/(2*a^5*x^4) + (3*c^4)/(a^3*x^2) + (2*c^4)/(a
^2*x) + c^4*x + (2*c^4*Log[x])/a

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fricas [A]  time = 0.41, size = 89, normalized size = 0.99 \[ \frac {210 \, a^{8} c^{4} x^{8} + 420 \, a^{7} c^{4} x^{7} \log \relax (x) + 420 \, a^{6} c^{4} x^{6} + 630 \, a^{5} c^{4} x^{5} - 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/210*(210*a^8*c^4*x^8 + 420*a^7*c^4*x^7*log(x) + 420*a^6*c^4*x^6 + 630*a^5*c^4*x^5 - 315*a^3*c^4*x^3 - 84*a^2
*c^4*x^2 + 70*a*c^4*x + 30*c^4)/(a^8*x^7)

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giac [A]  time = 0.12, size = 82, normalized size = 0.91 \[ c^{4} x + \frac {2 \, c^{4} \log \left ({\left | x \right |}\right )}{a} + \frac {420 \, a^{6} c^{4} x^{6} + 630 \, a^{5} c^{4} x^{5} - 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*x + 2*c^4*log(abs(x))/a + 1/210*(420*a^6*c^4*x^6 + 630*a^5*c^4*x^5 - 315*a^3*c^4*x^3 - 84*a^2*c^4*x^2 + 70
*a*c^4*x + 30*c^4)/(a^8*x^7)

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maple [A]  time = 0.04, size = 83, normalized size = 0.92 \[ \frac {c^{4}}{7 a^{8} x^{7}}+\frac {c^{4}}{3 a^{7} x^{6}}-\frac {2 c^{4}}{5 a^{6} x^{5}}-\frac {3 c^{4}}{2 a^{5} x^{4}}+\frac {3 c^{4}}{x^{2} a^{3}}+\frac {2 c^{4}}{a^{2} x}+c^{4} x +\frac {2 c^{4} \ln \relax (x )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c^4/a^8/x^7+1/3*c^4/a^7/x^6-2/5*c^4/a^6/x^5-3/2*c^4/a^5/x^4+3*c^4/x^2/a^3+2*c^4/a^2/x+c^4*x+2*c^4*ln(x)/a

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maxima [A]  time = 0.32, size = 81, normalized size = 0.90 \[ c^{4} x + \frac {2 \, c^{4} \log \relax (x)}{a} + \frac {420 \, a^{6} c^{4} x^{6} + 630 \, a^{5} c^{4} x^{5} - 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*x + 2*c^4*log(x)/a + 1/210*(420*a^6*c^4*x^6 + 630*a^5*c^4*x^5 - 315*a^3*c^4*x^3 - 84*a^2*c^4*x^2 + 70*a*c^
4*x + 30*c^4)/(a^8*x^7)

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mupad [B]  time = 1.24, size = 65, normalized size = 0.72 \[ \frac {c^4\,\left (\frac {a\,x}{3}-\frac {2\,a^2\,x^2}{5}-\frac {3\,a^3\,x^3}{2}+3\,a^5\,x^5+2\,a^6\,x^6+a^8\,x^8+2\,a^7\,x^7\,\ln \relax (x)+\frac {1}{7}\right )}{a^8\,x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^4*((a*x)/3 - (2*a^2*x^2)/5 - (3*a^3*x^3)/2 + 3*a^5*x^5 + 2*a^6*x^6 + a^8*x^8 + 2*a^7*x^7*log(x) + 1/7))/(a^
8*x^7)

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sympy [A]  time = 0.44, size = 88, normalized size = 0.98 \[ \frac {a^{8} c^{4} x + 2 a^{7} c^{4} \log {\relax (x )} + \frac {420 a^{6} c^{4} x^{6} + 630 a^{5} c^{4} x^{5} - 315 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} + 70 a c^{4} x + 30 c^{4}}{210 x^{7}}}{a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a**8*c**4*x + 2*a**7*c**4*log(x) + (420*a**6*c**4*x**6 + 630*a**5*c**4*x**5 - 315*a**3*c**4*x**3 - 84*a**2*c*
*4*x**2 + 70*a*c**4*x + 30*c**4)/(210*x**7))/a**8

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