3.100 \(\int x^8 (a+b \tanh ^{-1}(c x^3)) \, dx\)

Optimal. Leaf size=48 \[ \frac{1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^2 x^6\right )}{18 c^3}+\frac{b x^6}{18 c} \]

[Out]

(b*x^6)/(18*c) + (x^9*(a + b*ArcTanh[c*x^3]))/9 + (b*Log[1 - c^2*x^6])/(18*c^3)

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Rubi [A]  time = 0.0359128, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6097, 266, 43} \[ \frac{1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^2 x^6\right )}{18 c^3}+\frac{b x^6}{18 c} \]

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*ArcTanh[c*x^3]),x]

[Out]

(b*x^6)/(18*c) + (x^9*(a + b*ArcTanh[c*x^3]))/9 + (b*Log[1 - c^2*x^6])/(18*c^3)

Rule 6097

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
nh[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 - c^2*x^(2*n)), x], x
] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^8 \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{1}{3} (b c) \int \frac{x^{11}}{1-c^2 x^6} \, dx\\ &=\frac{1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^6\right )\\ &=\frac{1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^6\right )\\ &=\frac{b x^6}{18 c}+\frac{1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^2 x^6\right )}{18 c^3}\\ \end{align*}

Mathematica [A]  time = 0.0158537, size = 53, normalized size = 1.1 \[ \frac{a x^9}{9}+\frac{b \log \left (1-c^2 x^6\right )}{18 c^3}+\frac{b x^6}{18 c}+\frac{1}{9} b x^9 \tanh ^{-1}\left (c x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*ArcTanh[c*x^3]),x]

[Out]

(b*x^6)/(18*c) + (a*x^9)/9 + (b*x^9*ArcTanh[c*x^3])/9 + (b*Log[1 - c^2*x^6])/(18*c^3)

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Maple [A]  time = 0.008, size = 45, normalized size = 0.9 \begin{align*}{\frac{{x}^{9}a}{9}}+{\frac{b{x}^{9}{\it Artanh} \left ( c{x}^{3} \right ) }{9}}+{\frac{b{x}^{6}}{18\,c}}+{\frac{b\ln \left ({c}^{2}{x}^{6}-1 \right ) }{18\,{c}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(a+b*arctanh(c*x^3)),x)

[Out]

1/9*x^9*a+1/9*b*x^9*arctanh(c*x^3)+1/18*b*x^6/c+1/18*b/c^3*ln(c^2*x^6-1)

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Maxima [A]  time = 1.02131, size = 62, normalized size = 1.29 \begin{align*} \frac{1}{9} \, a x^{9} + \frac{1}{18} \,{\left (2 \, x^{9} \operatorname{artanh}\left (c x^{3}\right ) +{\left (\frac{x^{6}}{c^{2}} + \frac{\log \left (c^{2} x^{6} - 1\right )}{c^{4}}\right )} c\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(a+b*arctanh(c*x^3)),x, algorithm="maxima")

[Out]

1/9*a*x^9 + 1/18*(2*x^9*arctanh(c*x^3) + (x^6/c^2 + log(c^2*x^6 - 1)/c^4)*c)*b

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Fricas [A]  time = 1.91646, size = 134, normalized size = 2.79 \begin{align*} \frac{b c^{3} x^{9} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + 2 \, a c^{3} x^{9} + b c^{2} x^{6} + b \log \left (c^{2} x^{6} - 1\right )}{18 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(a+b*arctanh(c*x^3)),x, algorithm="fricas")

[Out]

1/18*(b*c^3*x^9*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 2*a*c^3*x^9 + b*c^2*x^6 + b*log(c^2*x^6 - 1))/c^3

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: KeyError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(a+b*atanh(c*x**3)),x)

[Out]

Exception raised: KeyError

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Giac [A]  time = 1.19606, size = 77, normalized size = 1.6 \begin{align*} \frac{1}{18} \, b x^{9} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + \frac{1}{9} \, a x^{9} + \frac{b x^{6}}{18 \, c} + \frac{b \log \left (c^{2} x^{6} - 1\right )}{18 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(a+b*arctanh(c*x^3)),x, algorithm="giac")

[Out]

1/18*b*x^9*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 1/9*a*x^9 + 1/18*b*x^6/c + 1/18*b*log(c^2*x^6 - 1)/c^3