3.351 \(\int \frac{x^6}{(1-a^2 x^2)^4 \tanh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=55 \[ \frac{15 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{32 a^7}-\frac{3 \text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{16 a^7}+\frac{\text{Chi}\left (6 \tanh ^{-1}(a x)\right )}{32 a^7}-\frac{5 \log \left (\tanh ^{-1}(a x)\right )}{16 a^7} \]

[Out]

(15*CoshIntegral[2*ArcTanh[a*x]])/(32*a^7) - (3*CoshIntegral[4*ArcTanh[a*x]])/(16*a^7) + CoshIntegral[6*ArcTan
h[a*x]]/(32*a^7) - (5*Log[ArcTanh[a*x]])/(16*a^7)

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Rubi [A]  time = 0.145157, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6034, 3312, 3301} \[ \frac{15 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{32 a^7}-\frac{3 \text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{16 a^7}+\frac{\text{Chi}\left (6 \tanh ^{-1}(a x)\right )}{32 a^7}-\frac{5 \log \left (\tanh ^{-1}(a x)\right )}{16 a^7} \]

Antiderivative was successfully verified.

[In]

Int[x^6/((1 - a^2*x^2)^4*ArcTanh[a*x]),x]

[Out]

(15*CoshIntegral[2*ArcTanh[a*x]])/(32*a^7) - (3*CoshIntegral[4*ArcTanh[a*x]])/(16*a^7) + CoshIntegral[6*ArcTan
h[a*x]]/(32*a^7) - (5*Log[ArcTanh[a*x]])/(16*a^7)

Rule 6034

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^q/c^(
m + 1), Subst[Int[((a + b*x)^p*Sinh[x]^m)/Cosh[x]^(m + 2*(q + 1)), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c,
d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \frac{x^6}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sinh ^6(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^7}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{5}{16 x}-\frac{15 \cosh (2 x)}{32 x}+\frac{3 \cosh (4 x)}{16 x}-\frac{\cosh (6 x)}{32 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^7}\\ &=-\frac{5 \log \left (\tanh ^{-1}(a x)\right )}{16 a^7}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (6 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a^7}-\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a^7}+\frac{15 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a^7}\\ &=\frac{15 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{32 a^7}-\frac{3 \text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{16 a^7}+\frac{\text{Chi}\left (6 \tanh ^{-1}(a x)\right )}{32 a^7}-\frac{5 \log \left (\tanh ^{-1}(a x)\right )}{16 a^7}\\ \end{align*}

Mathematica [A]  time = 0.155643, size = 40, normalized size = 0.73 \[ \frac{15 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )-6 \text{Chi}\left (4 \tanh ^{-1}(a x)\right )+\text{Chi}\left (6 \tanh ^{-1}(a x)\right )-10 \log \left (\tanh ^{-1}(a x)\right )}{32 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/((1 - a^2*x^2)^4*ArcTanh[a*x]),x]

[Out]

(15*CoshIntegral[2*ArcTanh[a*x]] - 6*CoshIntegral[4*ArcTanh[a*x]] + CoshIntegral[6*ArcTanh[a*x]] - 10*Log[ArcT
anh[a*x]])/(32*a^7)

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Maple [A]  time = 0.072, size = 48, normalized size = 0.9 \begin{align*}{\frac{15\,{\it Chi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{32\,{a}^{7}}}-{\frac{3\,{\it Chi} \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{16\,{a}^{7}}}+{\frac{{\it Chi} \left ( 6\,{\it Artanh} \left ( ax \right ) \right ) }{32\,{a}^{7}}}-{\frac{5\,\ln \left ({\it Artanh} \left ( ax \right ) \right ) }{16\,{a}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(-a^2*x^2+1)^4/arctanh(a*x),x)

[Out]

15/32*Chi(2*arctanh(a*x))/a^7-3/16*Chi(4*arctanh(a*x))/a^7+1/32*Chi(6*arctanh(a*x))/a^7-5/16*ln(arctanh(a*x))/
a^7

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname{artanh}\left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(-a^2*x^2+1)^4/arctanh(a*x),x, algorithm="maxima")

[Out]

integrate(x^6/((a^2*x^2 - 1)^4*arctanh(a*x)), x)

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Fricas [B]  time = 2.0381, size = 568, normalized size = 10.33 \begin{align*} -\frac{20 \, \log \left (\log \left (-\frac{a x + 1}{a x - 1}\right )\right ) - \logintegral \left (-\frac{a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) - \logintegral \left (-\frac{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}{a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}\right ) + 6 \, \logintegral \left (\frac{a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 6 \, \logintegral \left (\frac{a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) - 15 \, \logintegral \left (-\frac{a x + 1}{a x - 1}\right ) - 15 \, \logintegral \left (-\frac{a x - 1}{a x + 1}\right )}{64 \, a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(-a^2*x^2+1)^4/arctanh(a*x),x, algorithm="fricas")

[Out]

-1/64*(20*log(log(-(a*x + 1)/(a*x - 1))) - log_integral(-(a^3*x^3 + 3*a^2*x^2 + 3*a*x + 1)/(a^3*x^3 - 3*a^2*x^
2 + 3*a*x - 1)) - log_integral(-(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)/(a^3*x^3 + 3*a^2*x^2 + 3*a*x + 1)) + 6*log_i
ntegral((a^2*x^2 + 2*a*x + 1)/(a^2*x^2 - 2*a*x + 1)) + 6*log_integral((a^2*x^2 - 2*a*x + 1)/(a^2*x^2 + 2*a*x +
1)) - 15*log_integral(-(a*x + 1)/(a*x - 1)) - 15*log_integral(-(a*x - 1)/(a*x + 1)))/a^7

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4} \operatorname{atanh}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(-a**2*x**2+1)**4/atanh(a*x),x)

[Out]

Integral(x**6/((a*x - 1)**4*(a*x + 1)**4*atanh(a*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname{artanh}\left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(-a^2*x^2+1)^4/arctanh(a*x),x, algorithm="giac")

[Out]

integrate(x^6/((a^2*x^2 - 1)^4*arctanh(a*x)), x)