\(\int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^4} \, dx\) [821]

Optimal result

Integrand size = 22, antiderivative size = 143 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {x}{c^4}-\frac {1}{64 a c^4 (1-a x)^2}+\frac {11}{64 a c^4 (1-a x)}+\frac {1}{32 a c^4 (1+a x)^4}-\frac {13}{48 a c^4 (1+a x)^3}+\frac {35}{32 a c^4 (1+a x)^2}-\frac {99}{32 a c^4 (1+a x)}+\frac {47 \log (1-a x)}{128 a c^4}-\frac {303 \log (1+a x)}{128 a c^4} \]

[Out]

x/c^4-1/64/a/c^4/(-a*x+1)^2+11/64/a/c^4/(-a*x+1)+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)+47/128*ln(-a*x+1)/a/c^4-303/128*ln(a*x+1)/a/c^4

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6302, 6292, 6285, 90} \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {11}{64 a c^4 (1-a x)}-\frac {99}{32 a c^4 (a x+1)}-\frac {1}{64 a c^4 (1-a x)^2}+\frac {35}{32 a c^4 (a x+1)^2}-\frac {13}{48 a c^4 (a x+1)^3}+\frac {1}{32 a c^4 (a x+1)^4}+\frac {47 \log (1-a x)}{128 a c^4}-\frac {303 \log (a x+1)}{128 a c^4}+\frac {x}{c^4} \]

[In]

Int[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^4),x]

[Out]

x/c^4 - 1/(64*a*c^4*(1 - a*x)^2) + 11/(64*a*c^4*(1 - a*x)) + 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)) + (47*Log[1 - a*x])/(128*a*c^4) - (303*Log[1 + a*x])
/(128*a*c^4)

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*} \text {integral}& = -\int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx \\ & = -\frac {a^8 \int \frac {e^{-2 \text {arctanh}(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)^3 (1+a x)^5} \, dx}{c^4} \\ & = -\frac {a^8 \int \left (-\frac {1}{a^8}-\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)}+\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)}\right ) \, dx}{c^4} \\ & = \frac {x}{c^4}-\frac {1}{64 a c^4 (1-a x)^2}+\frac {11}{64 a c^4 (1-a x)}+\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 {47 \log (1-a x)}{128 a c^4}-\frac {303 \log (1+a x)}{128 a c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.87 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {2 \left (-400-275 a x+1258 a^2 x^2+866 a^3 x^3-1254 a^4 x^4-819 a^5 x^5+384 a^6 x^6+192 a^7 x^7\right )+141 (-1+a x)^2 (1+a x)^4 \log (1-a x)-909 (-1+a x)^2 (1+a x)^4 \log (1+a x)}{384 a (-1+a x)^2 (c+a c x)^4} \]

[In]

Integrate[1/(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) + 14
1*(-1 + a*x)^2*(1 + a*x)^4*Log[1 - a*x] - 909*(-1 + a*x)^2*(1 + a*x)^4*Log[1 + a*x])/(384*a*(-1 + a*x)^2*(c +
a*c*x)^4)

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.76

method	result	size
default	\(\frac {a^{8} \left (\frac {1}{32 a^{9} \left (a x +1\right )^{4}}-\frac {13}{48 a^{9} \left (a x +1\right )^{3}}+\frac {35}{32 a^{9} \left (a x +1\right )^{2}}-\frac {99}{32 a^{9} \left (a x +1\right )}-\frac {303 \ln \left (a x +1\right )}{128 a^{9}}+\frac {x}{a^{8}}-\frac {1}{64 a^{9} \left (a x -1\right )^{2}}-\frac {11}{64 a^{9} \left (a x -1\right )}+\frac {47 \ln \left (a x -1\right )}{128 a^{9}}\right )}{c^{4}}\)	\(108\)
risch	\(\frac {x}{c^{4}}+\frac {-\frac {209 a^{4} c^{4} x^{5}}{64}-\frac {81 a^{3} c^{4} x^{4}}{32}+\frac {529 a^{2} c^{4} x^{3}}{96}+\frac {437 a \,c^{4} x^{2}}{96}-\frac {467 c^{4} x}{192}-\frac {25 c^{4}}{12 a}}{c^{8} \left (a x +1\right )^{2} \left (a^{2} x^{2}-1\right )^{2}}+\frac {47 \ln \left (-a x +1\right )}{128 a \,c^{4}}-\frac {303 \ln \left (a x +1\right )}{128 a \,c^{4}}\)	\(115\)
norman	\(\frac {\frac {a^{7} x^{8}}{c}-\frac {175 x}{64 c}-\frac {111 a \,x^{2}}{64 c}+\frac {199 a^{2} x^{3}}{24 c}+\frac {115 a^{3} x^{4}}{24 c}-\frac {545 a^{4} x^{5}}{64 c}-\frac {803 a^{5} x^{6}}{192 c}+\frac {37 a^{6} x^{7}}{12 c}}{\left (a x -1\right )^{3} c^{3} \left (a x +1\right )^{4}}+\frac {47 \ln \left (a x -1\right )}{128 a \,c^{4}}-\frac {303 \ln \left (a x +1\right )}{128 a \,c^{4}}\)	\(129\)
parallelrisch	\(\frac {-1818 a \ln \left (a x +1\right ) x +909 a^{2} \ln \left (a x +1\right ) x^{2}-38 a^{5} x^{5}-1468 a^{3} x^{3}-1818 \ln \left (a x +1\right ) x^{5} a^{5}-909 \ln \left (a x +1\right ) x^{6} a^{6}+909 \ln \left (a x +1\right ) x^{4} a^{4}+141 \ln \left (a x -1\right ) x^{6} a^{6}+282 \ln \left (a x -1\right ) x^{5} a^{5}-141 \ln \left (a x -1\right ) x^{4} a^{4}+3636 a^{3} \ln \left (a x +1\right ) x^{3}+1568 a^{6} x^{6}+1050 a x -564 a^{3} \ln \left (a x -1\right ) x^{3}-141 a^{2} \ln \left (a x -1\right ) x^{2}+282 a \ln \left (a x -1\right ) x -3308 a^{4} x^{4}+141 \ln \left (a x -1\right )-909 \ln \left (a x +1\right )+384 a^{7} x^{7}+1716 a^{2} x^{2}}{384 c^{4} \left (a x +1\right )^{2} \left (a^{2} x^{2}-1\right )^{2} a}\)	\(256\)

[In]

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

[Out]

a^8/c^4*(1/32/a^9/(a*x+1)^4-13/48/a^9/(a*x+1)^3+35/32/a^9/(a*x+1)^2-99/32/a^9/(a*x+1)-303/128/a^9*ln(a*x+1)+1/
a^8*x-1/64/a^9/(a*x-1)^2-11/64/a^9/(a*x-1)+47/128/a^9*ln(a*x-1))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.63 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {384 \, a^{7} x^{7} + 768 \, a^{6} x^{6} - 1638 \, a^{5} x^{5} - 2508 \, a^{4} x^{4} + 1732 \, a^{3} x^{3} + 2516 \, a^{2} x^{2} - 550 \, a x - 909 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 141 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 800}{384 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]

[In]

integrate((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 - 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) + 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) - 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)

Sympy [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.09 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=a^{8} \left (\frac {- 627 a^{5} x^{5} - 486 a^{4} x^{4} + 1058 a^{3} x^{3} + 874 a^{2} x^{2} - 467 a x - 400}{192 a^{15} c^{4} x^{6} + 384 a^{14} c^{4} x^{5} - 192 a^{13} c^{4} x^{4} - 768 a^{12} c^{4} x^{3} - 192 a^{11} c^{4} x^{2} + 384 a^{10} c^{4} x + 192 a^{9} c^{4}} + \frac {x}{a^{8} c^{4}} + \frac {\frac {47 \log {\left (x - \frac {1}{a} \right )}}{128} - \frac {303 \log {\left (x + \frac {1}{a} \right )}}{128}}{a^{9} c^{4}}\right ) \]

[In]

integrate((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) + (47*log(x - 1/a)/128 - 303*log(x + 1/a)/128)/(a**9*c**4))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.01 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=-\frac {627 \, a^{5} x^{5} + 486 \, a^{4} x^{4} - 1058 \, a^{3} x^{3} - 874 \, a^{2} x^{2} + 467 \, a x + 400}{192 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac {x}{c^{4}} - \frac {303 \, \log \left (a x + 1\right )}{128 \, a c^{4}} + \frac {47 \, \log \left (a x - 1\right )}{128 \, a c^{4}} \]

[In]

integrate((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 - 303/128*log(a*x + 1)/(a*c^4) + 47/
128*log(a*x - 1)/(a*c^4)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.67 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {x}{c^{4}} - \frac {303 \, \log \left ({\left | a x + 1 \right |}\right )}{128 \, a c^{4}} + \frac {47 \, \log \left ({\left | a x - 1 \right |}\right )}{128 \, a c^{4}} - \frac {627 \, a^{5} x^{5} + 486 \, a^{4} x^{4} - 1058 \, a^{3} x^{3} - 874 \, a^{2} x^{2} + 467 \, a x + 400}{192 \, {\left (a x + 1\right )}^{4} {\left (a x - 1\right )}^{2} a c^{4}} \]

[In]

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

[Out]

x/c^4 - 303/128*log(abs(a*x + 1))/(a*c^4) + 47/128*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)^4*(a*x - 1)^2*a*c^4)

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {x}{c^4}-\frac {\frac {467\,x}{192}-\frac {437\,a\,x^2}{96}+\frac {25}{12\,a}-\frac {529\,a^2\,x^3}{96}+\frac {81\,a^3\,x^4}{32}+\frac {209\,a^4\,x^5}{64}}{a^6\,c^4\,x^6+2\,a^5\,c^4\,x^5-a^4\,c^4\,x^4-4\,a^3\,c^4\,x^3-a^2\,c^4\,x^2+2\,a\,c^4\,x+c^4}+\frac {47\,\ln \left (a\,x-1\right )}{128\,a\,c^4}-\frac {303\,\ln \left (a\,x+1\right )}{128\,a\,c^4} \]

[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)/(c^
4 - a^2*c^4*x^2 - 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) + (47*log(a*x - 1))/(
128*a*c^4) - (303*log(a*x + 1))/(128*a*c^4)