3.1075 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{x^3 (c-a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=134 \[ \frac {39 a^2}{16 c^3 (1-a x)}+\frac {a^2}{16 c^3 (a x+1)}+\frac {a^2}{2 c^3 (1-a x)^2}+\frac {a^2}{12 c^3 (1-a x)^3}+\frac {5 a^2 \log (x)}{c^3}-\frac {75 a^2 \log (1-a x)}{16 c^3}-\frac {5 a^2 \log (a x+1)}{16 c^3}-\frac {2 a}{c^3 x}-\frac {1}{2 c^3 x^2} \]

[Out]

-1/2/c^3/x^2-2*a/c^3/x+1/12*a^2/c^3/(-a*x+1)^3+1/2*a^2/c^3/(-a*x+1)^2+39/16*a^2/c^3/(-a*x+1)+1/16*a^2/c^3/(a*x
+1)+5*a^2*ln(x)/c^3-75/16*a^2*ln(-a*x+1)/c^3-5/16*a^2*ln(a*x+1)/c^3

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Rubi [A]  time = 0.15, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 88} \[ \frac {39 a^2}{16 c^3 (1-a x)}+\frac {a^2}{16 c^3 (a x+1)}+\frac {a^2}{2 c^3 (1-a x)^2}+\frac {a^2}{12 c^3 (1-a x)^3}+\frac {5 a^2 \log (x)}{c^3}-\frac {75 a^2 \log (1-a x)}{16 c^3}-\frac {5 a^2 \log (a x+1)}{16 c^3}-\frac {2 a}{c^3 x}-\frac {1}{2 c^3 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*c^3*x^2) - (2*a)/(c^3*x) + a^2/(12*c^3*(1 - a*x)^3) + a^2/(2*c^3*(1 - a*x)^2) + (39*a^2)/(16*c^3*(1 - a*
x)) + a^2/(16*c^3*(1 + a*x)) + (5*a^2*Log[x])/c^3 - (75*a^2*Log[1 - a*x])/(16*c^3) - (5*a^2*Log[1 + a*x])/(16*
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])

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {1}{x^3 (1-a x)^4 (1+a x)^2} \, dx}{c^3}\\ &=\frac {\int \left (\frac {1}{x^3}+\frac {2 a}{x^2}+\frac {5 a^2}{x}+\frac {a^3}{4 (-1+a x)^4}-\frac {a^3}{(-1+a x)^3}+\frac {39 a^3}{16 (-1+a x)^2}-\frac {75 a^3}{16 (-1+a x)}-\frac {a^3}{16 (1+a x)^2}-\frac {5 a^3}{16 (1+a x)}\right ) \, dx}{c^3}\\ &=-\frac {1}{2 c^3 x^2}-\frac {2 a}{c^3 x}+\frac {a^2}{12 c^3 (1-a x)^3}+\frac {a^2}{2 c^3 (1-a x)^2}+\frac {39 a^2}{16 c^3 (1-a x)}+\frac {a^2}{16 c^3 (1+a x)}+\frac {5 a^2 \log (x)}{c^3}-\frac {75 a^2 \log (1-a x)}{16 c^3}-\frac {5 a^2 \log (1+a x)}{16 c^3}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 98, normalized size = 0.73 \[ \frac {\frac {117 a^2}{1-a x}+\frac {3 a^2}{a x+1}+\frac {24 a^2}{(a x-1)^2}-\frac {4 a^2}{(a x-1)^3}+240 a^2 \log (x)-225 a^2 \log (1-a x)-15 a^2 \log (a x+1)-\frac {96 a}{x}-\frac {24}{x^2}}{48 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-24/x^2 - (96*a)/x + (117*a^2)/(1 - a*x) - (4*a^2)/(-1 + a*x)^3 + (24*a^2)/(-1 + a*x)^2 + (3*a^2)/(1 + a*x) +
240*a^2*Log[x] - 225*a^2*Log[1 - a*x] - 15*a^2*Log[1 + a*x])/(48*c^3)

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fricas [A]  time = 0.59, size = 197, normalized size = 1.47 \[ -\frac {210 \, a^{5} x^{5} - 300 \, a^{4} x^{4} - 170 \, a^{3} x^{3} + 340 \, a^{2} x^{2} - 48 \, a x + 15 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} + 2 \, a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x + 1\right ) + 225 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} + 2 \, a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 240 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} + 2 \, a^{3} x^{3} - a^{2} x^{2}\right )} \log \relax (x) - 24}{48 \, {\left (a^{4} c^{3} x^{6} - 2 \, a^{3} c^{3} x^{5} + 2 \, a c^{3} x^{3} - c^{3} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(210*a^5*x^5 - 300*a^4*x^4 - 170*a^3*x^3 + 340*a^2*x^2 - 48*a*x + 15*(a^6*x^6 - 2*a^5*x^5 + 2*a^3*x^3 -
a^2*x^2)*log(a*x + 1) + 225*(a^6*x^6 - 2*a^5*x^5 + 2*a^3*x^3 - a^2*x^2)*log(a*x - 1) - 240*(a^6*x^6 - 2*a^5*x^
5 + 2*a^3*x^3 - a^2*x^2)*log(x) - 24)/(a^4*c^3*x^6 - 2*a^3*c^3*x^5 + 2*a*c^3*x^3 - c^3*x^2)

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giac [A]  time = 0.19, size = 102, normalized size = 0.76 \[ -\frac {5 \, a^{2} \log \left ({\left | a x + 1 \right |}\right )}{16 \, c^{3}} - \frac {75 \, a^{2} \log \left ({\left | a x - 1 \right |}\right )}{16 \, c^{3}} + \frac {5 \, a^{2} \log \left ({\left | x \right |}\right )}{c^{3}} - \frac {105 \, a^{5} x^{5} - 150 \, a^{4} x^{4} - 85 \, a^{3} x^{3} + 170 \, a^{2} x^{2} - 24 \, a x - 12}{24 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{3} c^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/16*a^2*log(abs(a*x + 1))/c^3 - 75/16*a^2*log(abs(a*x - 1))/c^3 + 5*a^2*log(abs(x))/c^3 - 1/24*(105*a^5*x^5
- 150*a^4*x^4 - 85*a^3*x^3 + 170*a^2*x^2 - 24*a*x - 12)/((a*x + 1)*(a*x - 1)^3*c^3*x^2)

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maple [A]  time = 0.04, size = 117, normalized size = 0.87 \[ -\frac {1}{2 c^{3} x^{2}}-\frac {2 a}{c^{3} x}+\frac {5 a^{2} \ln \relax (x )}{c^{3}}-\frac {a^{2}}{12 c^{3} \left (a x -1\right )^{3}}+\frac {a^{2}}{2 c^{3} \left (a x -1\right )^{2}}-\frac {39 a^{2}}{16 c^{3} \left (a x -1\right )}-\frac {75 a^{2} \ln \left (a x -1\right )}{16 c^{3}}+\frac {a^{2}}{16 c^{3} \left (a x +1\right )}-\frac {5 a^{2} \ln \left (a x +1\right )}{16 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/c^3/x^2-2*a/c^3/x+5*a^2*ln(x)/c^3-1/12/c^3*a^2/(a*x-1)^3+1/2/c^3*a^2/(a*x-1)^2-39/16/c^3*a^2/(a*x-1)-75/1
6/c^3*a^2*ln(a*x-1)+1/16*a^2/c^3/(a*x+1)-5/16*a^2*ln(a*x+1)/c^3

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maxima [A]  time = 0.33, size = 120, normalized size = 0.90 \[ -\frac {105 \, a^{5} x^{5} - 150 \, a^{4} x^{4} - 85 \, a^{3} x^{3} + 170 \, a^{2} x^{2} - 24 \, a x - 12}{24 \, {\left (a^{4} c^{3} x^{6} - 2 \, a^{3} c^{3} x^{5} + 2 \, a c^{3} x^{3} - c^{3} x^{2}\right )}} - \frac {5 \, a^{2} \log \left (a x + 1\right )}{16 \, c^{3}} - \frac {75 \, a^{2} \log \left (a x - 1\right )}{16 \, c^{3}} + \frac {5 \, a^{2} \log \relax (x)}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(105*a^5*x^5 - 150*a^4*x^4 - 85*a^3*x^3 + 170*a^2*x^2 - 24*a*x - 12)/(a^4*c^3*x^6 - 2*a^3*c^3*x^5 + 2*a*
c^3*x^3 - c^3*x^2) - 5/16*a^2*log(a*x + 1)/c^3 - 75/16*a^2*log(a*x - 1)/c^3 + 5*a^2*log(x)/c^3

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mupad [B]  time = 1.01, size = 119, normalized size = 0.89 \[ \frac {5\,a^2\,\ln \relax (x)}{c^3}-\frac {-\frac {35\,a^5\,x^5}{8}+\frac {25\,a^4\,x^4}{4}+\frac {85\,a^3\,x^3}{24}-\frac {85\,a^2\,x^2}{12}+a\,x+\frac {1}{2}}{-a^4\,c^3\,x^6+2\,a^3\,c^3\,x^5-2\,a\,c^3\,x^3+c^3\,x^2}-\frac {75\,a^2\,\ln \left (a\,x-1\right )}{16\,c^3}-\frac {5\,a^2\,\ln \left (a\,x+1\right )}{16\,c^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*a^2*log(x))/c^3 - (a*x - (85*a^2*x^2)/12 + (85*a^3*x^3)/24 + (25*a^4*x^4)/4 - (35*a^5*x^5)/8 + 1/2)/(c^3*x^
2 - 2*a*c^3*x^3 + 2*a^3*c^3*x^5 - a^4*c^3*x^6) - (75*a^2*log(a*x - 1))/(16*c^3) - (5*a^2*log(a*x + 1))/(16*c^3
)

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sympy [A]  time = 0.88, size = 121, normalized size = 0.90 \[ \frac {- 105 a^{5} x^{5} + 150 a^{4} x^{4} + 85 a^{3} x^{3} - 170 a^{2} x^{2} + 24 a x + 12}{24 a^{4} c^{3} x^{6} - 48 a^{3} c^{3} x^{5} + 48 a c^{3} x^{3} - 24 c^{3} x^{2}} + \frac {5 a^{2} \log {\relax (x )} - \frac {75 a^{2} \log {\left (x - \frac {1}{a} \right )}}{16} - \frac {5 a^{2} \log {\left (x + \frac {1}{a} \right )}}{16}}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-105*a**5*x**5 + 150*a**4*x**4 + 85*a**3*x**3 - 170*a**2*x**2 + 24*a*x + 12)/(24*a**4*c**3*x**6 - 48*a**3*c**
3*x**5 + 48*a*c**3*x**3 - 24*c**3*x**2) + (5*a**2*log(x) - 75*a**2*log(x - 1/a)/16 - 5*a**2*log(x + 1/a)/16)/c
**3

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