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

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

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

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Rubi [A]  time = 0.14, antiderivative size = 116, 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 = {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*ArcTanh[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 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]

Rubi steps

\begin {align*} \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*}

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Mathematica [A]  time = 0.04, size = 116, normalized size = 1.00 \[ \frac {c^5}{9 a^{10} x^9}+\frac {c^5}{2 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}-\frac {14 c^5}{5 a^6 x^5}+\frac {14 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}+\frac {4 c^5 \log (x)}{a}+c^5 x \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcTanh[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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fricas [A]  time = 0.56, size = 111, normalized size = 0.96 \[ \frac {630 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \relax (x) - 1890 \, a^{8} c^{5} x^{8} + 2520 \, a^{7} c^{5} x^{7} + 2940 \, a^{6} c^{5} x^{6} - 1764 \, a^{4} c^{5} x^{4} - 840 \, a^{3} c^{5} x^{3} + 270 \, a^{2} c^{5} x^{2} + 315 \, a c^{5} x + 70 \, c^{5}}{630 \, a^{10} x^{9}} \]

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^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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giac [A]  time = 0.20, size = 104, normalized size = 0.90 \[ c^{5} x + \frac {4 \, c^{5} \log \left ({\left | x \right |}\right )}{a} - \frac {1890 \, a^{8} c^{5} x^{8} - 2520 \, a^{7} c^{5} x^{7} - 2940 \, a^{6} c^{5} x^{6} + 1764 \, a^{4} c^{5} x^{4} + 840 \, a^{3} c^{5} x^{3} - 270 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 70 \, c^{5}}{630 \, a^{10} x^{9}} \]

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^2/x^2)^5,x, algorithm="giac")

[Out]

c^5*x + 4*c^5*log(abs(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
+ 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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maple [A]  time = 0.04, size = 105, normalized size = 0.91 \[ \frac {c^{5}}{9 a^{10} x^{9}}+\frac {c^{5}}{2 a^{9} x^{8}}+\frac {3 c^{5}}{7 a^{8} x^{7}}-\frac {4 c^{5}}{3 a^{7} x^{6}}-\frac {14 c^{5}}{5 a^{6} x^{5}}+\frac {14 c^{5}}{3 a^{4} x^{3}}+\frac {4 c^{5}}{x^{2} a^{3}}-\frac {3 c^{5}}{a^{2} x}+c^{5} x +\frac {4 c^{5} \ln \relax (x )}{a} \]

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^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 = 0.31, size = 103, normalized size = 0.89 \[ c^{5} x + \frac {4 \, c^{5} \log \relax (x)}{a} - \frac {1890 \, a^{8} c^{5} x^{8} - 2520 \, a^{7} c^{5} x^{7} - 2940 \, a^{6} c^{5} x^{6} + 1764 \, a^{4} c^{5} x^{4} + 840 \, a^{3} c^{5} x^{3} - 270 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 70 \, c^{5}}{630 \, a^{10} x^{9}} \]

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^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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mupad [B]  time = 0.09, size = 81, normalized size = 0.70 \[ \frac {c^5\,\left (\frac {a\,x}{2}+\frac {3\,a^2\,x^2}{7}-\frac {4\,a^3\,x^3}{3}-\frac {14\,a^4\,x^4}{5}+\frac {14\,a^6\,x^6}{3}+4\,a^7\,x^7-3\,a^8\,x^8+a^{10}\,x^{10}+4\,a^9\,x^9\,\ln \relax (x)+\frac {1}{9}\right )}{a^{10}\,x^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.66, size = 112, normalized size = 0.97 \[ \frac {a^{10} c^{5} x + 4 a^{9} c^{5} \log {\relax (x )} + \frac {- 1890 a^{8} c^{5} x^{8} + 2520 a^{7} c^{5} x^{7} + 2940 a^{6} c^{5} x^{6} - 1764 a^{4} c^{5} x^{4} - 840 a^{3} c^{5} x^{3} + 270 a^{2} c^{5} x^{2} + 315 a c^{5} x + 70 c^{5}}{630 x^{9}}}{a^{10}} \]

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**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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