3.3.13 \(\int x^3 (a+b \tanh ^{-1}(c x^{3/2})) \, dx\) [213]

Optimal. Leaf size=190 \[ \frac {3 b x^{5/2}}{20 c}-\frac {\sqrt {3} b \text {ArcTan}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {\sqrt {3} b \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{4 c^{8/3}}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{16 c^{8/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{16 c^{8/3}} \]

[Out]

3/20*b*x^(5/2)/c+1/4*x^4*(a+b*arctanh(c*x^(3/2)))-1/4*b*arctanh(c^(1/3)*x^(1/2))/c^(8/3)+1/16*b*ln(1+c^(2/3)*x
-c^(1/3)*x^(1/2))/c^(8/3)-1/16*b*ln(1+c^(2/3)*x+c^(1/3)*x^(1/2))/c^(8/3)-1/8*b*arctan(1/3*(1-2*c^(1/3)*x^(1/2)
)*3^(1/2))*3^(1/2)/c^(8/3)+1/8*b*arctan(1/3*(1+2*c^(1/3)*x^(1/2))*3^(1/2))*3^(1/2)/c^(8/3)

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Rubi [A]
time = 0.25, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6037, 327, 335, 302, 648, 632, 210, 642, 212} \begin {gather*} \frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {\sqrt {3} b \text {ArcTan}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {\sqrt {3} b \text {ArcTan}\left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {b \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}{16 c^{8/3}}-\frac {b \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}{16 c^{8/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{4 c^{8/3}}+\frac {3 b x^{5/2}}{20 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*ArcTanh[c*x^(3/2)]),x]

[Out]

(3*b*x^(5/2))/(20*c) - (Sqrt[3]*b*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(8*c^(8/3)) + (Sqrt[3]*b*ArcTan[(1
+ 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(8*c^(8/3)) - (b*ArcTanh[c^(1/3)*Sqrt[x]])/(4*c^(8/3)) + (x^4*(a + b*ArcTanh[c*
x^(3/2)]))/4 + (b*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/(16*c^(8/3)) - (b*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x]
)/(16*c^(8/3))

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 302

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[-a/b, n]], s = Denominator[Rt[-
a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k*m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]
*x + s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[2*k*(Pi/n)]*x + s
^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m))*Int[1/(r^2 - s^2*x^2), x] + Dist[2*(r^(m + 1)/(a*n*s^m)), Sum[u, {k, 1, (
n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && 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 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && 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^{3/2}\right )\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {1}{8} (3 b c) \int \frac {x^{9/2}}{1-c^2 x^3} \, dx\\ &=\frac {3 b x^{5/2}}{20 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {(3 b) \int \frac {x^{3/2}}{1-c^2 x^3} \, dx}{8 c}\\ &=\frac {3 b x^{5/2}}{20 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {(3 b) \text {Subst}\left (\int \frac {x^4}{1-c^2 x^6} \, dx,x,\sqrt {x}\right )}{4 c}\\ &=\frac {3 b x^{5/2}}{20 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {b \text {Subst}\left (\int \frac {1}{1-c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{4 c^{7/3}}-\frac {b \text {Subst}\left (\int \frac {-\frac {1}{2}-\frac {\sqrt [3]{c} x}{2}}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{4 c^{7/3}}-\frac {b \text {Subst}\left (\int \frac {-\frac {1}{2}+\frac {\sqrt [3]{c} x}{2}}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{4 c^{7/3}}\\ &=\frac {3 b x^{5/2}}{20 c}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{4 c^{8/3}}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {b \text {Subst}\left (\int \frac {-\sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{16 c^{8/3}}-\frac {b \text {Subst}\left (\int \frac {\sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{16 c^{8/3}}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{16 c^{7/3}}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{16 c^{7/3}}\\ &=\frac {3 b x^{5/2}}{20 c}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{4 c^{8/3}}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{16 c^{8/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{16 c^{8/3}}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} \sqrt {x}\right )}{8 c^{8/3}}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} \sqrt {x}\right )}{8 c^{8/3}}\\ &=\frac {3 b x^{5/2}}{20 c}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{4 c^{8/3}}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{16 c^{8/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{16 c^{8/3}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 222, normalized size = 1.17 \begin {gather*} \frac {3 b x^{5/2}}{20 c}+\frac {a x^4}{4}+\frac {\sqrt {3} b \text {ArcTan}\left (\frac {-1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {\sqrt {3} b \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {1}{4} b x^4 \tanh ^{-1}\left (c x^{3/2}\right )+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}\right )}{8 c^{8/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}\right )}{8 c^{8/3}}+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{16 c^{8/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{16 c^{8/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*ArcTanh[c*x^(3/2)]),x]

[Out]

(3*b*x^(5/2))/(20*c) + (a*x^4)/4 + (Sqrt[3]*b*ArcTan[(-1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(8*c^(8/3)) + (Sqrt[3]
*b*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(8*c^(8/3)) + (b*x^4*ArcTanh[c*x^(3/2)])/4 + (b*Log[1 - c^(1/3)*Sq
rt[x]])/(8*c^(8/3)) - (b*Log[1 + c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + (b*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/(16*
c^(8/3)) - (b*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/(16*c^(8/3))

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Maple [A]
time = 0.05, size = 194, normalized size = 1.02

method	result	size
derivativedivides	\(\frac {x^{4} a}{4}+\frac {b \,x^{4} \arctanh \left (c \,x^{\frac {3}{2}}\right )}{4}+\frac {3 b \,x^{\frac {5}{2}}}{20 c}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}\)	\(194\)
default	\(\frac {x^{4} a}{4}+\frac {b \,x^{4} \arctanh \left (c \,x^{\frac {3}{2}}\right )}{4}+\frac {3 b \,x^{\frac {5}{2}}}{20 c}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}\)	\(194\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arctanh(c*x^(3/2))),x,method=_RETURNVERBOSE)

[Out]

1/4*x^4*a+1/4*b*x^4*arctanh(c*x^(3/2))+3/20*b*x^(5/2)/c-1/8*b/c^3/(1/c)^(1/3)*ln(x^(1/2)+(1/c)^(1/3))+1/16*b/c
^3/(1/c)^(1/3)*ln(x-(1/c)^(1/3)*x^(1/2)+(1/c)^(2/3))+1/8*b/c^3*3^(1/2)/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)
^(1/3)*x^(1/2)-1))+1/8*b/c^3/(1/c)^(1/3)*ln(x^(1/2)-(1/c)^(1/3))-1/16*b/c^3/(1/c)^(1/3)*ln(x+(1/c)^(1/3)*x^(1/
2)+(1/c)^(2/3))+1/8*b/c^3*3^(1/2)/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x^(1/2)+1))

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Maxima [A]
time = 0.47, size = 172, normalized size = 0.91 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {1}{80} \, {\left (20 \, x^{4} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + c {\left (\frac {12 \, x^{\frac {5}{2}}}{c^{2}} + \frac {10 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}} + \frac {10 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}} - \frac {5 \, \log \left (c^{\frac {2}{3}} x + c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {11}{3}}} + \frac {5 \, \log \left (c^{\frac {2}{3}} x - c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {11}{3}}} - \frac {10 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}} + \frac {10 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {11}{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^(3/2))),x, algorithm="maxima")

[Out]

1/4*a*x^4 + 1/80*(20*x^4*arctanh(c*x^(3/2)) + c*(12*x^(5/2)/c^2 + 10*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*sqr
t(x) + c^(1/3))/c^(1/3))/c^(11/3) + 10*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*sqrt(x) - c^(1/3))/c^(1/3))/c^(11
/3) - 5*log(c^(2/3)*x + c^(1/3)*sqrt(x) + 1)/c^(11/3) + 5*log(c^(2/3)*x - c^(1/3)*sqrt(x) + 1)/c^(11/3) - 10*l
og((c^(1/3)*sqrt(x) + 1)/c^(1/3))/c^(11/3) + 10*log((c^(1/3)*sqrt(x) - 1)/c^(1/3))/c^(11/3)))*b

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Fricas [C] Result contains complex when optimal does not.
time = 1.18, size = 1803, normalized size = 9.49 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arctanh(c*x^(3/2))),x, algorithm="fricas")

[Out]

1/160*(40*a*c*x^4 + 24*b*x^(5/2) - 20*sqrt(3)*sqrt(((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*s
qrt(3) + 1) + 2*b)^2 - 4*((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b + 4*b
^2)*c*arctan(1/24*(4*sqrt(3)*sqrt(((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b
)^2*b^2*c^5*sqrt(x) + 4*b^4*c^5*sqrt(x) + 4*b^4*c^2 + 4*b^4*x - 2*(2*b^3*c^5*sqrt(x) + b^3*c^2)*((1/2)^(1/3)*(
b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b))*sqrt(((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 +
b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 - 4*((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3)
+ 1) + 2*b)*b + 4*b^2)*c^3 - sqrt(3)*(((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) +
2*b)^2*c^8 - 4*((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b*c^8 + 4*b^2*c^
8 + 8*b^2*c^3*sqrt(x))*sqrt(((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 -
4*((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b + 4*b^2))/b^3) - 10*((1/2)^(
1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)*c*log(-1/4*((1/2)^(1/3)*(b^3 - (c^8 - 1)
*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2*c^5 + ((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3
)*(I*sqrt(3) + 1) + 2*b)*b*c^5 - b^2*c^5 + b^2*sqrt(x)) - 20*(4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/10
24*b^3/c^8)^(1/3)*(I*sqrt(3) + 1) - b)*c*log((4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/c^8)^(1/3
)*(I*sqrt(3) + 1) - b)^2*c^5 + 2*(4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/c^8)^(1/3)*(I*sqrt(3)
+ 1) - b)*b*c^5 + b^2*c^5 + b^2*sqrt(x)) - 40*sqrt(3*(4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/
c^8)^(1/3)*(I*sqrt(3) + 1) - b)^2 + 6*(4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/c^8)^(1/3)*(I*sq
rt(3) + 1) - b)*b + 3*b^2)*c*arctan(1/3*((4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/c^8)^(1/3)*(I
*sqrt(3) + 1) - b)^2*c^8 + 2*(4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/c^8)^(1/3)*(I*sqrt(3) + 1
) - b)*b*c^8 + b^2*c^8 - 2*b^2*c^3*sqrt(x) + sqrt(-4*(4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/c
^8)^(1/3)*(I*sqrt(3) + 1) - b)^2*b^2*c^5*sqrt(x) - 4*b^4*c^5*sqrt(x) + 4*b^4*c^2 + 4*b^4*x - 4*(2*b^3*c^5*sqrt
(x) - b^3*c^2)*(4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/c^8)^(1/3)*(I*sqrt(3) + 1) - b))*c^3)*s
qrt(3*(4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/c^8)^(1/3)*(I*sqrt(3) + 1) - b)^2 + 6*(4*(-1/102
4*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/c^8)^(1/3)*(I*sqrt(3) + 1) - b)*b + 3*b^2)/b^3) + 5*(((1/2)^(1/3
)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)*c - 6*b*c)*log(((1/2)^(1/3)*(b^3 - (c^8 - 1
)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2*b^2*c^5*sqrt(x) + 4*b^4*c^5*sqrt(x) + 4*b^4*c^2 + 4*b^4*x
- 2*(2*b^3*c^5*sqrt(x) + b^3*c^2)*((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b
)) + 10*((4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/c^8)^(1/3)*(I*sqrt(3) + 1) - b)*c + 3*b*c)*lo
g(-4*(4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/c^8)^(1/3)*(I*sqrt(3) + 1) - b)^2*b^2*c^5*sqrt(x)
- 4*b^4*c^5*sqrt(x) + 4*b^4*c^2 + 4*b^4*x - 4*(2*b^3*c^5*sqrt(x) - b^3*c^2)*(4*(-1/1024*b^3 + 1/1024*(c^8 - 1
)*b^3/c^8 + 1/1024*b^3/c^8)^(1/3)*(I*sqrt(3) + 1) - b)) + 20*(b*c*x^4 - b*c)*log(-(c^2*x^3 + 2*c*x^(3/2) + 1)/
(c^2*x^3 - 1)))/c

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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**(3/2))),x)

[Out]

Timed out

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Giac [C] Result contains complex when optimal does not.
time = 0.49, size = 227, normalized size = 1.19 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {1}{320} \, {\left (40 \, x^{4} \log \left (-\frac {c x^{\frac {3}{2}} + 1}{c x^{\frac {3}{2}} - 1}\right ) + c {\left (\frac {48 \, x^{\frac {5}{2}}}{c^{2}} - \frac {10 \, \sqrt {3} {\left (-i \, \sqrt {3} - 1\right )}^{2} {\left | c \right |}^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \sqrt {x} + \left (-\frac {1}{c}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {1}{c}\right )^{\frac {1}{3}}}\right )}{c^{5}} + \frac {5 \, {\left (-i \, \sqrt {3} - 1\right )}^{2} {\left | c \right |}^{\frac {4}{3}} \log \left (x + \sqrt {x} \left (-\frac {1}{c}\right )^{\frac {1}{3}} + \left (-\frac {1}{c}\right )^{\frac {2}{3}}\right )}{c^{5}} - \frac {40 \, \left (-\frac {1}{c}\right )^{\frac {2}{3}} \log \left ({\left | \sqrt {x} - \left (-\frac {1}{c}\right )^{\frac {1}{3}} \right |}\right )}{c^{3}} + \frac {40 \, \sqrt {3} {\left | c \right |}^{\frac {4}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} c^{\frac {1}{3}} {\left (2 \, \sqrt {x} + \frac {1}{c^{\frac {1}{3}}}\right )}\right )}{c^{5}} - \frac {20 \, {\left | c \right |}^{\frac {4}{3}} \log \left (x + \frac {\sqrt {x}}{c^{\frac {1}{3}}} + \frac {1}{c^{\frac {2}{3}}}\right )}{c^{5}} + \frac {40 \, \log \left ({\left | \sqrt {x} - \frac {1}{c^{\frac {1}{3}}} \right |}\right )}{c^{\frac {11}{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^(3/2))),x, algorithm="giac")

[Out]

1/4*a*x^4 + 1/320*(40*x^4*log(-(c*x^(3/2) + 1)/(c*x^(3/2) - 1)) + c*(48*x^(5/2)/c^2 - 10*sqrt(3)*(-I*sqrt(3) -
1)^2*abs(c)^(4/3)*arctan(1/3*sqrt(3)*(2*sqrt(x) + (-1/c)^(1/3))/(-1/c)^(1/3))/c^5 + 5*(-I*sqrt(3) - 1)^2*abs(
c)^(4/3)*log(x + sqrt(x)*(-1/c)^(1/3) + (-1/c)^(2/3))/c^5 - 40*(-1/c)^(2/3)*log(abs(sqrt(x) - (-1/c)^(1/3)))/c
^3 + 40*sqrt(3)*abs(c)^(4/3)*arctan(1/3*sqrt(3)*c^(1/3)*(2*sqrt(x) + 1/c^(1/3)))/c^5 - 20*abs(c)^(4/3)*log(x +
sqrt(x)/c^(1/3) + 1/c^(2/3))/c^5 + 40*log(abs(sqrt(x) - 1/c^(1/3)))/c^(11/3)))*b

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Mupad [B]
time = 13.38, size = 231, normalized size = 1.22 \begin {gather*} \frac {a\,x^4}{4}+\frac {3\,b\,x^{5/2}}{20\,c}+\frac {b\,\ln \left (\frac {c^{1/3}\,\sqrt {x}-1}{c^{1/3}\,\sqrt {x}+1}\right )}{8\,c^{8/3}}+\frac {\ln \left (1-c\,x^{3/2}\right )\,\left (\frac {b\,x^4}{4}-\frac {b\,c^2\,x^7}{4}\right )}{2\,c^2\,x^3-2}+\frac {b\,x^4\,\ln \left (c\,x^{3/2}+1\right )}{8}+\frac {b\,\ln \left (\frac {\sqrt {3}+c^{2/3}\,x\,1{}\mathrm {i}-c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}\,c^{2/3}\,x+1{}\mathrm {i}}{2\,c^{2/3}\,x+1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}{8\,c^{8/3}}+\frac {\sqrt {2}\,b\,\ln \left (\frac {\sqrt {3}\,c^{2/3}\,x+c^{2/3}\,x\,1{}\mathrm {i}+c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}+1{}\mathrm {i}}{2\,c^{2/3}\,x+1-\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{16\,c^{8/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*atanh(c*x^(3/2))),x)

[Out]

(a*x^4)/4 + (3*b*x^(5/2))/(20*c) + (b*log((c^(1/3)*x^(1/2) - 1)/(c^(1/3)*x^(1/2) + 1)))/(8*c^(8/3)) + (log(1 -
c*x^(3/2))*((b*x^4)/4 - (b*c^2*x^7)/4))/(2*c^2*x^3 - 2) + (b*x^4*log(c*x^(3/2) + 1))/8 + (b*log((3^(1/2) + c^
(2/3)*x*1i - c^(1/3)*x^(1/2)*4i - 3^(1/2)*c^(2/3)*x + 1i)/(3^(1/2)*1i + 2*c^(2/3)*x + 1))*((3^(1/2)*1i)/2 - 1/
2)^(1/2))/(8*c^(8/3)) + (2^(1/2)*b*log((c^(2/3)*x*1i - 3^(1/2) + c^(1/3)*x^(1/2)*4i + 3^(1/2)*c^(2/3)*x + 1i)/
(2*c^(2/3)*x - 3^(1/2)*1i + 1))*(3^(1/2)*1i + 1)^(1/2)*1i)/(16*c^(8/3))

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