3.8.97 \(\int e^{4 \coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^5 \, dx\) [797]

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

[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 = 0.11, antiderivative size = 116, 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 = {6302, 6292, 6285, 90} \begin {gather*} \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 \end {gather*}

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 90

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 6285

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 6292

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

Rule 6302

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^{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*}

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Mathematica [A]
time = 0.03, size = 116, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.42, size = 80, normalized size = 0.69

method	result	size
default	\(\frac {c^{5} \left (a^{10} x +\frac {a}{2 x^{8}}+\frac {3 a^{2}}{7 x^{7}}+\frac {1}{9 x^{9}}-\frac {3 a^{8}}{x}+\frac {4 a^{7}}{x^{2}}+4 a^{9} \ln \left (x \right )+\frac {14 a^{6}}{3 x^{3}}-\frac {4 a^{3}}{3 x^{6}}-\frac {14 a^{4}}{5 x^{5}}\right )}{a^{10}}\)	\(80\)
risch	\(c^{5} x +\frac {-3 a^{8} c^{5} x^{8}+4 a^{7} c^{5} x^{7}+\frac {14}{3} a^{6} c^{5} x^{6}-\frac {14}{5} a^{4} c^{5} x^{4}-\frac {4}{3} a^{3} c^{5} x^{3}+\frac {3}{7} a^{2} c^{5} x^{2}+\frac {1}{2} a \,c^{5} x +\frac {1}{9} c^{5}}{a^{10} x^{9}}+\frac {4 c^{5} \ln \left (x \right )}{a}\)	\(103\)
norman	\(\frac {-4 a^{9} c^{5} x^{10}+a^{10} c^{5} x^{11}-\frac {c^{5}}{9 a}-\frac {7 c^{5} x}{18}+\frac {a \,c^{5} x^{2}}{14}-\frac {14 a^{4} c^{5} x^{5}}{5}-\frac {14 a^{5} c^{5} x^{6}}{3}+\frac {2 a^{6} c^{5} x^{7}}{3}+7 a^{7} c^{5} x^{8}+\frac {37 c^{5} a^{2} x^{3}}{21}+\frac {22 c^{5} a^{3} x^{4}}{15}}{\left (a x -1\right ) a^{9} x^{9}}+\frac {4 c^{5} \ln \left (x \right )}{a}\)	\(137\)
meijerg	\(-\frac {c^{5} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}-\frac {4 c^{5} x}{-a x +1}-\frac {5 c^{5} \left (-\frac {3 a x}{-3 a x +3}+2 \ln \left (-a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (-a \right )+\frac {1}{a x}\right )}{a}+\frac {5 c^{5} \left (-\frac {7 a x}{-7 a x +7}+6 \ln \left (-a x +1\right )-1-6 \ln \left (x \right )-6 \ln \left (-a \right )+\frac {1}{5 x^{5} a^{5}}+\frac {1}{2 x^{4} a^{4}}+\frac {1}{x^{3} a^{3}}+\frac {2}{a^{2} x^{2}}+\frac {5}{a x}\right )}{a}-\frac {4 c^{5} \left (-\frac {9 a x}{-9 a x +9}+8 \ln \left (-a x +1\right )-1-8 \ln \left (x \right )-8 \ln \left (-a \right )+\frac {1}{7 x^{7} a^{7}}+\frac {1}{3 x^{6} a^{6}}+\frac {3}{5 x^{5} a^{5}}+\frac {1}{x^{4} a^{4}}+\frac {5}{3 x^{3} a^{3}}+\frac {3}{a^{2} x^{2}}+\frac {7}{a x}\right )}{a}+\frac {2 c^{5} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}-\frac {10 c^{5} \left (\frac {2 a x}{-2 a x +2}-\ln \left (-a x +1\right )+1+\ln \left (x \right )+\ln \left (-a \right )\right )}{a}+\frac {20 c^{5} \left (\frac {4 a x}{-4 a x +4}-3 \ln \left (-a x +1\right )+1+3 \ln \left (x \right )+3 \ln \left (-a \right )-\frac {1}{2 a^{2} x^{2}}-\frac {2}{a x}\right )}{a}-\frac {20 c^{5} \left (\frac {6 a x}{-6 a x +6}-5 \ln \left (-a x +1\right )+1+5 \ln \left (x \right )+5 \ln \left (-a \right )-\frac {1}{4 x^{4} a^{4}}-\frac {2}{3 x^{3} a^{3}}-\frac {3}{2 a^{2} x^{2}}-\frac {4}{a x}\right )}{a}+\frac {10 c^{5} \left (\frac {8 a x}{-8 a x +8}-7 \ln \left (-a x +1\right )+1+7 \ln \left (x \right )+7 \ln \left (-a \right )-\frac {1}{6 x^{6} a^{6}}-\frac {2}{5 x^{5} a^{5}}-\frac {3}{4 x^{4} a^{4}}-\frac {4}{3 x^{3} a^{3}}-\frac {5}{2 a^{2} x^{2}}-\frac {6}{a x}\right )}{a}-\frac {2 c^{5} \left (\frac {10 a x}{-10 a x +10}-9 \ln \left (-a x +1\right )+1+9 \ln \left (x \right )+9 \ln \left (-a \right )-\frac {1}{8 x^{8} a^{8}}-\frac {2}{7 x^{7} a^{7}}-\frac {1}{2 x^{6} a^{6}}-\frac {4}{5 x^{5} a^{5}}-\frac {5}{4 x^{4} a^{4}}-\frac {2}{x^{3} a^{3}}-\frac {7}{2 a^{2} x^{2}}-\frac {8}{a x}\right )}{a}+\frac {c^{5} \left (-\frac {11 a x}{-11 a x +11}+10 \ln \left (-a x +1\right )-1-10 \ln \left (x \right )-10 \ln \left (-a \right )+\frac {1}{9 x^{9} a^{9}}+\frac {1}{4 x^{8} a^{8}}+\frac {3}{7 x^{7} a^{7}}+\frac {2}{3 x^{6} a^{6}}+\frac {1}{x^{5} a^{5}}+\frac {3}{2 x^{4} a^{4}}+\frac {7}{3 x^{3} a^{3}}+\frac {4}{a^{2} x^{2}}+\frac {9}{a x}\right )}{a}\)	\(775\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 103, normalized size = 0.89 \begin {gather*} 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 {gather*}

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 = 0.38, size = 111, normalized size = 0.96 \begin {gather*} \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 {gather*}

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 = 0.38, size = 112, normalized size = 0.97 \begin {gather*} \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 {gather*}

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 = 0.40, size = 184, normalized size = 1.59 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 0.09, size = 81, normalized size = 0.70 \begin {gather*} \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 \left (x\right )+\frac {1}{9}\right )}{a^{10}\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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