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

Optimal. Leaf size=145 \[ \frac {99}{32 a c^4 (1-a x)}-\frac {11}{64 a c^4 (a x+1)}-\frac {35}{32 a c^4 (1-a x)^2}+\frac {1}{64 a c^4 (a x+1)^2}+\frac {13}{48 a c^4 (1-a x)^3}-\frac {1}{32 a c^4 (1-a x)^4}+\frac {303 \log (1-a x)}{128 a c^4}-\frac {47 \log (a x+1)}{128 a c^4}+\frac {x}{c^4} \]

[Out]

x/c^4-1/32/a/c^4/(-a*x+1)^4+13/48/a/c^4/(-a*x+1)^3-35/32/a/c^4/(-a*x+1)^2+99/32/a/c^4/(-a*x+1)+1/64/a/c^4/(a*x
+1)^2-11/64/a/c^4/(a*x+1)+303/128*ln(-a*x+1)/a/c^4-47/128*ln(a*x+1)/a/c^4

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Rubi [A]  time = 0.23, antiderivative size = 145, 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 = {6167, 6157, 6150, 88} \[ \frac {99}{32 a c^4 (1-a x)}-\frac {11}{64 a c^4 (a x+1)}-\frac {35}{32 a c^4 (1-a x)^2}+\frac {1}{64 a c^4 (a x+1)^2}+\frac {13}{48 a c^4 (1-a x)^3}-\frac {1}{32 a c^4 (1-a x)^4}+\frac {303 \log (1-a x)}{128 a c^4}-\frac {47 \log (a x+1)}{128 a c^4}+\frac {x}{c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/c^4 - 1/(32*a*c^4*(1 - a*x)^4) + 13/(48*a*c^4*(1 - a*x)^3) - 35/(32*a*c^4*(1 - a*x)^2) + 99/(32*a*c^4*(1 - a
*x)) + 1/(64*a*c^4*(1 + a*x)^2) - 11/(64*a*c^4*(1 + a*x)) + (303*Log[1 - a*x])/(128*a*c^4) - (47*Log[1 + a*x])
/(128*a*c^4)

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]

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]

Rubi steps

\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx\\ &=-\frac {a^8 \int \frac {e^{2 \tanh ^{-1}(a x)} x^8}{\left (1-a^2 x^2\right )^4} \, dx}{c^4}\\ &=-\frac {a^8 \int \frac {x^8}{(1-a x)^5 (1+a x)^3} \, dx}{c^4}\\ &=-\frac {a^8 \int \left (-\frac {1}{a^8}-\frac {1}{8 a^8 (-1+a x)^5}-\frac {13}{16 a^8 (-1+a x)^4}-\frac {35}{16 a^8 (-1+a x)^3}-\frac {99}{32 a^8 (-1+a x)^2}-\frac {303}{128 a^8 (-1+a x)}+\frac {1}{32 a^8 (1+a x)^3}-\frac {11}{64 a^8 (1+a x)^2}+\frac {47}{128 a^8 (1+a x)}\right ) \, dx}{c^4}\\ &=\frac {x}{c^4}-\frac {1}{32 a c^4 (1-a x)^4}+\frac {13}{48 a c^4 (1-a x)^3}-\frac {35}{32 a c^4 (1-a x)^2}+\frac {99}{32 a c^4 (1-a x)}+\frac {1}{64 a c^4 (1+a x)^2}-\frac {11}{64 a c^4 (1+a x)}+\frac {303 \log (1-a x)}{128 a c^4}-\frac {47 \log (1+a x)}{128 a c^4}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 98, normalized size = 0.68 \[ \frac {\frac {2 \left (192 a^7 x^7-384 a^6 x^6-819 a^5 x^5+1254 a^4 x^4+866 a^3 x^3-1258 a^2 x^2-275 a x+400\right )}{(a x-1)^4 (a x+1)^2}+909 \log (1-a x)-141 \log (a x+1)}{384 a c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(400 - 275*a*x - 1258*a^2*x^2 + 866*a^3*x^3 + 1254*a^4*x^4 - 819*a^5*x^5 - 384*a^6*x^6 + 192*a^7*x^7))/((-
1 + a*x)^4*(1 + a*x)^2) + 909*Log[1 - a*x] - 141*Log[1 + a*x])/(384*a*c^4)

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fricas [A]  time = 0.86, size = 233, normalized size = 1.61 \[ \frac {384 \, a^{7} x^{7} - 768 \, a^{6} x^{6} - 1638 \, a^{5} x^{5} + 2508 \, a^{4} x^{4} + 1732 \, a^{3} x^{3} - 2516 \, a^{2} x^{2} - 550 \, a x - 141 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 909 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 800}{384 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]

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)^4,x, algorithm="fricas")

[Out]

1/384*(384*a^7*x^7 - 768*a^6*x^6 - 1638*a^5*x^5 + 2508*a^4*x^4 + 1732*a^3*x^3 - 2516*a^2*x^2 - 550*a*x - 141*(
a^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + 909*(a^6*x^6 - 2*a^5*x^5 - a^4
*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)*log(a*x - 1) + 800)/(a^7*c^4*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4
*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)

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giac [A]  time = 0.14, size = 96, normalized size = 0.66 \[ \frac {x}{c^{4}} - \frac {47 \, \log \left ({\left | a x + 1 \right |}\right )}{128 \, a c^{4}} + \frac {303 \, \log \left ({\left | a x - 1 \right |}\right )}{128 \, a c^{4}} - \frac {627 \, a^{5} x^{5} - 486 \, a^{4} x^{4} - 1058 \, a^{3} x^{3} + 874 \, a^{2} x^{2} + 467 \, a x - 400}{192 \, {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{4} a c^{4}} \]

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)^4,x, algorithm="giac")

[Out]

x/c^4 - 47/128*log(abs(a*x + 1))/(a*c^4) + 303/128*log(abs(a*x - 1))/(a*c^4) - 1/192*(627*a^5*x^5 - 486*a^4*x^
4 - 1058*a^3*x^3 + 874*a^2*x^2 + 467*a*x - 400)/((a*x + 1)^2*(a*x - 1)^4*a*c^4)

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maple [A]  time = 0.05, size = 125, normalized size = 0.86 \[ \frac {x}{c^{4}}-\frac {1}{32 c^{4} a \left (a x -1\right )^{4}}-\frac {13}{48 c^{4} a \left (a x -1\right )^{3}}-\frac {35}{32 c^{4} a \left (a x -1\right )^{2}}-\frac {99}{32 a \,c^{4} \left (a x -1\right )}+\frac {303 \ln \left (a x -1\right )}{128 c^{4} a}+\frac {1}{64 a \,c^{4} \left (a x +1\right )^{2}}-\frac {11}{64 a \,c^{4} \left (a x +1\right )}-\frac {47 \ln \left (a x +1\right )}{128 a \,c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c^4-1/32/c^4/a/(a*x-1)^4-13/48/c^4/a/(a*x-1)^3-35/32/c^4/a/(a*x-1)^2-99/32/a/c^4/(a*x-1)+303/128/c^4/a*ln(a*
x-1)+1/64/a/c^4/(a*x+1)^2-11/64/a/c^4/(a*x+1)-47/128*ln(a*x+1)/a/c^4

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maxima [A]  time = 0.31, size = 145, normalized size = 1.00 \[ -\frac {627 \, a^{5} x^{5} - 486 \, a^{4} x^{4} - 1058 \, a^{3} x^{3} + 874 \, a^{2} x^{2} + 467 \, a x - 400}{192 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac {x}{c^{4}} - \frac {47 \, \log \left (a x + 1\right )}{128 \, a c^{4}} + \frac {303 \, \log \left (a x - 1\right )}{128 \, a c^{4}} \]

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)^4,x, algorithm="maxima")

[Out]

-1/192*(627*a^5*x^5 - 486*a^4*x^4 - 1058*a^3*x^3 + 874*a^2*x^2 + 467*a*x - 400)/(a^7*c^4*x^6 - 2*a^6*c^4*x^5 -
a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4) + x/c^4 - 47/128*log(a*x + 1)/(a*c^4) + 303/
128*log(a*x - 1)/(a*c^4)

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mupad [B]  time = 1.45, size = 142, normalized size = 0.98 \[ \frac {x}{c^4}+\frac {\frac {467\,x}{192}+\frac {437\,a\,x^2}{96}-\frac {25}{12\,a}-\frac {529\,a^2\,x^3}{96}-\frac {81\,a^3\,x^4}{32}+\frac {209\,a^4\,x^5}{64}}{-a^6\,c^4\,x^6+2\,a^5\,c^4\,x^5+a^4\,c^4\,x^4-4\,a^3\,c^4\,x^3+a^2\,c^4\,x^2+2\,a\,c^4\,x-c^4}+\frac {303\,\ln \left (a\,x-1\right )}{128\,a\,c^4}-\frac {47\,\ln \left (a\,x+1\right )}{128\,a\,c^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c^4 + ((467*x)/192 + (437*a*x^2)/96 - 25/(12*a) - (529*a^2*x^3)/96 - (81*a^3*x^4)/32 + (209*a^4*x^5)/64)/(a^
2*c^4*x^2 - c^4 - 4*a^3*c^4*x^3 + a^4*c^4*x^4 + 2*a^5*c^4*x^5 - a^6*c^4*x^6 + 2*a*c^4*x) + (303*log(a*x - 1))/
(128*a*c^4) - (47*log(a*x + 1))/(128*a*c^4)

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sympy [A]  time = 0.84, size = 156, normalized size = 1.08 \[ a^{8} \left (\frac {- 627 a^{5} x^{5} + 486 a^{4} x^{4} + 1058 a^{3} x^{3} - 874 a^{2} x^{2} - 467 a x + 400}{192 a^{15} c^{4} x^{6} - 384 a^{14} c^{4} x^{5} - 192 a^{13} c^{4} x^{4} + 768 a^{12} c^{4} x^{3} - 192 a^{11} c^{4} x^{2} - 384 a^{10} c^{4} x + 192 a^{9} c^{4}} + \frac {x}{a^{8} c^{4}} + \frac {\frac {303 \log {\left (x - \frac {1}{a} \right )}}{128} - \frac {47 \log {\left (x + \frac {1}{a} \right )}}{128}}{a^{9} c^{4}}\right ) \]

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)**4,x)

[Out]

a**8*((-627*a**5*x**5 + 486*a**4*x**4 + 1058*a**3*x**3 - 874*a**2*x**2 - 467*a*x + 400)/(192*a**15*c**4*x**6 -
384*a**14*c**4*x**5 - 192*a**13*c**4*x**4 + 768*a**12*c**4*x**3 - 192*a**11*c**4*x**2 - 384*a**10*c**4*x + 19
2*a**9*c**4) + x/(a**8*c**4) + (303*log(x - 1/a)/128 - 47*log(x + 1/a)/128)/(a**9*c**4))

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