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

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

[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 = 0.02, 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 = {6037, 308, 212} \begin {gather*} \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} \end {gather*}

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 212

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

Rule 308

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 6037

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTanh[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))
), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 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 \begin {gather*} \frac {b x}{6 c^5}+\frac {b x^3}{18 c^3}+\frac {b x^5}{30 c}+\frac {a x^6}{6}+\frac {1}{6} b x^6 \tanh ^{-1}(c x)+\frac {b \log (1-c x)}{12 c^6}-\frac {b \log (1+c x)}{12 c^6} \end {gather*}

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 = 0.06, size = 69, normalized size = 1.17

method	result	size
derivativedivides	\(\frac {\frac {c^{6} x^{6} a}{6}+\frac {b \,c^{6} x^{6} \arctanh \left (c x \right )}{6}+\frac {c^{5} x^{5} b}{30}+\frac {b \,c^{3} x^{3}}{18}+\frac {b c x}{6}+\frac {b \ln \left (c x -1\right )}{12}-\frac {b \ln \left (c x +1\right )}{12}}{c^{6}}\)	\(69\)
default	\(\frac {\frac {c^{6} x^{6} a}{6}+\frac {b \,c^{6} x^{6} \arctanh \left (c x \right )}{6}+\frac {c^{5} x^{5} b}{30}+\frac {b \,c^{3} x^{3}}{18}+\frac {b c x}{6}+\frac {b \ln \left (c x -1\right )}{12}-\frac {b \ln \left (c x +1\right )}{12}}{c^{6}}\)	\(69\)
risch	\(\frac {x^{6} b \ln \left (c x +1\right )}{12}-\frac {x^{6} b \ln \left (-c x +1\right )}{12}+\frac {x^{6} a}{6}+\frac {b \,x^{5}}{30 c}+\frac {b \,x^{3}}{18 c^{3}}+\frac {b x}{6 c^{5}}+\frac {b \ln \left (-c x +1\right )}{12 c^{6}}-\frac {b \ln \left (c x +1\right )}{12 c^{6}}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 70, normalized size = 1.19 \begin {gather*} \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 {gather*}

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 = 0.33, size = 67, normalized size = 1.14 \begin {gather*} \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 {gather*}

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 = 0.39, size = 63, normalized size = 1.07 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (49) = 98\).
time = 0.46, size = 442, normalized size = 7.49 \begin {gather*} \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 )} \end {gather*}

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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Mupad [B]
time = 0.84, size = 52, normalized size = 0.88 \begin {gather*} \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} \end {gather*}

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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