3.2.7 \(\int x^3 (a+b \tanh ^{-1}(c x^3)) \, dx\) [107]

Optimal. Leaf size=174 \[ \frac {3 b x}{4 c}+\frac {\sqrt {3} b \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{8 c^{4/3}}-\frac {\sqrt {3} b \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{8 c^{4/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{4 c^{4/3}}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}}-\frac {b \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}} \]

[Out]

3/4*b*x/c-1/4*b*arctanh(c^(1/3)*x)/c^(4/3)+1/4*x^4*(a+b*arctanh(c*x^3))+1/16*b*ln(1-c^(1/3)*x+c^(2/3)*x^2)/c^(
4/3)-1/16*b*ln(1+c^(1/3)*x+c^(2/3)*x^2)/c^(4/3)-1/8*b*arctan(-1/3*3^(1/2)+2/3*c^(1/3)*x*3^(1/2))*3^(1/2)/c^(4/
3)-1/8*b*arctan(1/3*3^(1/2)+2/3*c^(1/3)*x*3^(1/2))*3^(1/2)/c^(4/3)

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Rubi [A]
time = 0.16, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6037, 327, 216, 648, 632, 210, 642, 212} \begin {gather*} \frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {\sqrt {3} b \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{8 c^{4/3}}-\frac {\sqrt {3} b \text {ArcTan}\left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{8 c^{4/3}}+\frac {b \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{16 c^{4/3}}-\frac {b \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{16 c^{4/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{4 c^{4/3}}+\frac {3 b x}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*ArcTanh[c*x^3]),x]

[Out]

(3*b*x)/(4*c) + (Sqrt[3]*b*ArcTan[1/Sqrt[3] - (2*c^(1/3)*x)/Sqrt[3]])/(8*c^(4/3)) - (Sqrt[3]*b*ArcTan[1/Sqrt[3
] + (2*c^(1/3)*x)/Sqrt[3]])/(8*c^(4/3)) - (b*ArcTanh[c^(1/3)*x])/(4*c^(4/3)) + (x^4*(a + b*ArcTanh[c*x^3]))/4
+ (b*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(16*c^(4/3)) - (b*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(16*c^(4/3))

Rule 210

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

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 216

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n
]], k, u}, Simp[u = Int[(r - s*Cos[(2*k*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*C
os[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n))*Int[1/(r^2 - s^2*x^2), x] + Dis
t[2*(r/(a*n)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac {1}{4} (3 b c) \int \frac {x^6}{1-c^2 x^6} \, dx\\ &=\frac {3 b x}{4 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac {(3 b) \int \frac {1}{1-c^2 x^6} \, dx}{4 c}\\ &=\frac {3 b x}{4 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac {b \int \frac {1}{1-c^{2/3} x^2} \, dx}{4 c}-\frac {b \int \frac {1-\frac {\sqrt [3]{c} x}{2}}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 c}-\frac {b \int \frac {1+\frac {\sqrt [3]{c} x}{2}}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 c}\\ &=\frac {3 b x}{4 c}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{4 c^{4/3}}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \int \frac {-\sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{16 c^{4/3}}-\frac {b \int \frac {\sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{16 c^{4/3}}-\frac {(3 b) \int \frac {1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{16 c}-\frac {(3 b) \int \frac {1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{16 c}\\ &=\frac {3 b x}{4 c}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{4 c^{4/3}}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}}-\frac {b \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} x\right )}{8 c^{4/3}}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} x\right )}{8 c^{4/3}}\\ &=\frac {3 b x}{4 c}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{8 c^{4/3}}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{8 c^{4/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{4 c^{4/3}}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}}-\frac {b \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 196, normalized size = 1.13 \begin {gather*} \frac {3 b x}{4 c}+\frac {a x^4}{4}-\frac {\sqrt {3} b \text {ArcTan}\left (\frac {-1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{8 c^{4/3}}-\frac {\sqrt {3} b \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{8 c^{4/3}}+\frac {1}{4} b x^4 \tanh ^{-1}\left (c x^3\right )+\frac {b \log \left (1-\sqrt [3]{c} x\right )}{8 c^{4/3}}-\frac {b \log \left (1+\sqrt [3]{c} x\right )}{8 c^{4/3}}+\frac {b \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}}-\frac {b \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*x)/(4*c) + (a*x^4)/4 - (Sqrt[3]*b*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/(8*c^(4/3)) - (Sqrt[3]*b*ArcTan[(1
+ 2*c^(1/3)*x)/Sqrt[3]])/(8*c^(4/3)) + (b*x^4*ArcTanh[c*x^3])/4 + (b*Log[1 - c^(1/3)*x])/(8*c^(4/3)) - (b*Log[
1 + c^(1/3)*x])/(8*c^(4/3)) + (b*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(16*c^(4/3)) - (b*Log[1 + c^(1/3)*x + c^(2/
3)*x^2])/(16*c^(4/3))

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Maple [A]
time = 0.04, size = 184, normalized size = 1.06

method	result	size
default	\(\frac {x^{4} a}{4}+\frac {b \,x^{4} \arctanh \left (c \,x^{3}\right )}{4}+\frac {3 b x}{4 c}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}\)	\(184\)
risch	\(\frac {x^{4} b \ln \left (c \,x^{3}+1\right )}{8}+\frac {x^{4} a}{4}-\frac {b \,x^{4} \ln \left (-c \,x^{3}+1\right )}{8}+\frac {3 b x}{4 c}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arctanh(c*x^3)),x,method=_RETURNVERBOSE)

[Out]

1/4*x^4*a+1/4*b*x^4*arctanh(c*x^3)+3/4*b*x/c-1/8*b/c^2/(1/c)^(2/3)*ln(x+(1/c)^(1/3))+1/16*b/c^2/(1/c)^(2/3)*ln
(x^2-(1/c)^(1/3)*x+(1/c)^(2/3))-1/8*b/c^2/(1/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x-1))+1/8*b/c^
2/(1/c)^(2/3)*ln(x-(1/c)^(1/3))-1/16*b/c^2/(1/c)^(2/3)*ln(x^2+(1/c)^(1/3)*x+(1/c)^(2/3))-1/8*b/c^2/(1/c)^(2/3)
*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x+1))

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Maxima [A]
time = 0.47, size = 162, normalized size = 0.93 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {1}{16} \, {\left (4 \, x^{4} \operatorname {artanh}\left (c x^{3}\right ) - c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {7}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {7}{3}}} - \frac {12 \, x}{c^{2}} + \frac {\log \left (c^{\frac {2}{3}} x^{2} + c^{\frac {1}{3}} x + 1\right )}{c^{\frac {7}{3}}} - \frac {\log \left (c^{\frac {2}{3}} x^{2} - c^{\frac {1}{3}} x + 1\right )}{c^{\frac {7}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} x + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {7}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} x - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {7}{3}}}\right )}\right )} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*x^4 + 1/16*(4*x^4*arctanh(c*x^3) - c*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*x + c^(1/3))/c^(1/3))/c^(7
/3) + 2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*x - c^(1/3))/c^(1/3))/c^(7/3) - 12*x/c^2 + log(c^(2/3)*x^2 + c^(
1/3)*x + 1)/c^(7/3) - log(c^(2/3)*x^2 - c^(1/3)*x + 1)/c^(7/3) + 2*log((c^(1/3)*x + 1)/c^(1/3))/c^(7/3) - 2*lo
g((c^(1/3)*x - 1)/c^(1/3))/c^(7/3)))*b

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Fricas [A]
time = 0.40, size = 981, normalized size = 5.64 \begin {gather*} \left [\frac {2 \, b c^{2} x^{4} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a c^{2} x^{4} + \sqrt {3} b c \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} \log \left (\frac {2 \, c x^{3} - \sqrt {3} {\left (2 \, c x^{2} + \left (-c\right )^{\frac {2}{3}} x + \left (-c\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} + 3 \, \left (-c\right )^{\frac {1}{3}} x - 1}{c x^{3} + 1}\right ) + \sqrt {3} b c \sqrt {-\frac {1}{c^{\frac {2}{3}}}} \log \left (\frac {2 \, c x^{3} - \sqrt {3} {\left (2 \, c x^{2} - c^{\frac {2}{3}} x - c^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} - 3 \, c^{\frac {1}{3}} x + 1}{c x^{3} - 1}\right ) + 12 \, b c x + b \left (-c\right )^{\frac {2}{3}} \log \left (c x^{2} - \left (-c\right )^{\frac {2}{3}} x - \left (-c\right )^{\frac {1}{3}}\right ) - b c^{\frac {2}{3}} \log \left (c x^{2} + c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right ) - 2 \, b \left (-c\right )^{\frac {2}{3}} \log \left (c x + \left (-c\right )^{\frac {2}{3}}\right ) + 2 \, b c^{\frac {2}{3}} \log \left (c x - c^{\frac {2}{3}}\right )}{16 \, c^{2}}, \frac {2 \, b c^{2} x^{4} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a c^{2} x^{4} - 2 \, \sqrt {3} b c \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (-c\right )^{\frac {2}{3}} x + \left (-c\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}}\right ) + \sqrt {3} b c \sqrt {-\frac {1}{c^{\frac {2}{3}}}} \log \left (\frac {2 \, c x^{3} - \sqrt {3} {\left (2 \, c x^{2} - c^{\frac {2}{3}} x - c^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} - 3 \, c^{\frac {1}{3}} x + 1}{c x^{3} - 1}\right ) + 12 \, b c x + b \left (-c\right )^{\frac {2}{3}} \log \left (c x^{2} - \left (-c\right )^{\frac {2}{3}} x - \left (-c\right )^{\frac {1}{3}}\right ) - b c^{\frac {2}{3}} \log \left (c x^{2} + c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right ) - 2 \, b \left (-c\right )^{\frac {2}{3}} \log \left (c x + \left (-c\right )^{\frac {2}{3}}\right ) + 2 \, b c^{\frac {2}{3}} \log \left (c x - c^{\frac {2}{3}}\right )}{16 \, c^{2}}, \frac {2 \, b c^{2} x^{4} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a c^{2} x^{4} + \sqrt {3} b c \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} \log \left (\frac {2 \, c x^{3} - \sqrt {3} {\left (2 \, c x^{2} + \left (-c\right )^{\frac {2}{3}} x + \left (-c\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} + 3 \, \left (-c\right )^{\frac {1}{3}} x - 1}{c x^{3} + 1}\right ) - 2 \, \sqrt {3} b c^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) + 12 \, b c x + b \left (-c\right )^{\frac {2}{3}} \log \left (c x^{2} - \left (-c\right )^{\frac {2}{3}} x - \left (-c\right )^{\frac {1}{3}}\right ) - b c^{\frac {2}{3}} \log \left (c x^{2} + c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right ) - 2 \, b \left (-c\right )^{\frac {2}{3}} \log \left (c x + \left (-c\right )^{\frac {2}{3}}\right ) + 2 \, b c^{\frac {2}{3}} \log \left (c x - c^{\frac {2}{3}}\right )}{16 \, c^{2}}, \frac {2 \, b c^{2} x^{4} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a c^{2} x^{4} - 2 \, \sqrt {3} b c \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (-c\right )^{\frac {2}{3}} x + \left (-c\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}}\right ) - 2 \, \sqrt {3} b c^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) + 12 \, b c x + b \left (-c\right )^{\frac {2}{3}} \log \left (c x^{2} - \left (-c\right )^{\frac {2}{3}} x - \left (-c\right )^{\frac {1}{3}}\right ) - b c^{\frac {2}{3}} \log \left (c x^{2} + c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right ) - 2 \, b \left (-c\right )^{\frac {2}{3}} \log \left (c x + \left (-c\right )^{\frac {2}{3}}\right ) + 2 \, b c^{\frac {2}{3}} \log \left (c x - c^{\frac {2}{3}}\right )}{16 \, c^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(2*b*c^2*x^4*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 4*a*c^2*x^4 + sqrt(3)*b*c*sqrt((-c)^(1/3)/c)*log((2*c*x^3 -
sqrt(3)*(2*c*x^2 + (-c)^(2/3)*x + (-c)^(1/3))*sqrt((-c)^(1/3)/c) + 3*(-c)^(1/3)*x - 1)/(c*x^3 + 1)) + sqrt(3)
*b*c*sqrt(-1/c^(2/3))*log((2*c*x^3 - sqrt(3)*(2*c*x^2 - c^(2/3)*x - c^(1/3))*sqrt(-1/c^(2/3)) - 3*c^(1/3)*x +
1)/(c*x^3 - 1)) + 12*b*c*x + b*(-c)^(2/3)*log(c*x^2 - (-c)^(2/3)*x - (-c)^(1/3)) - b*c^(2/3)*log(c*x^2 + c^(2/
3)*x + c^(1/3)) - 2*b*(-c)^(2/3)*log(c*x + (-c)^(2/3)) + 2*b*c^(2/3)*log(c*x - c^(2/3)))/c^2, 1/16*(2*b*c^2*x^
4*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 4*a*c^2*x^4 - 2*sqrt(3)*b*c*sqrt(-(-c)^(1/3)/c)*arctan(1/3*sqrt(3)*(2*(-c)^(
2/3)*x + (-c)^(1/3))*sqrt(-(-c)^(1/3)/c)) + sqrt(3)*b*c*sqrt(-1/c^(2/3))*log((2*c*x^3 - sqrt(3)*(2*c*x^2 - c^(
2/3)*x - c^(1/3))*sqrt(-1/c^(2/3)) - 3*c^(1/3)*x + 1)/(c*x^3 - 1)) + 12*b*c*x + b*(-c)^(2/3)*log(c*x^2 - (-c)^
(2/3)*x - (-c)^(1/3)) - b*c^(2/3)*log(c*x^2 + c^(2/3)*x + c^(1/3)) - 2*b*(-c)^(2/3)*log(c*x + (-c)^(2/3)) + 2*
b*c^(2/3)*log(c*x - c^(2/3)))/c^2, 1/16*(2*b*c^2*x^4*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 4*a*c^2*x^4 + sqrt(3)*b*c
*sqrt((-c)^(1/3)/c)*log((2*c*x^3 - sqrt(3)*(2*c*x^2 + (-c)^(2/3)*x + (-c)^(1/3))*sqrt((-c)^(1/3)/c) + 3*(-c)^(
1/3)*x - 1)/(c*x^3 + 1)) - 2*sqrt(3)*b*c^(2/3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*x + c^(1/3))/c^(1/3)) + 12*b*c*x
+ b*(-c)^(2/3)*log(c*x^2 - (-c)^(2/3)*x - (-c)^(1/3)) - b*c^(2/3)*log(c*x^2 + c^(2/3)*x + c^(1/3)) - 2*b*(-c)^
(2/3)*log(c*x + (-c)^(2/3)) + 2*b*c^(2/3)*log(c*x - c^(2/3)))/c^2, 1/16*(2*b*c^2*x^4*log(-(c*x^3 + 1)/(c*x^3 -
1)) + 4*a*c^2*x^4 - 2*sqrt(3)*b*c*sqrt(-(-c)^(1/3)/c)*arctan(1/3*sqrt(3)*(2*(-c)^(2/3)*x + (-c)^(1/3))*sqrt(-
(-c)^(1/3)/c)) - 2*sqrt(3)*b*c^(2/3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*x + c^(1/3))/c^(1/3)) + 12*b*c*x + b*(-c)^(
2/3)*log(c*x^2 - (-c)^(2/3)*x - (-c)^(1/3)) - b*c^(2/3)*log(c*x^2 + c^(2/3)*x + c^(1/3)) - 2*b*(-c)^(2/3)*log(
c*x + (-c)^(2/3)) + 2*b*c^(2/3)*log(c*x - c^(2/3)))/c^2]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.47, size = 207, normalized size = 1.19 \begin {gather*} \frac {1}{16} \, b c^{7} {\left (\frac {2 \, \left (-\frac {1}{c}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {1}{c}\right )^{\frac {1}{3}} \right |}\right )}{c^{8}} - \frac {2 \, \sqrt {3} {\left | c \right |}^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} c^{\frac {1}{3}} {\left (2 \, x + \frac {1}{c^{\frac {1}{3}}}\right )}\right )}{c^{9}} - \frac {2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {1}{c}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {1}{c}\right )^{\frac {1}{3}}}\right )}{c^{9}} - \frac {{\left | c \right |}^{\frac {2}{3}} \log \left (x^{2} + \frac {x}{c^{\frac {1}{3}}} + \frac {1}{c^{\frac {2}{3}}}\right )}{c^{9}} + \frac {2 \, \log \left ({\left | x - \frac {1}{c^{\frac {1}{3}}} \right |}\right )}{c^{\frac {25}{3}}} - \frac {\left (-c^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {1}{c}\right )^{\frac {1}{3}} + \left (-\frac {1}{c}\right )^{\frac {2}{3}}\right )}{c^{9}}\right )} + \frac {1}{8} \, b x^{4} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + \frac {1}{4} \, a x^{4} + \frac {3 \, b x}{4 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*b*c^7*(2*(-1/c)^(1/3)*log(abs(x - (-1/c)^(1/3)))/c^8 - 2*sqrt(3)*abs(c)^(2/3)*arctan(1/3*sqrt(3)*c^(1/3)*
(2*x + 1/c^(1/3)))/c^9 - 2*sqrt(3)*(-c^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-1/c)^(1/3))/(-1/c)^(1/3))/c^9 - ab
s(c)^(2/3)*log(x^2 + x/c^(1/3) + 1/c^(2/3))/c^9 + 2*log(abs(x - 1/c^(1/3)))/c^(25/3) - (-c^2)^(1/3)*log(x^2 +
x*(-1/c)^(1/3) + (-1/c)^(2/3))/c^9) + 1/8*b*x^4*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 1/4*a*x^4 + 3/4*b*x/c

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Mupad [B]
time = 1.57, size = 125, normalized size = 0.72 \begin {gather*} \frac {a\,x^4}{4}+\frac {b\,\left (-\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )}{2}+\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )}{2}+\mathrm {atan}\left (c^{1/3}\,x\,1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{4\,c^{4/3}}+\frac {3\,b\,x}{4\,c}+\frac {b\,x^4\,\ln \left (c\,x^3+1\right )}{8}-\frac {b\,x^4\,\ln \left (1-c\,x^3\right )}{8}-\frac {\sqrt {3}\,b\,\left (\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )+\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )\right )}{8\,c^{4/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*atanh(c*x^3)),x)

[Out]

(a*x^4)/4 + (b*(atan((c^(1/3)*x*(3^(1/2) + 1i))/2)/2 - atan((c^(1/3)*x*(3^(1/2) - 1i))/2)/2 + atan(c^(1/3)*x*1
i))*1i)/(4*c^(4/3)) + (3*b*x)/(4*c) + (b*x^4*log(c*x^3 + 1))/8 - (b*x^4*log(1 - c*x^3))/8 - (3^(1/2)*b*(atan((
c^(1/3)*x*(3^(1/2) - 1i))/2) + atan((c^(1/3)*x*(3^(1/2) + 1i))/2)))/(8*c^(4/3))

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