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

Optimal. Leaf size=127 \[ \frac{4 c^5}{a^3 x^2}-\frac{2 c^5}{3 a^4 x^3}-\frac{3 c^5}{a^5 x^4}-\frac{2 c^5}{5 a^6 x^5}+\frac{4 c^5}{3 a^7 x^6}+\frac{3 c^5}{7 a^8 x^7}-\frac{c^5}{4 a^9 x^8}-\frac{c^5}{9 a^{10} x^9}+\frac{3 c^5}{a^2 x}+\frac{2 c^5 \log (x)}{a}+c^5 x \]

[Out]

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

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Rubi [A]  time = 0.175185, antiderivative size = 127, 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} \[ \frac{4 c^5}{a^3 x^2}-\frac{2 c^5}{3 a^4 x^3}-\frac{3 c^5}{a^5 x^4}-\frac{2 c^5}{5 a^6 x^5}+\frac{4 c^5}{3 a^7 x^6}+\frac{3 c^5}{7 a^8 x^7}-\frac{c^5}{4 a^9 x^8}-\frac{c^5}{9 a^{10} x^9}+\frac{3 c^5}{a^2 x}+\frac{2 c^5 \log (x)}{a}+c^5 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c^5/(9*a^10*x^9) - c^5/(4*a^9*x^8) + (3*c^5)/(7*a^8*x^7) + (4*c^5)/(3*a^7*x^6) - (2*c^5)/(5*a^6*x^5) - (3*c^5
)/(a^5*x^4) - (2*c^5)/(3*a^4*x^3) + (4*c^5)/(a^3*x^2) + (3*c^5)/(a^2*x) + c^5*x + (2*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^{2 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^5 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^5 \, dx\\ &=\frac{c^5 \int \frac{e^{2 \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)^4 (1+a x)^6}{x^{10}} \, dx}{a^{10}}\\ &=\frac{c^5 \int \left (a^{10}+\frac{1}{x^{10}}+\frac{2 a}{x^9}-\frac{3 a^2}{x^8}-\frac{8 a^3}{x^7}+\frac{2 a^4}{x^6}+\frac{12 a^5}{x^5}+\frac{2 a^6}{x^4}-\frac{8 a^7}{x^3}-\frac{3 a^8}{x^2}+\frac{2 a^9}{x}\right ) \, dx}{a^{10}}\\ &=-\frac{c^5}{9 a^{10} x^9}-\frac{c^5}{4 a^9 x^8}+\frac{3 c^5}{7 a^8 x^7}+\frac{4 c^5}{3 a^7 x^6}-\frac{2 c^5}{5 a^6 x^5}-\frac{3 c^5}{a^5 x^4}-\frac{2 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{2 c^5 \log (x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.0429798, size = 127, normalized size = 1. \[ \frac{4 c^5}{a^3 x^2}-\frac{2 c^5}{3 a^4 x^3}-\frac{3 c^5}{a^5 x^4}-\frac{2 c^5}{5 a^6 x^5}+\frac{4 c^5}{3 a^7 x^6}+\frac{3 c^5}{7 a^8 x^7}-\frac{c^5}{4 a^9 x^8}-\frac{c^5}{9 a^{10} x^9}+\frac{3 c^5}{a^2 x}+\frac{2 c^5 \log (x)}{a}+c^5 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.055, size = 116, normalized size = 0.9 \begin{align*} -{\frac{{c}^{5}}{9\,{a}^{10}{x}^{9}}}-{\frac{{c}^{5}}{4\,{a}^{9}{x}^{8}}}+{\frac{3\,{c}^{5}}{7\,{a}^{8}{x}^{7}}}+{\frac{4\,{c}^{5}}{3\,{a}^{7}{x}^{6}}}-{\frac{2\,{c}^{5}}{5\,{a}^{6}{x}^{5}}}-3\,{\frac{{c}^{5}}{{a}^{5}{x}^{4}}}-{\frac{2\,{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+2\,{\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)/(a*x-1)*(c-c/a^2/x^2)^5,x)

[Out]

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

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Maxima [A]  time = 1.0479, size = 154, normalized size = 1.21 \begin{align*} c^{5} x + \frac{2 \, c^{5} \log \left (x\right )}{a} + \frac{3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \end{align*}

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)^5,x, algorithm="maxima")

[Out]

c^5*x + 2*c^5*log(x)/a + 1/1260*(3780*a^8*c^5*x^8 + 5040*a^7*c^5*x^7 - 840*a^6*c^5*x^6 - 3780*a^5*c^5*x^5 - 50
4*a^4*c^5*x^4 + 1680*a^3*c^5*x^3 + 540*a^2*c^5*x^2 - 315*a*c^5*x - 140*c^5)/(a^10*x^9)

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Fricas [A]  time = 1.20419, size = 297, normalized size = 2.34 \begin{align*} \frac{1260 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \left (x\right ) + 3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \end{align*}

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)^5,x, algorithm="fricas")

[Out]

1/1260*(1260*a^10*c^5*x^10 + 2520*a^9*c^5*x^9*log(x) + 3780*a^8*c^5*x^8 + 5040*a^7*c^5*x^7 - 840*a^6*c^5*x^6 -
 3780*a^5*c^5*x^5 - 504*a^4*c^5*x^4 + 1680*a^3*c^5*x^3 + 540*a^2*c^5*x^2 - 315*a*c^5*x - 140*c^5)/(a^10*x^9)

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Sympy [A]  time = 0.900577, size = 124, normalized size = 0.98 \begin{align*} \frac{a^{10} c^{5} x + 2 a^{9} c^{5} \log{\left (x \right )} + \frac{3780 a^{8} c^{5} x^{8} + 5040 a^{7} c^{5} x^{7} - 840 a^{6} c^{5} x^{6} - 3780 a^{5} c^{5} x^{5} - 504 a^{4} c^{5} x^{4} + 1680 a^{3} c^{5} x^{3} + 540 a^{2} c^{5} x^{2} - 315 a c^{5} x - 140 c^{5}}{1260 x^{9}}}{a^{10}} \end{align*}

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)**5,x)

[Out]

(a**10*c**5*x + 2*a**9*c**5*log(x) + (3780*a**8*c**5*x**8 + 5040*a**7*c**5*x**7 - 840*a**6*c**5*x**6 - 3780*a*
*5*c**5*x**5 - 504*a**4*c**5*x**4 + 1680*a**3*c**5*x**3 + 540*a**2*c**5*x**2 - 315*a*c**5*x - 140*c**5)/(1260*
x**9))/a**10

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Giac [A]  time = 1.1442, size = 155, normalized size = 1.22 \begin{align*} c^{5} x + \frac{2 \, c^{5} \log \left ({\left | x \right |}\right )}{a} + \frac{3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \end{align*}

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)^5,x, algorithm="giac")

[Out]

c^5*x + 2*c^5*log(abs(x))/a + 1/1260*(3780*a^8*c^5*x^8 + 5040*a^7*c^5*x^7 - 840*a^6*c^5*x^6 - 3780*a^5*c^5*x^5
 - 504*a^4*c^5*x^4 + 1680*a^3*c^5*x^3 + 540*a^2*c^5*x^2 - 315*a*c^5*x - 140*c^5)/(a^10*x^9)