3.1067 \(\int \frac{e^{2 \tanh ^{-1}(a x)} x^5}{(c-a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=105 \[ \frac{23}{16 a^6 c^3 (1-a x)}+\frac{1}{16 a^6 c^3 (a x+1)}-\frac{1}{2 a^6 c^3 (1-a x)^2}+\frac{1}{12 a^6 c^3 (1-a x)^3}+\frac{13 \log (1-a x)}{16 a^6 c^3}+\frac{3 \log (a x+1)}{16 a^6 c^3} \]

[Out]

1/(12*a^6*c^3*(1 - a*x)^3) - 1/(2*a^6*c^3*(1 - a*x)^2) + 23/(16*a^6*c^3*(1 - a*x)) + 1/(16*a^6*c^3*(1 + a*x))
+ (13*Log[1 - a*x])/(16*a^6*c^3) + (3*Log[1 + a*x])/(16*a^6*c^3)

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Rubi [A]  time = 0.133259, antiderivative size = 105, 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} \[ \frac{23}{16 a^6 c^3 (1-a x)}+\frac{1}{16 a^6 c^3 (a x+1)}-\frac{1}{2 a^6 c^3 (1-a x)^2}+\frac{1}{12 a^6 c^3 (1-a x)^3}+\frac{13 \log (1-a x)}{16 a^6 c^3}+\frac{3 \log (a x+1)}{16 a^6 c^3} \]

Antiderivative was successfully verified.

[In]

Int[(E^(2*ArcTanh[a*x])*x^5)/(c - a^2*c*x^2)^3,x]

[Out]

1/(12*a^6*c^3*(1 - a*x)^3) - 1/(2*a^6*c^3*(1 - a*x)^2) + 23/(16*a^6*c^3*(1 - a*x)) + 1/(16*a^6*c^3*(1 + a*x))
+ (13*Log[1 - a*x])/(16*a^6*c^3) + (3*Log[1 + a*x])/(16*a^6*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^5}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x^5}{(1-a x)^4 (1+a x)^2} \, dx}{c^3}\\ &=\frac{\int \left (\frac{1}{4 a^5 (-1+a x)^4}+\frac{1}{a^5 (-1+a x)^3}+\frac{23}{16 a^5 (-1+a x)^2}+\frac{13}{16 a^5 (-1+a x)}-\frac{1}{16 a^5 (1+a x)^2}+\frac{3}{16 a^5 (1+a x)}\right ) \, dx}{c^3}\\ &=\frac{1}{12 a^6 c^3 (1-a x)^3}-\frac{1}{2 a^6 c^3 (1-a x)^2}+\frac{23}{16 a^6 c^3 (1-a x)}+\frac{1}{16 a^6 c^3 (1+a x)}+\frac{13 \log (1-a x)}{16 a^6 c^3}+\frac{3 \log (1+a x)}{16 a^6 c^3}\\ \end{align*}

Mathematica [A]  time = 0.053934, size = 87, normalized size = 0.83 \[ \frac{-66 a^3 x^3+36 a^2 x^2+74 a x+39 (a x-1)^3 (a x+1) \log (1-a x)+9 (a x-1)^3 (a x+1) \log (a x+1)-52}{48 a^6 c^3 (a x-1)^3 (a x+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*ArcTanh[a*x])*x^5)/(c - a^2*c*x^2)^3,x]

[Out]

(-52 + 74*a*x + 36*a^2*x^2 - 66*a^3*x^3 + 39*(-1 + a*x)^3*(1 + a*x)*Log[1 - a*x] + 9*(-1 + a*x)^3*(1 + a*x)*Lo
g[1 + a*x])/(48*a^6*c^3*(-1 + a*x)^3*(1 + a*x))

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Maple [A]  time = 0.039, size = 90, normalized size = 0.9 \begin{align*}{\frac{1}{16\,{a}^{6}{c}^{3} \left ( ax+1 \right ) }}+{\frac{3\,\ln \left ( ax+1 \right ) }{16\,{a}^{6}{c}^{3}}}-{\frac{1}{2\,{a}^{6}{c}^{3} \left ( ax-1 \right ) ^{2}}}-{\frac{1}{12\,{a}^{6}{c}^{3} \left ( ax-1 \right ) ^{3}}}-{\frac{23}{16\,{a}^{6}{c}^{3} \left ( ax-1 \right ) }}+{\frac{13\,\ln \left ( ax-1 \right ) }{16\,{a}^{6}{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^5/(-a^2*c*x^2+c)^3,x)

[Out]

1/16/a^6/c^3/(a*x+1)+3/16*ln(a*x+1)/a^6/c^3-1/2/c^3/a^6/(a*x-1)^2-1/12/c^3/a^6/(a*x-1)^3-23/16/c^3/a^6/(a*x-1)
+13/16/c^3/a^6*ln(a*x-1)

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Maxima [A]  time = 0.959164, size = 127, normalized size = 1.21 \begin{align*} -\frac{33 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 37 \, a x + 26}{24 \,{\left (a^{10} c^{3} x^{4} - 2 \, a^{9} c^{3} x^{3} + 2 \, a^{7} c^{3} x - a^{6} c^{3}\right )}} + \frac{3 \, \log \left (a x + 1\right )}{16 \, a^{6} c^{3}} + \frac{13 \, \log \left (a x - 1\right )}{16 \, a^{6} 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^5/(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/24*(33*a^3*x^3 - 18*a^2*x^2 - 37*a*x + 26)/(a^10*c^3*x^4 - 2*a^9*c^3*x^3 + 2*a^7*c^3*x - a^6*c^3) + 3/16*lo
g(a*x + 1)/(a^6*c^3) + 13/16*log(a*x - 1)/(a^6*c^3)

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Fricas [A]  time = 2.37255, size = 277, normalized size = 2.64 \begin{align*} -\frac{66 \, a^{3} x^{3} - 36 \, a^{2} x^{2} - 74 \, a x - 9 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x + 1\right ) - 39 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x - 1\right ) + 52}{48 \,{\left (a^{10} c^{3} x^{4} - 2 \, a^{9} c^{3} x^{3} + 2 \, a^{7} c^{3} x - a^{6} c^{3}\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^5/(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

-1/48*(66*a^3*x^3 - 36*a^2*x^2 - 74*a*x - 9*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x + 1) - 39*(a^4*x^4 - 2*a
^3*x^3 + 2*a*x - 1)*log(a*x - 1) + 52)/(a^10*c^3*x^4 - 2*a^9*c^3*x^3 + 2*a^7*c^3*x - a^6*c^3)

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Sympy [A]  time = 0.786879, size = 92, normalized size = 0.88 \begin{align*} - \frac{33 a^{3} x^{3} - 18 a^{2} x^{2} - 37 a x + 26}{24 a^{10} c^{3} x^{4} - 48 a^{9} c^{3} x^{3} + 48 a^{7} c^{3} x - 24 a^{6} c^{3}} + \frac{\frac{13 \log{\left (x - \frac{1}{a} \right )}}{16} + \frac{3 \log{\left (x + \frac{1}{a} \right )}}{16}}{a^{6} 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**5/(-a**2*c*x**2+c)**3,x)

[Out]

-(33*a**3*x**3 - 18*a**2*x**2 - 37*a*x + 26)/(24*a**10*c**3*x**4 - 48*a**9*c**3*x**3 + 48*a**7*c**3*x - 24*a**
6*c**3) + (13*log(x - 1/a)/16 + 3*log(x + 1/a)/16)/(a**6*c**3)

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Giac [A]  time = 1.15004, size = 101, normalized size = 0.96 \begin{align*} \frac{3 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{6} c^{3}} + \frac{13 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{6} c^{3}} - \frac{33 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 37 \, a x + 26}{24 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{3} a^{6} 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^5/(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

3/16*log(abs(a*x + 1))/(a^6*c^3) + 13/16*log(abs(a*x - 1))/(a^6*c^3) - 1/24*(33*a^3*x^3 - 18*a^2*x^2 - 37*a*x
+ 26)/((a*x + 1)*(a*x - 1)^3*a^6*c^3)