3.1 \(\int x^5 (a+b \tanh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=59 \[ \frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{b x^3}{18 c^3}+\frac{b x}{6 c^5}-\frac{b \tanh ^{-1}(c x)}{6 c^6}+\frac{b x^5}{30 c} \]

[Out]

(b*x)/(6*c^5) + (b*x^3)/(18*c^3) + (b*x^5)/(30*c) - (b*ArcTanh[c*x])/(6*c^6) + (x^6*(a + b*ArcTanh[c*x]))/6

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Rubi [A]  time = 0.0329148, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5916, 302, 206} \[ \frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{b x^3}{18 c^3}+\frac{b x}{6 c^5}-\frac{b \tanh ^{-1}(c x)}{6 c^6}+\frac{b x^5}{30 c} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*ArcTanh[c*x]),x]

[Out]

(b*x)/(6*c^5) + (b*x^3)/(18*c^3) + (b*x^5)/(30*c) - (b*ArcTanh[c*x])/(6*c^6) + (x^6*(a + b*ArcTanh[c*x]))/6

Rule 5916

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcT
anh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x^5 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{6} (b c) \int \frac{x^6}{1-c^2 x^2} \, dx\\ &=\frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{6} (b c) \int \left (-\frac{1}{c^6}-\frac{x^2}{c^4}-\frac{x^4}{c^2}+\frac{1}{c^6 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac{b x}{6 c^5}+\frac{b x^3}{18 c^3}+\frac{b x^5}{30 c}+\frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac{b \int \frac{1}{1-c^2 x^2} \, dx}{6 c^5}\\ &=\frac{b x}{6 c^5}+\frac{b x^3}{18 c^3}+\frac{b x^5}{30 c}-\frac{b \tanh ^{-1}(c x)}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0095314, size = 81, normalized size = 1.37 \[ \frac{a x^6}{6}+\frac{b x^3}{18 c^3}+\frac{b x}{6 c^5}+\frac{b \log (1-c x)}{12 c^6}-\frac{b \log (c x+1)}{12 c^6}+\frac{b x^5}{30 c}+\frac{1}{6} b x^6 \tanh ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*ArcTanh[c*x]),x]

[Out]

(b*x)/(6*c^5) + (b*x^3)/(18*c^3) + (b*x^5)/(30*c) + (a*x^6)/6 + (b*x^6*ArcTanh[c*x])/6 + (b*Log[1 - c*x])/(12*
c^6) - (b*Log[1 + c*x])/(12*c^6)

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Maple [A]  time = 0.007, size = 67, normalized size = 1.1 \begin{align*}{\frac{{x}^{6}a}{6}}+{\frac{b{x}^{6}{\it Artanh} \left ( cx \right ) }{6}}+{\frac{b{x}^{5}}{30\,c}}+{\frac{b{x}^{3}}{18\,{c}^{3}}}+{\frac{bx}{6\,{c}^{5}}}+{\frac{b\ln \left ( cx-1 \right ) }{12\,{c}^{6}}}-{\frac{b\ln \left ( cx+1 \right ) }{12\,{c}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arctanh(c*x)),x)

[Out]

1/6*x^6*a+1/6*b*x^6*arctanh(c*x)+1/30*b*x^5/c+1/18*b*x^3/c^3+1/6*b*x/c^5+1/12/c^6*b*ln(c*x-1)-1/12/c^6*b*ln(c*
x+1)

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Maxima [A]  time = 0.969474, size = 95, normalized size = 1.61 \begin{align*} \frac{1}{6} \, a x^{6} + \frac{1}{180} \,{\left (30 \, x^{6} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \,{\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac{15 \, \log \left (c x + 1\right )}{c^{7}} + \frac{15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctanh(c*x)),x, algorithm="maxima")

[Out]

1/6*a*x^6 + 1/180*(30*x^6*arctanh(c*x) + c*(2*(3*c^4*x^5 + 5*c^2*x^3 + 15*x)/c^6 - 15*log(c*x + 1)/c^7 + 15*lo
g(c*x - 1)/c^7))*b

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Fricas [A]  time = 1.95422, size = 153, normalized size = 2.59 \begin{align*} \frac{30 \, a c^{6} x^{6} + 6 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 30 \, b c x + 15 \,{\left (b c^{6} x^{6} - b\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{180 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctanh(c*x)),x, algorithm="fricas")

[Out]

1/180*(30*a*c^6*x^6 + 6*b*c^5*x^5 + 10*b*c^3*x^3 + 30*b*c*x + 15*(b*c^6*x^6 - b)*log(-(c*x + 1)/(c*x - 1)))/c^
6

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Sympy [A]  time = 2.13222, size = 63, normalized size = 1.07 \begin{align*} \begin{cases} \frac{a x^{6}}{6} + \frac{b x^{6} \operatorname{atanh}{\left (c x \right )}}{6} + \frac{b x^{5}}{30 c} + \frac{b x^{3}}{18 c^{3}} + \frac{b x}{6 c^{5}} - \frac{b \operatorname{atanh}{\left (c x \right )}}{6 c^{6}} & \text{for}\: c \neq 0 \\\frac{a x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*atanh(c*x)),x)

[Out]

Piecewise((a*x**6/6 + b*x**6*atanh(c*x)/6 + b*x**5/(30*c) + b*x**3/(18*c**3) + b*x/(6*c**5) - b*atanh(c*x)/(6*
c**6), Ne(c, 0)), (a*x**6/6, True))

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Giac [A]  time = 1.173, size = 104, normalized size = 1.76 \begin{align*} \frac{1}{12} \, b x^{6} \log \left (-\frac{c x + 1}{c x - 1}\right ) + \frac{1}{6} \, a x^{6} + \frac{b x^{5}}{30 \, c} + \frac{b x^{3}}{18 \, c^{3}} + \frac{b x}{6 \, c^{5}} - \frac{b \log \left (c x + 1\right )}{12 \, c^{6}} + \frac{b \log \left (c x - 1\right )}{12 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctanh(c*x)),x, algorithm="giac")

[Out]

1/12*b*x^6*log(-(c*x + 1)/(c*x - 1)) + 1/6*a*x^6 + 1/30*b*x^5/c + 1/18*b*x^3/c^3 + 1/6*b*x/c^5 - 1/12*b*log(c*
x + 1)/c^6 + 1/12*b*log(c*x - 1)/c^6