3.676 \(\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^3} \, dx\)

Optimal. Leaf size=109 \[ -\frac {1}{16 a c^3 (1-a x)}+\frac {39}{16 a c^3 (a x+1)}-\frac {5}{8 a c^3 (a x+1)^2}+\frac {1}{12 a c^3 (a x+1)^3}-\frac {\log (1-a x)}{4 a c^3}+\frac {9 \log (a x+1)}{4 a c^3}-\frac {x}{c^3} \]

[Out]

-x/c^3-1/16/a/c^3/(-a*x+1)+1/12/a/c^3/(a*x+1)^3-5/8/a/c^3/(a*x+1)^2+39/16/a/c^3/(a*x+1)-1/4*ln(-a*x+1)/a/c^3+9
/4*ln(a*x+1)/a/c^3

________________________________________________________________________________________

Rubi [A]  time = 0.17, antiderivative size = 109, 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} \[ -\frac {1}{16 a c^3 (1-a x)}+\frac {39}{16 a c^3 (a x+1)}-\frac {5}{8 a c^3 (a x+1)^2}+\frac {1}{12 a c^3 (a x+1)^3}-\frac {\log (1-a x)}{4 a c^3}+\frac {9 \log (a x+1)}{4 a c^3}-\frac {x}{c^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^3),x]

[Out]

-(x/c^3) - 1/(16*a*c^3*(1 - a*x)) + 1/(12*a*c^3*(1 + a*x)^3) - 5/(8*a*c^3*(1 + a*x)^2) + 39/(16*a*c^3*(1 + a*x
)) - Log[1 - a*x]/(4*a*c^3) + (9*Log[1 + a*x])/(4*a*c^3)

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 \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx &=-\frac {a^6 \int \frac {e^{-2 \tanh ^{-1}(a x)} x^6}{\left (1-a^2 x^2\right )^3} \, dx}{c^3}\\ &=-\frac {a^6 \int \frac {x^6}{(1-a x)^2 (1+a x)^4} \, dx}{c^3}\\ &=-\frac {a^6 \int \left (\frac {1}{a^6}+\frac {1}{16 a^6 (-1+a x)^2}+\frac {1}{4 a^6 (-1+a x)}+\frac {1}{4 a^6 (1+a x)^4}-\frac {5}{4 a^6 (1+a x)^3}+\frac {39}{16 a^6 (1+a x)^2}-\frac {9}{4 a^6 (1+a x)}\right ) \, dx}{c^3}\\ &=-\frac {x}{c^3}-\frac {1}{16 a c^3 (1-a x)}+\frac {1}{12 a c^3 (1+a x)^3}-\frac {5}{8 a c^3 (1+a x)^2}+\frac {39}{16 a c^3 (1+a x)}-\frac {\log (1-a x)}{4 a c^3}+\frac {9 \log (1+a x)}{4 a c^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 104, normalized size = 0.95 \[ \frac {-2 \left (6 a^5 x^5+12 a^4 x^4-15 a^3 x^3-24 a^2 x^2+7 a x+11\right )-3 (a x-1) (a x+1)^3 \log (1-a x)+27 (a x-1) (a x+1)^3 \log (a x+1)}{12 a (a x-1) (a c x+c)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^3),x]

[Out]

(-2*(11 + 7*a*x - 24*a^2*x^2 - 15*a^3*x^3 + 12*a^4*x^4 + 6*a^5*x^5) - 3*(-1 + a*x)*(1 + a*x)^3*Log[1 - a*x] +
27*(-1 + a*x)*(1 + a*x)^3*Log[1 + a*x])/(12*a*(-1 + a*x)*(c + a*c*x)^3)

________________________________________________________________________________________

fricas [A]  time = 0.64, size = 137, normalized size = 1.26 \[ -\frac {12 \, a^{5} x^{5} + 24 \, a^{4} x^{4} - 30 \, a^{3} x^{3} - 48 \, a^{2} x^{2} + 14 \, a x - 27 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (a x - 1\right ) + 22}{12 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^3,x, algorithm="fricas")

[Out]

-1/12*(12*a^5*x^5 + 24*a^4*x^4 - 30*a^3*x^3 - 48*a^2*x^2 + 14*a*x - 27*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*log(a
*x + 1) + 3*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*log(a*x - 1) + 22)/(a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^2*c^3*x -
a*c^3)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 140, normalized size = 1.28 \[ -\frac {{\left (a x + 1\right )} {\left (\frac {65}{a x + 1} - 32\right )}}{32 \, a c^{3} {\left (\frac {2}{a x + 1} - 1\right )}} - \frac {2 \, \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a c^{3}} - \frac {\log \left ({\left | -\frac {2}{a x + 1} + 1 \right |}\right )}{4 \, a c^{3}} + \frac {\frac {117 \, a^{11} c^{6}}{a x + 1} - \frac {30 \, a^{11} c^{6}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, a^{11} c^{6}}{{\left (a x + 1\right )}^{3}}}{48 \, a^{12} c^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^3,x, algorithm="giac")

[Out]

-1/32*(a*x + 1)*(65/(a*x + 1) - 32)/(a*c^3*(2/(a*x + 1) - 1)) - 2*log(abs(a*x + 1)/((a*x + 1)^2*abs(a)))/(a*c^
3) - 1/4*log(abs(-2/(a*x + 1) + 1))/(a*c^3) + 1/48*(117*a^11*c^6/(a*x + 1) - 30*a^11*c^6/(a*x + 1)^2 + 4*a^11*
c^6/(a*x + 1)^3)/(a^12*c^9)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 96, normalized size = 0.88 \[ -\frac {x}{c^{3}}+\frac {1}{16 a \,c^{3} \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{4 a \,c^{3}}+\frac {1}{12 a \,c^{3} \left (a x +1\right )^{3}}-\frac {5}{8 a \,c^{3} \left (a x +1\right )^{2}}+\frac {39}{16 a \,c^{3} \left (a x +1\right )}+\frac {9 \ln \left (a x +1\right )}{4 a \,c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^3,x)

[Out]

-x/c^3+1/16/a/c^3/(a*x-1)-1/4/a/c^3*ln(a*x-1)+1/12/a/c^3/(a*x+1)^3-5/8/a/c^3/(a*x+1)^2+39/16/a/c^3/(a*x+1)+9/4
*ln(a*x+1)/a/c^3

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 98, normalized size = 0.90 \[ \frac {15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 13 \, a x - 11}{6 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} - \frac {x}{c^{3}} + \frac {9 \, \log \left (a x + 1\right )}{4 \, a c^{3}} - \frac {\log \left (a x - 1\right )}{4 \, a c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^3,x, algorithm="maxima")

[Out]

1/6*(15*a^3*x^3 + 12*a^2*x^2 - 13*a*x - 11)/(a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^2*c^3*x - a*c^3) - x/c^3 + 9/4*
log(a*x + 1)/(a*c^3) - 1/4*log(a*x - 1)/(a*c^3)

________________________________________________________________________________________

mupad [B]  time = 0.11, size = 94, normalized size = 0.86 \[ \frac {\frac {13\,x}{6}-2\,a\,x^2+\frac {11}{6\,a}-\frac {5\,a^2\,x^3}{2}}{-a^4\,c^3\,x^4-2\,a^3\,c^3\,x^3+2\,a\,c^3\,x+c^3}-\frac {x}{c^3}-\frac {\ln \left (a\,x-1\right )}{4\,a\,c^3}+\frac {9\,\ln \left (a\,x+1\right )}{4\,a\,c^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(a^2*x^2 - 1)/((c - c/(a^2*x^2))^3*(a*x + 1)^2),x)

[Out]

((13*x)/6 - 2*a*x^2 + 11/(6*a) - (5*a^2*x^3)/2)/(c^3 - 2*a^3*c^3*x^3 - a^4*c^3*x^4 + 2*a*c^3*x) - x/c^3 - log(
a*x - 1)/(4*a*c^3) + (9*log(a*x + 1))/(4*a*c^3)

________________________________________________________________________________________

sympy [A]  time = 0.63, size = 104, normalized size = 0.95 \[ - a^{6} \left (\frac {- 15 a^{3} x^{3} - 12 a^{2} x^{2} + 13 a x + 11}{6 a^{11} c^{3} x^{4} + 12 a^{10} c^{3} x^{3} - 12 a^{8} c^{3} x - 6 a^{7} c^{3}} + \frac {x}{a^{6} c^{3}} + \frac {\frac {\log {\left (x - \frac {1}{a} \right )}}{4} - \frac {9 \log {\left (x + \frac {1}{a} \right )}}{4}}{a^{7} c^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a**2/x**2)**3,x)

[Out]

-a**6*((-15*a**3*x**3 - 12*a**2*x**2 + 13*a*x + 11)/(6*a**11*c**3*x**4 + 12*a**10*c**3*x**3 - 12*a**8*c**3*x -
6*a**7*c**3) + x/(a**6*c**3) + (log(x - 1/a)/4 - 9*log(x + 1/a)/4)/(a**7*c**3))

________________________________________________________________________________________