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

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 81, normalized size = 1.37 \[ \frac {a x^6}{6}+\frac {b \log (1-c x)}{12 c^6}-\frac {b \log (c x+1)}{12 c^6}+\frac {b x}{6 c^5}+\frac {b x^3}{18 c^3}+\frac {1}{6} b x^6 \tanh ^{-1}(c x)+\frac {b x^5}{30 c} \]

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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fricas [A]  time = 1.33, size = 67, normalized size = 1.14 \[ \frac {30 \, a c^{6} x^{6} + 6 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 30 \, b c x + 15 \, {\left (b c^{6} x^{6} - b\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{180 \, c^{6}} \]

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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giac [B]  time = 0.31, size = 442, normalized size = 7.49 \[ \frac {1}{45} \, c {\left (\frac {15 \, {\left (\frac {3 \, {\left (c x + 1\right )}^{5} b}{{\left (c x - 1\right )}^{5}} + \frac {10 \, {\left (c x + 1\right )}^{3} b}{{\left (c x - 1\right )}^{3}} + \frac {3 \, {\left (c x + 1\right )} b}{c x - 1}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{6} c^{7}}{{\left (c x - 1\right )}^{6}} - \frac {6 \, {\left (c x + 1\right )}^{5} c^{7}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4} c^{7}}{{\left (c x - 1\right )}^{4}} - \frac {20 \, {\left (c x + 1\right )}^{3} c^{7}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} c^{7}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} c^{7}}{c x - 1} + c^{7}} + \frac {\frac {90 \, {\left (c x + 1\right )}^{5} a}{{\left (c x - 1\right )}^{5}} + \frac {300 \, {\left (c x + 1\right )}^{3} a}{{\left (c x - 1\right )}^{3}} + \frac {90 \, {\left (c x + 1\right )} a}{c x - 1} + \frac {45 \, {\left (c x + 1\right )}^{5} b}{{\left (c x - 1\right )}^{5}} - \frac {135 \, {\left (c x + 1\right )}^{4} b}{{\left (c x - 1\right )}^{4}} + \frac {230 \, {\left (c x + 1\right )}^{3} b}{{\left (c x - 1\right )}^{3}} - \frac {210 \, {\left (c x + 1\right )}^{2} b}{{\left (c x - 1\right )}^{2}} + \frac {93 \, {\left (c x + 1\right )} b}{c x - 1} - 23 \, b}{\frac {{\left (c x + 1\right )}^{6} c^{7}}{{\left (c x - 1\right )}^{6}} - \frac {6 \, {\left (c x + 1\right )}^{5} c^{7}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4} c^{7}}{{\left (c x - 1\right )}^{4}} - \frac {20 \, {\left (c x + 1\right )}^{3} c^{7}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} c^{7}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} c^{7}}{c x - 1} + c^{7}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45*c*(15*(3*(c*x + 1)^5*b/(c*x - 1)^5 + 10*(c*x + 1)^3*b/(c*x - 1)^3 + 3*(c*x + 1)*b/(c*x - 1))*log(-(c*x +
1)/(c*x - 1))/((c*x + 1)^6*c^7/(c*x - 1)^6 - 6*(c*x + 1)^5*c^7/(c*x - 1)^5 + 15*(c*x + 1)^4*c^7/(c*x - 1)^4 -
20*(c*x + 1)^3*c^7/(c*x - 1)^3 + 15*(c*x + 1)^2*c^7/(c*x - 1)^2 - 6*(c*x + 1)*c^7/(c*x - 1) + c^7) + (90*(c*x
+ 1)^5*a/(c*x - 1)^5 + 300*(c*x + 1)^3*a/(c*x - 1)^3 + 90*(c*x + 1)*a/(c*x - 1) + 45*(c*x + 1)^5*b/(c*x - 1)^5
- 135*(c*x + 1)^4*b/(c*x - 1)^4 + 230*(c*x + 1)^3*b/(c*x - 1)^3 - 210*(c*x + 1)^2*b/(c*x - 1)^2 + 93*(c*x + 1
)*b/(c*x - 1) - 23*b)/((c*x + 1)^6*c^7/(c*x - 1)^6 - 6*(c*x + 1)^5*c^7/(c*x - 1)^5 + 15*(c*x + 1)^4*c^7/(c*x -
1)^4 - 20*(c*x + 1)^3*c^7/(c*x - 1)^3 + 15*(c*x + 1)^2*c^7/(c*x - 1)^2 - 6*(c*x + 1)*c^7/(c*x - 1) + c^7))

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maple [A]  time = 0.01, size = 67, normalized size = 1.14 \[ \frac {x^{6} a}{6}+\frac {b \,x^{6} \arctanh \left (c x \right )}{6}+\frac {b \,x^{5}}{30 c}+\frac {b \,x^{3}}{18 c^{3}}+\frac {b x}{6 c^{5}}+\frac {b \ln \left (c x -1\right )}{12 c^{6}}-\frac {b \ln \left (c x +1\right )}{12 c^{6}} \]

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 = 0.31, size = 70, normalized size = 1.19 \[ \frac {1}{6} \, a x^{6} + \frac {1}{180} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b \]

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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mupad [B]  time = 0.84, size = 52, normalized size = 0.88 \[ \frac {\frac {b\,c^3\,x^3}{18}-\frac {b\,\mathrm {atanh}\left (c\,x\right )}{6}+\frac {b\,c^5\,x^5}{30}+\frac {b\,c\,x}{6}}{c^6}+\frac {a\,x^6}{6}+\frac {b\,x^6\,\mathrm {atanh}\left (c\,x\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*atanh(c*x)),x)

[Out]

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

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sympy [A]  time = 1.62, size = 63, normalized size = 1.07 \[ \begin {cases} \frac {a x^{6}}{6} + \frac {b x^{6} \operatorname {atanh}{\left (c x \right )}}{6} + \frac {b x^{5}}{30 c} + \frac {b x^{3}}{18 c^{3}} + \frac {b x}{6 c^{5}} - \frac {b \operatorname {atanh}{\left (c x \right )}}{6 c^{6}} & \text {for}\: c \neq 0 \\\frac {a x^{6}}{6} & \text {otherwise} \end {cases} \]

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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