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) + 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)

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Rubi [A]  time = 0.151327, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, 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 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 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]))

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*}

Mathematica [A]  time = 0.108706, 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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Maple [A]  time = 0.042, size = 117, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{c}^{3}{x}^{2}}}-2\,{\frac{a}{{c}^{3}x}}+5\,{\frac{{a}^{2}\ln \left ( x \right ) }{{c}^{3}}}+{\frac{{a}^{2}}{16\,{c}^{3} \left ( ax+1 \right ) }}-{\frac{5\,{a}^{2}\ln \left ( ax+1 \right ) }{16\,{c}^{3}}}-{\frac{{a}^{2}}{12\,{c}^{3} \left ( ax-1 \right ) ^{3}}}+{\frac{{a}^{2}}{2\,{c}^{3} \left ( ax-1 \right ) ^{2}}}-{\frac{39\,{a}^{2}}{16\,{c}^{3} \left ( ax-1 \right ) }}-{\frac{75\,{a}^{2}\ln \left ( ax-1 \right ) }{16\,{c}^{3}}} \end{align*}

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

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Maxima [A]  time = 0.968231, size = 162, normalized size = 1.21 \begin{align*} -\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 \left (x\right )}{c^{3}} \end{align*}

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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Fricas [A]  time = 3.10859, size = 423, normalized size = 3.16 \begin{align*} -\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 \left (x\right ) - 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 )}} \end{align*}

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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Sympy [A]  time = 1.43152, size = 121, normalized size = 0.9 \begin{align*} - \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{\left (x \right )} - \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}} \end{align*}

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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Giac [A]  time = 1.14353, size = 138, normalized size = 1.03 \begin{align*} -\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}} \end{align*}

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)