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

Optimal. Leaf size=71 \[ \frac {3 a}{4 (1-a x)}-\frac {a}{8 (a x+1)}+\frac {a}{8 (1-a x)^2}+a \log (x)-\frac {23}{16} a \log (1-a x)+\frac {7}{16} a \log (a x+1)-\frac {1}{x} \]

[Out]

-1/x+1/8*a/(-a*x+1)^2+3/4*a/(-a*x+1)-1/8*a/(a*x+1)+a*ln(x)-23/16*a*ln(-a*x+1)+7/16*a*ln(a*x+1)

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 71, normalized size of antiderivative = 1.00, 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} \[ \frac {3 a}{4 (1-a x)}-\frac {a}{8 (a x+1)}+\frac {a}{8 (1-a x)^2}+a \log (x)-\frac {23}{16} a \log (1-a x)+\frac {7}{16} a \log (a x+1)-\frac {1}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x^(-1) + a/(8*(1 - a*x)^2) + (3*a)/(4*(1 - a*x)) - a/(8*(1 + a*x)) + a*Log[x] - (23*a*Log[1 - a*x])/16 + (7*a
*Log[1 + a*x])/16

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^{\tanh ^{-1}(a x)}}{x^2 \left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac {1}{x^2 (1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {a}{x}-\frac {a^2}{4 (-1+a x)^3}+\frac {3 a^2}{4 (-1+a x)^2}-\frac {23 a^2}{16 (-1+a x)}+\frac {a^2}{8 (1+a x)^2}+\frac {7 a^2}{16 (1+a x)}\right ) \, dx\\ &=-\frac {1}{x}+\frac {a}{8 (1-a x)^2}+\frac {3 a}{4 (1-a x)}-\frac {a}{8 (1+a x)}+a \log (x)-\frac {23}{16} a \log (1-a x)+\frac {7}{16} a \log (1+a x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 65, normalized size = 0.92 \[ \frac {1}{16} \left (\frac {12 a}{1-a x}-\frac {2 a}{a x+1}+\frac {2 a}{(a x-1)^2}+16 a \log (x)-23 a \log (1-a x)+7 a \log (a x+1)-\frac {16}{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16/x + (12*a)/(1 - a*x) + (2*a)/(-1 + a*x)^2 - (2*a)/(1 + a*x) + 16*a*Log[x] - 23*a*Log[1 - a*x] + 7*a*Log[1
+ a*x])/16

________________________________________________________________________________________

fricas [B]  time = 0.65, size = 150, normalized size = 2.11 \[ -\frac {30 \, a^{3} x^{3} - 22 \, a^{2} x^{2} - 28 \, a x - 7 \, {\left (a^{4} x^{4} - a^{3} x^{3} - a^{2} x^{2} + a x\right )} \log \left (a x + 1\right ) + 23 \, {\left (a^{4} x^{4} - a^{3} x^{3} - a^{2} x^{2} + a x\right )} \log \left (a x - 1\right ) - 16 \, {\left (a^{4} x^{4} - a^{3} x^{3} - a^{2} x^{2} + a x\right )} \log \relax (x) + 16}{16 \, {\left (a^{3} x^{4} - a^{2} x^{3} - a x^{2} + x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(30*a^3*x^3 - 22*a^2*x^2 - 28*a*x - 7*(a^4*x^4 - a^3*x^3 - a^2*x^2 + a*x)*log(a*x + 1) + 23*(a^4*x^4 - a
^3*x^3 - a^2*x^2 + a*x)*log(a*x - 1) - 16*(a^4*x^4 - a^3*x^3 - a^2*x^2 + a*x)*log(x) + 16)/(a^3*x^4 - a^2*x^3
- a*x^2 + x)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 67, normalized size = 0.94 \[ \frac {7}{16} \, a \log \left ({\left | a x + 1 \right |}\right ) - \frac {23}{16} \, a \log \left ({\left | a x - 1 \right |}\right ) + a \log \left ({\left | x \right |}\right ) - \frac {15 \, a^{3} x^{3} - 11 \, a^{2} x^{2} - 14 \, a x + 8}{8 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/16*a*log(abs(a*x + 1)) - 23/16*a*log(abs(a*x - 1)) + a*log(abs(x)) - 1/8*(15*a^3*x^3 - 11*a^2*x^2 - 14*a*x +
8)/((a*x + 1)*(a*x - 1)^2*x)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 59, normalized size = 0.83 \[ -\frac {1}{x}+a \ln \relax (x )+\frac {a}{8 \left (a x -1\right )^{2}}-\frac {3 a}{4 \left (a x -1\right )}-\frac {23 a \ln \left (a x -1\right )}{16}-\frac {a}{8 \left (a x +1\right )}+\frac {7 a \ln \left (a x +1\right )}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/x+a*ln(x)+1/8*a/(a*x-1)^2-3/4*a/(a*x-1)-23/16*a*ln(a*x-1)-1/8*a/(a*x+1)+7/16*a*ln(a*x+1)

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 72, normalized size = 1.01 \[ \frac {7}{16} \, a \log \left (a x + 1\right ) - \frac {23}{16} \, a \log \left (a x - 1\right ) + a \log \relax (x) - \frac {15 \, a^{3} x^{3} - 11 \, a^{2} x^{2} - 14 \, a x + 8}{8 \, {\left (a^{3} x^{4} - a^{2} x^{3} - a x^{2} + x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/16*a*log(a*x + 1) - 23/16*a*log(a*x - 1) + a*log(x) - 1/8*(15*a^3*x^3 - 11*a^2*x^2 - 14*a*x + 8)/(a^3*x^4 -
a^2*x^3 - a*x^2 + x)

________________________________________________________________________________________

mupad [B]  time = 0.94, size = 71, normalized size = 1.00 \[ \frac {-\frac {15\,a^3\,x^3}{8}+\frac {11\,a^2\,x^2}{8}+\frac {7\,a\,x}{4}-1}{a^3\,x^4-a^2\,x^3-a\,x^2+x}+a\,\ln \relax (x)-\frac {23\,a\,\ln \left (a\,x-1\right )}{16}+\frac {7\,a\,\ln \left (a\,x+1\right )}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((7*a*x)/4 + (11*a^2*x^2)/8 - (15*a^3*x^3)/8 - 1)/(x - a*x^2 - a^2*x^3 + a^3*x^4) + a*log(x) - (23*a*log(a*x -
1))/16 + (7*a*log(a*x + 1))/16

________________________________________________________________________________________

sympy [A]  time = 0.52, size = 78, normalized size = 1.10 \[ a \log {\relax (x )} - \frac {23 a \log {\left (x - \frac {1}{a} \right )}}{16} + \frac {7 a \log {\left (x + \frac {1}{a} \right )}}{16} - \frac {15 a^{3} x^{3} - 11 a^{2} x^{2} - 14 a x + 8}{8 a^{3} x^{4} - 8 a^{2} x^{3} - 8 a x^{2} + 8 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x) - 23*a*log(x - 1/a)/16 + 7*a*log(x + 1/a)/16 - (15*a**3*x**3 - 11*a**2*x**2 - 14*a*x + 8)/(8*a**3*x**
4 - 8*a**2*x**3 - 8*a*x**2 + 8*x)

________________________________________________________________________________________