3.1064 \(\int \frac{e^{2 \tanh ^{-1}(a x)}}{x^2 (c-a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=78 \[ \frac{5 a}{4 c^2 (1-a x)}+\frac{a}{4 c^2 (1-a x)^2}+\frac{2 a \log (x)}{c^2}-\frac{17 a \log (1-a x)}{8 c^2}+\frac{a \log (a x+1)}{8 c^2}-\frac{1}{c^2 x} \]

[Out]

-(1/(c^2*x)) + a/(4*c^2*(1 - a*x)^2) + (5*a)/(4*c^2*(1 - a*x)) + (2*a*Log[x])/c^2 - (17*a*Log[1 - a*x])/(8*c^2
) + (a*Log[1 + a*x])/(8*c^2)

________________________________________________________________________________________

Rubi [A]  time = 0.111547, antiderivative size = 78, 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} \[ \frac{5 a}{4 c^2 (1-a x)}+\frac{a}{4 c^2 (1-a x)^2}+\frac{2 a \log (x)}{c^2}-\frac{17 a \log (1-a x)}{8 c^2}+\frac{a \log (a x+1)}{8 c^2}-\frac{1}{c^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(c^2*x)) + a/(4*c^2*(1 - a*x)^2) + (5*a)/(4*c^2*(1 - a*x)) + (2*a*Log[x])/c^2 - (17*a*Log[1 - a*x])/(8*c^2
) + (a*Log[1 + a*x])/(8*c^2)

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^2 \left (c-a^2 c x^2\right )^2} \, dx &=\frac{\int \frac{1}{x^2 (1-a x)^3 (1+a x)} \, dx}{c^2}\\ &=\frac{\int \left (\frac{1}{x^2}+\frac{2 a}{x}-\frac{a^2}{2 (-1+a x)^3}+\frac{5 a^2}{4 (-1+a x)^2}-\frac{17 a^2}{8 (-1+a x)}+\frac{a^2}{8 (1+a x)}\right ) \, dx}{c^2}\\ &=-\frac{1}{c^2 x}+\frac{a}{4 c^2 (1-a x)^2}+\frac{5 a}{4 c^2 (1-a x)}+\frac{2 a \log (x)}{c^2}-\frac{17 a \log (1-a x)}{8 c^2}+\frac{a \log (1+a x)}{8 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0509784, size = 57, normalized size = 0.73 \[ \frac{\frac{10 a}{1-a x}+\frac{2 a}{(a x-1)^2}+16 a \log (x)-17 a \log (1-a x)+a \log (a x+1)-\frac{8}{x}}{8 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8/x + (10*a)/(1 - a*x) + (2*a)/(-1 + a*x)^2 + 16*a*Log[x] - 17*a*Log[1 - a*x] + a*Log[1 + a*x])/(8*c^2)

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 68, normalized size = 0.9 \begin{align*} -{\frac{1}{x{c}^{2}}}+2\,{\frac{a\ln \left ( x \right ) }{{c}^{2}}}+{\frac{a\ln \left ( ax+1 \right ) }{8\,{c}^{2}}}+{\frac{a}{4\,{c}^{2} \left ( ax-1 \right ) ^{2}}}-{\frac{5\,a}{4\,{c}^{2} \left ( ax-1 \right ) }}-{\frac{17\,a\ln \left ( ax-1 \right ) }{8\,{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/x/c^2+2*a*ln(x)/c^2+1/8*a*ln(a*x+1)/c^2+1/4/c^2*a/(a*x-1)^2-5/4/c^2*a/(a*x-1)-17/8/c^2*a*ln(a*x-1)

________________________________________________________________________________________

Maxima [A]  time = 0.955182, size = 103, normalized size = 1.32 \begin{align*} -\frac{9 \, a^{2} x^{2} - 14 \, a x + 4}{4 \,{\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right )}} + \frac{a \log \left (a x + 1\right )}{8 \, c^{2}} - \frac{17 \, a \log \left (a x - 1\right )}{8 \, c^{2}} + \frac{2 \, a \log \left (x\right )}{c^{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^2/(-a^2*c*x^2+c)^2,x, algorithm="maxima")

[Out]

-1/4*(9*a^2*x^2 - 14*a*x + 4)/(a^2*c^2*x^3 - 2*a*c^2*x^2 + c^2*x) + 1/8*a*log(a*x + 1)/c^2 - 17/8*a*log(a*x -
1)/c^2 + 2*a*log(x)/c^2

________________________________________________________________________________________

Fricas [A]  time = 2.30388, size = 266, normalized size = 3.41 \begin{align*} -\frac{18 \, a^{2} x^{2} - 28 \, a x -{\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \log \left (a x + 1\right ) + 17 \,{\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \log \left (a x - 1\right ) - 16 \,{\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \log \left (x\right ) + 8}{8 \,{\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\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^2/(-a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

-1/8*(18*a^2*x^2 - 28*a*x - (a^3*x^3 - 2*a^2*x^2 + a*x)*log(a*x + 1) + 17*(a^3*x^3 - 2*a^2*x^2 + a*x)*log(a*x
- 1) - 16*(a^3*x^3 - 2*a^2*x^2 + a*x)*log(x) + 8)/(a^2*c^2*x^3 - 2*a*c^2*x^2 + c^2*x)

________________________________________________________________________________________

Sympy [A]  time = 0.795853, size = 76, normalized size = 0.97 \begin{align*} - \frac{9 a^{2} x^{2} - 14 a x + 4}{4 a^{2} c^{2} x^{3} - 8 a c^{2} x^{2} + 4 c^{2} x} - \frac{- 2 a \log{\left (x \right )} + \frac{17 a \log{\left (x - \frac{1}{a} \right )}}{8} - \frac{a \log{\left (x + \frac{1}{a} \right )}}{8}}{c^{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**2/(-a**2*c*x**2+c)**2,x)

[Out]

-(9*a**2*x**2 - 14*a*x + 4)/(4*a**2*c**2*x**3 - 8*a*c**2*x**2 + 4*c**2*x) - (-2*a*log(x) + 17*a*log(x - 1/a)/8
- a*log(x + 1/a)/8)/c**2

________________________________________________________________________________________

Giac [A]  time = 1.14039, size = 88, normalized size = 1.13 \begin{align*} \frac{a \log \left ({\left | a x + 1 \right |}\right )}{8 \, c^{2}} - \frac{17 \, a \log \left ({\left | a x - 1 \right |}\right )}{8 \, c^{2}} + \frac{2 \, a \log \left ({\left | x \right |}\right )}{c^{2}} - \frac{9 \, a^{2} x^{2} - 14 \, a x + 4}{4 \,{\left (a x - 1\right )}^{2} c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*log(abs(a*x + 1))/c^2 - 17/8*a*log(abs(a*x - 1))/c^2 + 2*a*log(abs(x))/c^2 - 1/4*(9*a^2*x^2 - 14*a*x + 4
)/((a*x - 1)^2*c^2*x)