3.797 \(\int e^{4 \coth ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^5 \, dx\)

Optimal. Leaf size=116 \[ \frac{4 c^5}{a^3 x^2}+\frac{14 c^5}{3 a^4 x^3}-\frac{14 c^5}{5 a^6 x^5}-\frac{4 c^5}{3 a^7 x^6}+\frac{3 c^5}{7 a^8 x^7}+\frac{c^5}{2 a^9 x^8}+\frac{c^5}{9 a^{10} x^9}-\frac{3 c^5}{a^2 x}+\frac{4 c^5 \log (x)}{a}+c^5 x \]

[Out]

c^5/(9*a^10*x^9) + c^5/(2*a^9*x^8) + (3*c^5)/(7*a^8*x^7) - (4*c^5)/(3*a^7*x^6) - (14*c^5)/(5*a^6*x^5) + (14*c^
5)/(3*a^4*x^3) + (4*c^5)/(a^3*x^2) - (3*c^5)/(a^2*x) + c^5*x + (4*c^5*Log[x])/a

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Rubi [A]  time = 0.170696, antiderivative size = 116, normalized size of antiderivative = 1., 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} \[ \frac{4 c^5}{a^3 x^2}+\frac{14 c^5}{3 a^4 x^3}-\frac{14 c^5}{5 a^6 x^5}-\frac{4 c^5}{3 a^7 x^6}+\frac{3 c^5}{7 a^8 x^7}+\frac{c^5}{2 a^9 x^8}+\frac{c^5}{9 a^{10} x^9}-\frac{3 c^5}{a^2 x}+\frac{4 c^5 \log (x)}{a}+c^5 x \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcCoth[a*x])*(c - c/(a^2*x^2))^5,x]

[Out]

c^5/(9*a^10*x^9) + c^5/(2*a^9*x^8) + (3*c^5)/(7*a^8*x^7) - (4*c^5)/(3*a^7*x^6) - (14*c^5)/(5*a^6*x^5) + (14*c^
5)/(3*a^4*x^3) + (4*c^5)/(a^3*x^2) - (3*c^5)/(a^2*x) + c^5*x + (4*c^5*Log[x])/a

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 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 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 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]))

Rubi steps

\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^5 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^5 \, dx\\ &=-\frac{c^5 \int \frac{e^{4 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^5}{x^{10}} \, dx}{a^{10}}\\ &=-\frac{c^5 \int \frac{(1-a x)^3 (1+a x)^7}{x^{10}} \, dx}{a^{10}}\\ &=-\frac{c^5 \int \left (-a^{10}+\frac{1}{x^{10}}+\frac{4 a}{x^9}+\frac{3 a^2}{x^8}-\frac{8 a^3}{x^7}-\frac{14 a^4}{x^6}+\frac{14 a^6}{x^4}+\frac{8 a^7}{x^3}-\frac{3 a^8}{x^2}-\frac{4 a^9}{x}\right ) \, dx}{a^{10}}\\ &=\frac{c^5}{9 a^{10} x^9}+\frac{c^5}{2 a^9 x^8}+\frac{3 c^5}{7 a^8 x^7}-\frac{4 c^5}{3 a^7 x^6}-\frac{14 c^5}{5 a^6 x^5}+\frac{14 c^5}{3 a^4 x^3}+\frac{4 c^5}{a^3 x^2}-\frac{3 c^5}{a^2 x}+c^5 x+\frac{4 c^5 \log (x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.041808, size = 116, normalized size = 1. \[ \frac{4 c^5}{a^3 x^2}+\frac{14 c^5}{3 a^4 x^3}-\frac{14 c^5}{5 a^6 x^5}-\frac{4 c^5}{3 a^7 x^6}+\frac{3 c^5}{7 a^8 x^7}+\frac{c^5}{2 a^9 x^8}+\frac{c^5}{9 a^{10} x^9}-\frac{3 c^5}{a^2 x}+\frac{4 c^5 \log (x)}{a}+c^5 x \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcCoth[a*x])*(c - c/(a^2*x^2))^5,x]

[Out]

c^5/(9*a^10*x^9) + c^5/(2*a^9*x^8) + (3*c^5)/(7*a^8*x^7) - (4*c^5)/(3*a^7*x^6) - (14*c^5)/(5*a^6*x^5) + (14*c^
5)/(3*a^4*x^3) + (4*c^5)/(a^3*x^2) - (3*c^5)/(a^2*x) + c^5*x + (4*c^5*Log[x])/a

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Maple [A]  time = 0.046, size = 105, normalized size = 0.9 \begin{align*}{\frac{{c}^{5}}{9\,{a}^{10}{x}^{9}}}+{\frac{{c}^{5}}{2\,{a}^{9}{x}^{8}}}+{\frac{3\,{c}^{5}}{7\,{a}^{8}{x}^{7}}}-{\frac{4\,{c}^{5}}{3\,{a}^{7}{x}^{6}}}-{\frac{14\,{c}^{5}}{5\,{a}^{6}{x}^{5}}}+{\frac{14\,{c}^{5}}{3\,{a}^{4}{x}^{3}}}+4\,{\frac{{c}^{5}}{{x}^{2}{a}^{3}}}-3\,{\frac{{c}^{5}}{{a}^{2}x}}+{c}^{5}x+4\,{\frac{{c}^{5}\ln \left ( x \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^5/a^10/x^9+1/2*c^5/a^9/x^8+3/7*c^5/a^8/x^7-4/3*c^5/a^7/x^6-14/5*c^5/a^6/x^5+14/3*c^5/a^4/x^3+4*c^5/x^2/a
^3-3*c^5/a^2/x+c^5*x+4*c^5*ln(x)/a

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Maxima [A]  time = 1.00432, size = 139, normalized size = 1.2 \begin{align*} c^{5} x + \frac{4 \, c^{5} \log \left (x\right )}{a} - \frac{1890 \, a^{8} c^{5} x^{8} - 2520 \, a^{7} c^{5} x^{7} - 2940 \, a^{6} c^{5} x^{6} + 1764 \, a^{4} c^{5} x^{4} + 840 \, a^{3} c^{5} x^{3} - 270 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 70 \, c^{5}}{630 \, a^{10} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^5*x + 4*c^5*log(x)/a - 1/630*(1890*a^8*c^5*x^8 - 2520*a^7*c^5*x^7 - 2940*a^6*c^5*x^6 + 1764*a^4*c^5*x^4 + 84
0*a^3*c^5*x^3 - 270*a^2*c^5*x^2 - 315*a*c^5*x - 70*c^5)/(a^10*x^9)

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Fricas [A]  time = 1.54395, size = 269, normalized size = 2.32 \begin{align*} \frac{630 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \left (x\right ) - 1890 \, a^{8} c^{5} x^{8} + 2520 \, a^{7} c^{5} x^{7} + 2940 \, a^{6} c^{5} x^{6} - 1764 \, a^{4} c^{5} x^{4} - 840 \, a^{3} c^{5} x^{3} + 270 \, a^{2} c^{5} x^{2} + 315 \, a c^{5} x + 70 \, c^{5}}{630 \, a^{10} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/630*(630*a^10*c^5*x^10 + 2520*a^9*c^5*x^9*log(x) - 1890*a^8*c^5*x^8 + 2520*a^7*c^5*x^7 + 2940*a^6*c^5*x^6 -
1764*a^4*c^5*x^4 - 840*a^3*c^5*x^3 + 270*a^2*c^5*x^2 + 315*a*c^5*x + 70*c^5)/(a^10*x^9)

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Sympy [A]  time = 0.900921, size = 112, normalized size = 0.97 \begin{align*} \frac{a^{10} c^{5} x + 4 a^{9} c^{5} \log{\left (x \right )} - \frac{1890 a^{8} c^{5} x^{8} - 2520 a^{7} c^{5} x^{7} - 2940 a^{6} c^{5} x^{6} + 1764 a^{4} c^{5} x^{4} + 840 a^{3} c^{5} x^{3} - 270 a^{2} c^{5} x^{2} - 315 a c^{5} x - 70 c^{5}}{630 x^{9}}}{a^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a**10*c**5*x + 4*a**9*c**5*log(x) - (1890*a**8*c**5*x**8 - 2520*a**7*c**5*x**7 - 2940*a**6*c**5*x**6 + 1764*a
**4*c**5*x**4 + 840*a**3*c**5*x**3 - 270*a**2*c**5*x**2 - 315*a*c**5*x - 70*c**5)/(630*x**9))/a**10

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Giac [A]  time = 1.12725, size = 248, normalized size = 2.14 \begin{align*} -\frac{4 \, c^{5} \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{4 \, c^{5} \log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right )}{a} + \frac{{\left (630 \, c^{5} + \frac{4049 \, c^{5}}{a x - 1} + \frac{6201 \, c^{5}}{{\left (a x - 1\right )}^{2}} - \frac{18036 \, c^{5}}{{\left (a x - 1\right )}^{3}} - \frac{89124 \, c^{5}}{{\left (a x - 1\right )}^{4}} - \frac{160146 \, c^{5}}{{\left (a x - 1\right )}^{5}} - \frac{153090 \, c^{5}}{{\left (a x - 1\right )}^{6}} - \frac{80220 \, c^{5}}{{\left (a x - 1\right )}^{7}} - \frac{21420 \, c^{5}}{{\left (a x - 1\right )}^{8}} - \frac{2520 \, c^{5}}{{\left (a x - 1\right )}^{9}}\right )}{\left (a x - 1\right )}}{630 \, a{\left (\frac{1}{a x - 1} + 1\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*c^5*log(abs(a*x - 1)/((a*x - 1)^2*abs(a)))/a + 4*c^5*log(abs(-1/(a*x - 1) - 1))/a + 1/630*(630*c^5 + 4049*c
^5/(a*x - 1) + 6201*c^5/(a*x - 1)^2 - 18036*c^5/(a*x - 1)^3 - 89124*c^5/(a*x - 1)^4 - 160146*c^5/(a*x - 1)^5 -
 153090*c^5/(a*x - 1)^6 - 80220*c^5/(a*x - 1)^7 - 21420*c^5/(a*x - 1)^8 - 2520*c^5/(a*x - 1)^9)*(a*x - 1)/(a*(
1/(a*x - 1) + 1)^9)