3.948 \(\int \frac{e^{\tanh ^{-1}(a x)}}{x^4 (1-a^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=102 \[ \frac{5 a^3}{4 (1-a x)}-\frac{a^3}{8 (a x+1)}+\frac{a^3}{8 (1-a x)^2}-\frac{3 a^2}{x}+3 a^3 \log (x)-\frac{59}{16} a^3 \log (1-a x)+\frac{11}{16} a^3 \log (a x+1)-\frac{a}{2 x^2}-\frac{1}{3 x^3} \]

[Out]

-1/(3*x^3) - a/(2*x^2) - (3*a^2)/x + a^3/(8*(1 - a*x)^2) + (5*a^3)/(4*(1 - a*x)) - a^3/(8*(1 + a*x)) + 3*a^3*L
og[x] - (59*a^3*Log[1 - a*x])/16 + (11*a^3*Log[1 + a*x])/16

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Rubi [A]  time = 0.145136, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6150, 88} \[ \frac{5 a^3}{4 (1-a x)}-\frac{a^3}{8 (a x+1)}+\frac{a^3}{8 (1-a x)^2}-\frac{3 a^2}{x}+3 a^3 \log (x)-\frac{59}{16} a^3 \log (1-a x)+\frac{11}{16} a^3 \log (a x+1)-\frac{a}{2 x^2}-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(x^4*(1 - a^2*x^2)^(5/2)),x]

[Out]

-1/(3*x^3) - a/(2*x^2) - (3*a^2)/x + a^3/(8*(1 - a*x)^2) + (5*a^3)/(4*(1 - a*x)) - a^3/(8*(1 + a*x)) + 3*a^3*L
og[x] - (59*a^3*Log[1 - a*x])/16 + (11*a^3*Log[1 + a*x])/16

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^{\tanh ^{-1}(a x)}}{x^4 \left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac{1}{x^4 (1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (\frac{1}{x^4}+\frac{a}{x^3}+\frac{3 a^2}{x^2}+\frac{3 a^3}{x}-\frac{a^4}{4 (-1+a x)^3}+\frac{5 a^4}{4 (-1+a x)^2}-\frac{59 a^4}{16 (-1+a x)}+\frac{a^4}{8 (1+a x)^2}+\frac{11 a^4}{16 (1+a x)}\right ) \, dx\\ &=-\frac{1}{3 x^3}-\frac{a}{2 x^2}-\frac{3 a^2}{x}+\frac{a^3}{8 (1-a x)^2}+\frac{5 a^3}{4 (1-a x)}-\frac{a^3}{8 (1+a x)}+3 a^3 \log (x)-\frac{59}{16} a^3 \log (1-a x)+\frac{11}{16} a^3 \log (1+a x)\\ \end{align*}

Mathematica [A]  time = 0.0830081, size = 91, normalized size = 0.89 \[ \frac{1}{48} \left (\frac{60 a^3}{1-a x}-\frac{6 a^3}{a x+1}+\frac{6 a^3}{(a x-1)^2}-\frac{144 a^2}{x}+144 a^3 \log (x)-177 a^3 \log (1-a x)+33 a^3 \log (a x+1)-\frac{24 a}{x^2}-\frac{16}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]/(x^4*(1 - a^2*x^2)^(5/2)),x]

[Out]

(-16/x^3 - (24*a)/x^2 - (144*a^2)/x + (60*a^3)/(1 - a*x) + (6*a^3)/(-1 + a*x)^2 - (6*a^3)/(1 + a*x) + 144*a^3*
Log[x] - 177*a^3*Log[1 - a*x] + 33*a^3*Log[1 + a*x])/48

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Maple [A]  time = 0.042, size = 86, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,{x}^{3}}}-{\frac{a}{2\,{x}^{2}}}-3\,{\frac{{a}^{2}}{x}}+3\,{a}^{3}\ln \left ( x \right ) -{\frac{{a}^{3}}{8\,ax+8}}+{\frac{11\,{a}^{3}\ln \left ( ax+1 \right ) }{16}}+{\frac{{a}^{3}}{8\, \left ( ax-1 \right ) ^{2}}}-{\frac{5\,{a}^{3}}{4\,ax-4}}-{\frac{59\,{a}^{3}\ln \left ( ax-1 \right ) }{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^3/x^4,x)

[Out]

-1/3/x^3-1/2*a/x^2-3*a^2/x+3*a^3*ln(x)-1/8*a^3/(a*x+1)+11/16*a^3*ln(a*x+1)+1/8*a^3/(a*x-1)^2-5/4*a^3/(a*x-1)-5
9/16*a^3*ln(a*x-1)

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Maxima [A]  time = 0.963698, size = 131, normalized size = 1.28 \begin{align*} \frac{11}{16} \, a^{3} \log \left (a x + 1\right ) - \frac{59}{16} \, a^{3} \log \left (a x - 1\right ) + 3 \, a^{3} \log \left (x\right ) - \frac{105 \, a^{5} x^{5} - 69 \, a^{4} x^{4} - 106 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 4 \, a x + 8}{24 \,{\left (a^{3} x^{6} - a^{2} x^{5} - a x^{4} + x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^3/x^4,x, algorithm="maxima")

[Out]

11/16*a^3*log(a*x + 1) - 59/16*a^3*log(a*x - 1) + 3*a^3*log(x) - 1/24*(105*a^5*x^5 - 69*a^4*x^4 - 106*a^3*x^3
+ 52*a^2*x^2 + 4*a*x + 8)/(a^3*x^6 - a^2*x^5 - a*x^4 + x^3)

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Fricas [B]  time = 2.03898, size = 378, normalized size = 3.71 \begin{align*} -\frac{210 \, a^{5} x^{5} - 138 \, a^{4} x^{4} - 212 \, a^{3} x^{3} + 104 \, a^{2} x^{2} + 8 \, a x - 33 \,{\left (a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (a x + 1\right ) + 177 \,{\left (a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (a x - 1\right ) - 144 \,{\left (a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (x\right ) + 16}{48 \,{\left (a^{3} x^{6} - a^{2} x^{5} - a x^{4} + x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^3/x^4,x, algorithm="fricas")

[Out]

-1/48*(210*a^5*x^5 - 138*a^4*x^4 - 212*a^3*x^3 + 104*a^2*x^2 + 8*a*x - 33*(a^6*x^6 - a^5*x^5 - a^4*x^4 + a^3*x
^3)*log(a*x + 1) + 177*(a^6*x^6 - a^5*x^5 - a^4*x^4 + a^3*x^3)*log(a*x - 1) - 144*(a^6*x^6 - a^5*x^5 - a^4*x^4
+ a^3*x^3)*log(x) + 16)/(a^3*x^6 - a^2*x^5 - a*x^4 + x^3)

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Sympy [A]  time = 1.17001, size = 104, normalized size = 1.02 \begin{align*} 3 a^{3} \log{\left (x \right )} - \frac{59 a^{3} \log{\left (x - \frac{1}{a} \right )}}{16} + \frac{11 a^{3} \log{\left (x + \frac{1}{a} \right )}}{16} - \frac{105 a^{5} x^{5} - 69 a^{4} x^{4} - 106 a^{3} x^{3} + 52 a^{2} x^{2} + 4 a x + 8}{24 a^{3} x^{6} - 24 a^{2} x^{5} - 24 a x^{4} + 24 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a**2*x**2+1)**3/x**4,x)

[Out]

3*a**3*log(x) - 59*a**3*log(x - 1/a)/16 + 11*a**3*log(x + 1/a)/16 - (105*a**5*x**5 - 69*a**4*x**4 - 106*a**3*x
**3 + 52*a**2*x**2 + 4*a*x + 8)/(24*a**3*x**6 - 24*a**2*x**5 - 24*a*x**4 + 24*x**3)

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Giac [A]  time = 1.17724, size = 122, normalized size = 1.2 \begin{align*} \frac{11}{16} \, a^{3} \log \left ({\left | a x + 1 \right |}\right ) - \frac{59}{16} \, a^{3} \log \left ({\left | a x - 1 \right |}\right ) + 3 \, a^{3} \log \left ({\left | x \right |}\right ) - \frac{105 \, a^{5} x^{5} - 69 \, a^{4} x^{4} - 106 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 4 \, a x + 8}{24 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^3/x^4,x, algorithm="giac")

[Out]

11/16*a^3*log(abs(a*x + 1)) - 59/16*a^3*log(abs(a*x - 1)) + 3*a^3*log(abs(x)) - 1/24*(105*a^5*x^5 - 69*a^4*x^4
- 106*a^3*x^3 + 52*a^2*x^2 + 4*a*x + 8)/((a*x + 1)*(a*x - 1)^2*x^3)