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

Optimal. Leaf size=91 \[ \frac{3 c^4}{a^3 x^2}-\frac{3 c^4}{2 a^5 x^4}+\frac{2 c^4}{5 a^6 x^5}+\frac{c^4}{3 a^7 x^6}-\frac{c^4}{7 a^8 x^7}-\frac{2 c^4}{a^2 x}+\frac{2 c^4 \log (x)}{a}+c^4 (-x) \]

[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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Rubi [A]  time = 0.139025, antiderivative size = 91, 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 = {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[(c - c/(a^2*x^2))^4/E^(2*ArcTanh[a*x]),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 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 \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)^5 (1+a x)^3}{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*}

Mathematica [A]  time = 0.0264886, size = 91, normalized size = 1. \[ \frac{3 c^4}{a^3 x^2}-\frac{3 c^4}{2 a^5 x^4}+\frac{2 c^4}{5 a^6 x^5}+\frac{c^4}{3 a^7 x^6}-\frac{c^4}{7 a^8 x^7}-\frac{2 c^4}{a^2 x}+\frac{2 c^4 \log (x)}{a}+c^4 (-x) \]

Antiderivative was successfully verified.

[In]

Integrate[(c - c/(a^2*x^2))^4/E^(2*ArcTanh[a*x]),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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Maple [A]  time = 0.042, size = 84, normalized size = 0.9 \begin{align*} -{\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}}}+3\,{\frac{{c}^{4}}{{x}^{2}{a}^{3}}}-2\,{\frac{{c}^{4}}{{a}^{2}x}}-{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((c-c/a^2/x^2)^4/(a*x+1)^2*(-a^2*x^2+1),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 = 0.961714, size = 111, normalized size = 1.22 \begin{align*} -c^{4} x + \frac{2 \, c^{4} \log \left (x\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^4/(a*x+1)^2*(-a^2*x^2+1),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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Fricas [A]  time = 1.92597, size = 208, normalized size = 2.29 \begin{align*} -\frac{210 \, a^{8} c^{4} x^{8} - 420 \, a^{7} c^{4} x^{7} \log \left (x\right ) + 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^4/(a*x+1)^2*(-a^2*x^2+1),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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Sympy [A]  time = 3.56531, size = 88, normalized size = 0.97 \begin{align*} \frac{- a^{8} c^{4} x + 2 a^{7} c^{4} \log{\left (x \right )} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a**2/x**2)**4/(a*x+1)**2*(-a**2*x**2+1),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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Giac [A]  time = 1.17059, size = 216, normalized size = 2.37 \begin{align*} -\frac{2 \, c^{4} \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{2 \, c^{4} \log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right )}{a} + \frac{{\left (210 \, c^{4} - \frac{719 \, c^{4}}{a x + 1} - \frac{427 \, c^{4}}{{\left (a x + 1\right )}^{2}} + \frac{5271 \, c^{4}}{{\left (a x + 1\right )}^{3}} - \frac{9485 \, c^{4}}{{\left (a x + 1\right )}^{4}} + \frac{7490 \, c^{4}}{{\left (a x + 1\right )}^{5}} - \frac{2730 \, c^{4}}{{\left (a x + 1\right )}^{6}} + \frac{420 \, c^{4}}{{\left (a x + 1\right )}^{7}}\right )}{\left (a x + 1\right )}}{210 \, a{\left (\frac{1}{a x + 1} - 1\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c^4*log(abs(a*x + 1)/((a*x + 1)^2*abs(a)))/a + 2*c^4*log(abs(-1/(a*x + 1) + 1))/a + 1/210*(210*c^4 - 719*c^
4/(a*x + 1) - 427*c^4/(a*x + 1)^2 + 5271*c^4/(a*x + 1)^3 - 9485*c^4/(a*x + 1)^4 + 7490*c^4/(a*x + 1)^5 - 2730*
c^4/(a*x + 1)^6 + 420*c^4/(a*x + 1)^7)*(a*x + 1)/(a*(1/(a*x + 1) - 1)^7)