3.1059 \(\int \frac {e^{2 \tanh ^{-1}(a x)} x^3}{(c-a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=70 \[ -\frac {5}{4 a^4 c^2 (1-a x)}+\frac {1}{4 a^4 c^2 (1-a x)^2}-\frac {7 \log (1-a x)}{8 a^4 c^2}-\frac {\log (a x+1)}{8 a^4 c^2} \]

[Out]

1/4/a^4/c^2/(-a*x+1)^2-5/4/a^4/c^2/(-a*x+1)-7/8*ln(-a*x+1)/a^4/c^2-1/8*ln(a*x+1)/a^4/c^2

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 70, 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} \[ -\frac {5}{4 a^4 c^2 (1-a x)}+\frac {1}{4 a^4 c^2 (1-a x)^2}-\frac {7 \log (1-a x)}{8 a^4 c^2}-\frac {\log (a x+1)}{8 a^4 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(4*a^4*c^2*(1 - a*x)^2) - 5/(4*a^4*c^2*(1 - a*x)) - (7*Log[1 - a*x])/(8*a^4*c^2) - Log[1 + a*x]/(8*a^4*c^2)

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

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 53, normalized size = 0.76 \[ -\frac {-10 a x+7 (a x-1)^2 \log (1-a x)+(a x-1)^2 \log (a x+1)+8}{8 a^4 c^2 (a x-1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(8 - 10*a*x + 7*(-1 + a*x)^2*Log[1 - a*x] + (-1 + a*x)^2*Log[1 + a*x])/(a^4*c^2*(-1 + a*x)^2)

________________________________________________________________________________________

fricas [A]  time = 0.50, size = 79, normalized size = 1.13 \[ \frac {10 \, a x - {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) - 7 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 8}{8 \, {\left (a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

1/8*(10*a*x - (a^2*x^2 - 2*a*x + 1)*log(a*x + 1) - 7*(a^2*x^2 - 2*a*x + 1)*log(a*x - 1) - 8)/(a^6*c^2*x^2 - 2*
a^5*c^2*x + a^4*c^2)

________________________________________________________________________________________

giac [A]  time = 0.34, size = 52, normalized size = 0.74 \[ -\frac {\log \left ({\left | a x + 1 \right |}\right )}{8 \, a^{4} c^{2}} - \frac {7 \, \log \left ({\left | a x - 1 \right |}\right )}{8 \, a^{4} c^{2}} + \frac {5 \, a x - 4}{4 \, {\left (a x - 1\right )}^{2} a^{4} c^{2}} \]

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)^2,x, algorithm="giac")

[Out]

-1/8*log(abs(a*x + 1))/(a^4*c^2) - 7/8*log(abs(a*x - 1))/(a^4*c^2) + 1/4*(5*a*x - 4)/((a*x - 1)^2*a^4*c^2)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 60, normalized size = 0.86 \[ \frac {1}{4 c^{2} a^{4} \left (a x -1\right )^{2}}+\frac {5}{4 c^{2} a^{4} \left (a x -1\right )}-\frac {7 \ln \left (a x -1\right )}{8 c^{2} a^{4}}-\frac {\ln \left (a x +1\right )}{8 a^{4} c^{2}} \]

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)^2,x)

[Out]

1/4/c^2/a^4/(a*x-1)^2+5/4/c^2/a^4/(a*x-1)-7/8/c^2/a^4*ln(a*x-1)-1/8*ln(a*x+1)/a^4/c^2

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 66, normalized size = 0.94 \[ \frac {5 \, a x - 4}{4 \, {\left (a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right )}} - \frac {\log \left (a x + 1\right )}{8 \, a^{4} c^{2}} - \frac {7 \, \log \left (a x - 1\right )}{8 \, a^{4} c^{2}} \]

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)^2,x, algorithm="maxima")

[Out]

1/4*(5*a*x - 4)/(a^6*c^2*x^2 - 2*a^5*c^2*x + a^4*c^2) - 1/8*log(a*x + 1)/(a^4*c^2) - 7/8*log(a*x - 1)/(a^4*c^2
)

________________________________________________________________________________________

mupad [B]  time = 0.21, size = 65, normalized size = 0.93 \[ \frac {\frac {5\,x}{4\,a^3}-\frac {1}{a^4}}{a^2\,c^2\,x^2-2\,a\,c^2\,x+c^2}-\frac {7\,\ln \left (a\,x-1\right )}{8\,a^4\,c^2}-\frac {\ln \left (a\,x+1\right )}{8\,a^4\,c^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*x)/(4*a^3) - 1/a^4)/(c^2 + a^2*c^2*x^2 - 2*a*c^2*x) - (7*log(a*x - 1))/(8*a^4*c^2) - log(a*x + 1)/(8*a^4*c
^2)

________________________________________________________________________________________

sympy [A]  time = 0.35, size = 63, normalized size = 0.90 \[ - \frac {- 5 a x + 4}{4 a^{6} c^{2} x^{2} - 8 a^{5} c^{2} x + 4 a^{4} c^{2}} - \frac {\frac {7 \log {\left (x - \frac {1}{a} \right )}}{8} + \frac {\log {\left (x + \frac {1}{a} \right )}}{8}}{a^{4} c^{2}} \]

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)**2,x)

[Out]

-(-5*a*x + 4)/(4*a**6*c**2*x**2 - 8*a**5*c**2*x + 4*a**4*c**2) - (7*log(x - 1/a)/8 + log(x + 1/a)/8)/(a**4*c**
2)

________________________________________________________________________________________