3.2.18 \(\int x^5 (a+b \tanh ^{-1}(c x^3))^2 \, dx\) [118]

Optimal. Leaf size=91 \[ \frac {a b x^3}{3 c}+\frac {b^2 x^3 \tanh ^{-1}\left (c x^3\right )}{3 c}-\frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{6 c^2}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2+\frac {b^2 \log \left (1-c^2 x^6\right )}{6 c^2} \]

[Out]

1/3*a*b*x^3/c+1/3*b^2*x^3*arctanh(c*x^3)/c-1/6*(a+b*arctanh(c*x^3))^2/c^2+1/6*x^6*(a+b*arctanh(c*x^3))^2+1/6*b
^2*ln(-c^2*x^6+1)/c^2

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Rubi [A]
time = 0.11, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6039, 6037, 6127, 6021, 266, 6095} \begin {gather*} -\frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{6 c^2}+\frac {a b x^3}{3 c}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2+\frac {b^2 \log \left (1-c^2 x^6\right )}{6 c^2}+\frac {b^2 x^3 \tanh ^{-1}\left (c x^3\right )}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*ArcTanh[c*x^3])^2,x]

[Out]

(a*b*x^3)/(3*c) + (b^2*x^3*ArcTanh[c*x^3])/(3*c) - (a + b*ArcTanh[c*x^3])^2/(6*c^2) + (x^6*(a + b*ArcTanh[c*x^
3])^2)/6 + (b^2*Log[1 - c^2*x^6])/(6*c^2)

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 6021

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Dist[b
*c*n*p, Int[x^n*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p
, 0] && (EqQ[n, 1] || EqQ[p, 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
]

Rule 6039

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m
+ 1)/n] - 1)*(a + b*ArcTanh[c*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[S
implify[(m + 1)/n]]

Rule 6095

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

Rule 6127

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2
/e, Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcTanh[c*x])
^p/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rubi steps

\begin {align*} \int x^5 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2 \, dx &=\int \left (\frac {1}{4} x^5 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {1}{2} b x^5 \left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac {1}{4} b^2 x^5 \log ^2\left (1+c x^3\right )\right ) \, dx\\ &=\frac {1}{4} \int x^5 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \, dx-\frac {1}{2} b \int x^5 \left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right ) \, dx+\frac {1}{4} b^2 \int x^5 \log ^2\left (1+c x^3\right ) \, dx\\ &=\frac {1}{12} \text {Subst}\left (\int x (2 a-b \log (1-c x))^2 \, dx,x,x^3\right )-\frac {1}{6} b \text {Subst}\left (\int x (-2 a+b \log (1-c x)) \log (1+c x) \, dx,x,x^3\right )+\frac {1}{12} b^2 \text {Subst}\left (\int x \log ^2(1+c x) \, dx,x,x^3\right )\\ &=\frac {1}{12} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac {1}{12} \text {Subst}\left (\int \left (\frac {(2 a-b \log (1-c x))^2}{c}-\frac {(1-c x) (2 a-b \log (1-c x))^2}{c}\right ) \, dx,x,x^3\right )+\frac {1}{12} b^2 \text {Subst}\left (\int \left (-\frac {\log ^2(1+c x)}{c}+\frac {(1+c x) \log ^2(1+c x)}{c}\right ) \, dx,x,x^3\right )+\frac {1}{12} (b c) \text {Subst}\left (\int \frac {x^2 (-2 a+b \log (1-c x))}{1+c x} \, dx,x,x^3\right )-\frac {1}{12} \left (b^2 c\right ) \text {Subst}\left (\int \frac {x^2 \log (1+c x)}{1-c x} \, dx,x,x^3\right )\\ &=\frac {1}{12} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac {\text {Subst}\left (\int (2 a-b \log (1-c x))^2 \, dx,x,x^3\right )}{12 c}-\frac {\text {Subst}\left (\int (1-c x) (2 a-b \log (1-c x))^2 \, dx,x,x^3\right )}{12 c}-\frac {b^2 \text {Subst}\left (\int \log ^2(1+c x) \, dx,x,x^3\right )}{12 c}+\frac {b^2 \text {Subst}\left (\int (1+c x) \log ^2(1+c x) \, dx,x,x^3\right )}{12 c}+\frac {1}{12} (b c) \text {Subst}\left (\int \left (-\frac {-2 a+b \log (1-c x)}{c^2}+\frac {x (-2 a+b \log (1-c x))}{c}+\frac {-2 a+b \log (1-c x)}{c^2 (1+c x)}\right ) \, dx,x,x^3\right )-\frac {1}{12} \left (b^2 c\right ) \text {Subst}\left (\int \left (-\frac {\log (1+c x)}{c^2}-\frac {x \log (1+c x)}{c}-\frac {\log (1+c x)}{c^2 (-1+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac {1}{12} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac {1}{12} b \text {Subst}\left (\int x (-2 a+b \log (1-c x)) \, dx,x,x^3\right )+\frac {1}{12} b^2 \text {Subst}\left (\int x \log (1+c x) \, dx,x,x^3\right )-\frac {\text {Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-c x^3\right )}{12 c^2}+\frac {\text {Subst}\left (\int x (2 a-b \log (x))^2 \, dx,x,1-c x^3\right )}{12 c^2}-\frac {b^2 \text {Subst}\left (\int \log ^2(x) \, dx,x,1+c x^3\right )}{12 c^2}+\frac {b^2 \text {Subst}\left (\int x \log ^2(x) \, dx,x,1+c x^3\right )}{12 c^2}-\frac {b \text {Subst}\left (\int (-2 a+b \log (1-c x)) \, dx,x,x^3\right )}{12 c}+\frac {b \text {Subst}\left (\int \frac {-2 a+b \log (1-c x)}{1+c x} \, dx,x,x^3\right )}{12 c}+\frac {b^2 \text {Subst}\left (\int \log (1+c x) \, dx,x,x^3\right )}{12 c}+\frac {b^2 \text {Subst}\left (\int \frac {\log (1+c x)}{-1+c x} \, dx,x,x^3\right )}{12 c}\\ &=\frac {a b x^3}{6 c}-\frac {1}{24} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c^2}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 c^2}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{12 c^2}+\frac {1}{24} b^2 x^6 \log \left (1+c x^3\right )+\frac {b^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 c^2}+\frac {1}{12} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac {b^2 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{12 c^2}+\frac {b^2 \left (1+c x^3\right )^2 \log ^2\left (1+c x^3\right )}{24 c^2}+\frac {b \text {Subst}\left (\int x (2 a-b \log (x)) \, dx,x,1-c x^3\right )}{12 c^2}-\frac {b \text {Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-c x^3\right )}{6 c^2}+\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{12 c^2}-\frac {b^2 \text {Subst}\left (\int x \log (x) \, dx,x,1+c x^3\right )}{12 c^2}+\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{6 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^3\right )}{12 c}-\frac {b^2 \text {Subst}\left (\int \log (1-c x) \, dx,x,x^3\right )}{12 c}+\frac {b^2 \text {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )}{12 c}+\frac {1}{24} \left (b^2 c\right ) \text {Subst}\left (\int \frac {x^2}{1-c x} \, dx,x,x^3\right )-\frac {1}{24} \left (b^2 c\right ) \text {Subst}\left (\int \frac {x^2}{1+c x} \, dx,x,x^3\right )\\ &=\frac {a b x^3}{2 c}-\frac {b^2 x^3}{4 c}+\frac {b^2 \left (1-c x^3\right )^2}{48 c^2}+\frac {b^2 \left (1+c x^3\right )^2}{48 c^2}-\frac {1}{24} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )+\frac {b \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{24 c^2}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c^2}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 c^2}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{12 c^2}+\frac {1}{24} b^2 x^6 \log \left (1+c x^3\right )+\frac {b^2 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{4 c^2}-\frac {b^2 \left (1+c x^3\right )^2 \log \left (1+c x^3\right )}{24 c^2}+\frac {b^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 c^2}+\frac {1}{12} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac {b^2 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{12 c^2}+\frac {b^2 \left (1+c x^3\right )^2 \log ^2\left (1+c x^3\right )}{24 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{12 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{12 c^2}+\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{12 c^2}+\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{6 c^2}+\frac {1}{24} \left (b^2 c\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {x}{c}-\frac {1}{c^2 (-1+c x)}\right ) \, dx,x,x^3\right )-\frac {1}{24} \left (b^2 c\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2}+\frac {x}{c}+\frac {1}{c^2 (1+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac {a b x^3}{2 c}-\frac {b^2 x^6}{24}+\frac {b^2 \left (1-c x^3\right )^2}{48 c^2}+\frac {b^2 \left (1+c x^3\right )^2}{48 c^2}-\frac {b^2 \log \left (1-c x^3\right )}{24 c^2}+\frac {b^2 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{4 c^2}-\frac {1}{24} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )+\frac {b \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{24 c^2}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c^2}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 c^2}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{12 c^2}-\frac {b^2 \log \left (1+c x^3\right )}{24 c^2}+\frac {1}{24} b^2 x^6 \log \left (1+c x^3\right )+\frac {b^2 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{4 c^2}-\frac {b^2 \left (1+c x^3\right )^2 \log \left (1+c x^3\right )}{24 c^2}+\frac {b^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 c^2}+\frac {1}{12} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac {b^2 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{12 c^2}+\frac {b^2 \left (1+c x^3\right )^2 \log ^2\left (1+c x^3\right )}{24 c^2}+\frac {b^2 \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )}{12 c^2}+\frac {b^2 \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )}{12 c^2}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 106, normalized size = 1.16 \begin {gather*} \frac {2 a b c x^3+a^2 c^2 x^6+2 b c x^3 \left (b+a c x^3\right ) \tanh ^{-1}\left (c x^3\right )+b^2 \left (-1+c^2 x^6\right ) \tanh ^{-1}\left (c x^3\right )^2+b (a+b) \log \left (1-c x^3\right )-a b \log \left (1+c x^3\right )+b^2 \log \left (1+c x^3\right )}{6 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*ArcTanh[c*x^3])^2,x]

[Out]

(2*a*b*c*x^3 + a^2*c^2*x^6 + 2*b*c*x^3*(b + a*c*x^3)*ArcTanh[c*x^3] + b^2*(-1 + c^2*x^6)*ArcTanh[c*x^3]^2 + b*
(a + b)*Log[1 - c*x^3] - a*b*Log[1 + c*x^3] + b^2*Log[1 + c*x^3])/(6*c^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(286\) vs. \(2(81)=162\).
time = 0.22, size = 287, normalized size = 3.15

method	result	size
risch	\(\frac {b^{2} \left (c^{2} x^{6}-1\right ) \ln \left (c \,x^{3}+1\right )^{2}}{24 c^{2}}+\frac {b \left (-2 x^{6} b \ln \left (-c \,x^{3}+1\right ) a \,c^{2}+4 a^{2} c^{2} x^{6}+4 a b c \,x^{3}+2 b \ln \left (-c \,x^{3}+1\right ) a +b^{2}\right ) \ln \left (c \,x^{3}+1\right )}{24 a \,c^{2}}+\frac {b^{2} x^{6} \ln \left (-c \,x^{3}+1\right )^{2}}{24}-\frac {a b \,x^{6} \ln \left (-c \,x^{3}+1\right )}{6}+\frac {x^{6} a^{2}}{6}-\frac {b^{2} x^{3} \ln \left (-c \,x^{3}+1\right )}{6 c}+\frac {a b \,x^{3}}{3 c}-\frac {b^{2} \ln \left (-c \,x^{3}+1\right )^{2}}{24 c^{2}}-\frac {a \ln \left (-c \,x^{3}-1\right ) b}{6 c^{2}}+\frac {\ln \left (-c \,x^{3}-1\right ) b^{2}}{6 c^{2}}-\frac {\ln \left (-c \,x^{3}-1\right ) b^{3}}{24 a \,c^{2}}+\frac {a \ln \left (-c \,x^{3}+1\right ) b}{6 c^{2}}+\frac {b^{2} \ln \left (-c \,x^{3}+1\right )}{6 c^{2}}+\frac {b^{2}}{6 c^{2}}\)	\(287\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arctanh(c*x^3))^2,x,method=_RETURNVERBOSE)

[Out]

1/24*b^2*(c^2*x^6-1)/c^2*ln(c*x^3+1)^2+1/24*b*(-2*x^6*b*ln(-c*x^3+1)*a*c^2+4*a^2*c^2*x^6+4*a*b*c*x^3+2*b*ln(-c
*x^3+1)*a+b^2)/a/c^2*ln(c*x^3+1)+1/24*b^2*x^6*ln(-c*x^3+1)^2-1/6*a*b*x^6*ln(-c*x^3+1)+1/6*x^6*a^2-1/6/c*b^2*x^
3*ln(-c*x^3+1)+1/3*a*b*x^3/c-1/24/c^2*b^2*ln(-c*x^3+1)^2-1/6*a/c^2*ln(-c*x^3-1)*b+1/6/c^2*ln(-c*x^3-1)*b^2-1/2
4/a/c^2*ln(-c*x^3-1)*b^3+1/6*a/c^2*ln(-c*x^3+1)*b+1/6/c^2*b^2*ln(-c*x^3+1)+1/6*b^2/c^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (81) = 162\).
time = 0.26, size = 186, normalized size = 2.04 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} \operatorname {artanh}\left (c x^{3}\right )^{2} + \frac {1}{6} \, a^{2} x^{6} + \frac {1}{6} \, {\left (2 \, x^{6} \operatorname {artanh}\left (c x^{3}\right ) + c {\left (\frac {2 \, x^{3}}{c^{2}} - \frac {\log \left (c x^{3} + 1\right )}{c^{3}} + \frac {\log \left (c x^{3} - 1\right )}{c^{3}}\right )}\right )} a b + \frac {1}{24} \, {\left (4 \, c {\left (\frac {2 \, x^{3}}{c^{2}} - \frac {\log \left (c x^{3} + 1\right )}{c^{3}} + \frac {\log \left (c x^{3} - 1\right )}{c^{3}}\right )} \operatorname {artanh}\left (c x^{3}\right ) - \frac {2 \, {\left (\log \left (c x^{3} - 1\right ) - 2\right )} \log \left (c x^{3} + 1\right ) - \log \left (c x^{3} + 1\right )^{2} - \log \left (c x^{3} - 1\right )^{2} - 4 \, \log \left (c x^{3} - 1\right )}{c^{2}}\right )} b^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctanh(c*x^3))^2,x, algorithm="maxima")

[Out]

1/6*b^2*x^6*arctanh(c*x^3)^2 + 1/6*a^2*x^6 + 1/6*(2*x^6*arctanh(c*x^3) + c*(2*x^3/c^2 - log(c*x^3 + 1)/c^3 + l
og(c*x^3 - 1)/c^3))*a*b + 1/24*(4*c*(2*x^3/c^2 - log(c*x^3 + 1)/c^3 + log(c*x^3 - 1)/c^3)*arctanh(c*x^3) - (2*
(log(c*x^3 - 1) - 2)*log(c*x^3 + 1) - log(c*x^3 + 1)^2 - log(c*x^3 - 1)^2 - 4*log(c*x^3 - 1))/c^2)*b^2

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Fricas [A]
time = 0.37, size = 138, normalized size = 1.52 \begin {gather*} \frac {4 \, a^{2} c^{2} x^{6} + 8 \, a b c x^{3} + {\left (b^{2} c^{2} x^{6} - b^{2}\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )^{2} - 4 \, {\left (a b - b^{2}\right )} \log \left (c x^{3} + 1\right ) + 4 \, {\left (a b + b^{2}\right )} \log \left (c x^{3} - 1\right ) + 4 \, {\left (a b c^{2} x^{6} + b^{2} c x^{3}\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{24 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctanh(c*x^3))^2,x, algorithm="fricas")

[Out]

1/24*(4*a^2*c^2*x^6 + 8*a*b*c*x^3 + (b^2*c^2*x^6 - b^2)*log(-(c*x^3 + 1)/(c*x^3 - 1))^2 - 4*(a*b - b^2)*log(c*
x^3 + 1) + 4*(a*b + b^2)*log(c*x^3 - 1) + 4*(a*b*c^2*x^6 + b^2*c*x^3)*log(-(c*x^3 + 1)/(c*x^3 - 1)))/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**5*(a+b*atanh(c*x**3))**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (81) = 162\).
time = 0.48, size = 361, normalized size = 3.97 \begin {gather*} \frac {1}{6} \, {\left (\frac {{\left (c x^{3} + 1\right )} b^{2} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )^{2}}{{\left (c x^{3} - 1\right )} {\left (\frac {{\left (c x^{3} + 1\right )}^{2} c^{3}}{{\left (c x^{3} - 1\right )}^{2}} - \frac {2 \, {\left (c x^{3} + 1\right )} c^{3}}{c x^{3} - 1} + c^{3}\right )}} + \frac {2 \, {\left (\frac {2 \, {\left (c x^{3} + 1\right )} a b}{c x^{3} - 1} + \frac {{\left (c x^{3} + 1\right )} b^{2}}{c x^{3} - 1} - b^{2}\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{\frac {{\left (c x^{3} + 1\right )}^{2} c^{3}}{{\left (c x^{3} - 1\right )}^{2}} - \frac {2 \, {\left (c x^{3} + 1\right )} c^{3}}{c x^{3} - 1} + c^{3}} + \frac {4 \, {\left (\frac {{\left (c x^{3} + 1\right )} a^{2}}{c x^{3} - 1} + \frac {{\left (c x^{3} + 1\right )} a b}{c x^{3} - 1} - a b\right )}}{\frac {{\left (c x^{3} + 1\right )}^{2} c^{3}}{{\left (c x^{3} - 1\right )}^{2}} - \frac {2 \, {\left (c x^{3} + 1\right )} c^{3}}{c x^{3} - 1} + c^{3}} - \frac {2 \, b^{2} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1} + 1\right )}{c^{3}} + \frac {2 \, b^{2} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{c^{3}}\right )} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctanh(c*x^3))^2,x, algorithm="giac")

[Out]

1/6*((c*x^3 + 1)*b^2*log(-(c*x^3 + 1)/(c*x^3 - 1))^2/((c*x^3 - 1)*((c*x^3 + 1)^2*c^3/(c*x^3 - 1)^2 - 2*(c*x^3
+ 1)*c^3/(c*x^3 - 1) + c^3)) + 2*(2*(c*x^3 + 1)*a*b/(c*x^3 - 1) + (c*x^3 + 1)*b^2/(c*x^3 - 1) - b^2)*log(-(c*x
^3 + 1)/(c*x^3 - 1))/((c*x^3 + 1)^2*c^3/(c*x^3 - 1)^2 - 2*(c*x^3 + 1)*c^3/(c*x^3 - 1) + c^3) + 4*((c*x^3 + 1)*
a^2/(c*x^3 - 1) + (c*x^3 + 1)*a*b/(c*x^3 - 1) - a*b)/((c*x^3 + 1)^2*c^3/(c*x^3 - 1)^2 - 2*(c*x^3 + 1)*c^3/(c*x
^3 - 1) + c^3) - 2*b^2*log(-(c*x^3 + 1)/(c*x^3 - 1) + 1)/c^3 + 2*b^2*log(-(c*x^3 + 1)/(c*x^3 - 1))/c^3)*c

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Mupad [B]
time = 1.25, size = 275, normalized size = 3.02 \begin {gather*} \frac {a^2\,x^6}{6}+\frac {b^2\,\ln \left (c\,x^3-1\right )}{6\,c^2}+\frac {b^2\,\ln \left (c\,x^3+1\right )}{6\,c^2}-\frac {b^2\,{\ln \left (c\,x^3+1\right )}^2}{24\,c^2}-\frac {b^2\,{\ln \left (1-c\,x^3\right )}^2}{24\,c^2}+\frac {b^2\,x^6\,{\ln \left (c\,x^3+1\right )}^2}{24}+\frac {b^2\,x^6\,{\ln \left (1-c\,x^3\right )}^2}{24}+\frac {b^2\,x^3\,\ln \left (c\,x^3+1\right )}{6\,c}-\frac {b^2\,x^3\,\ln \left (1-c\,x^3\right )}{6\,c}+\frac {a\,b\,\ln \left (c\,x^3-1\right )}{6\,c^2}-\frac {a\,b\,\ln \left (c\,x^3+1\right )}{6\,c^2}+\frac {a\,b\,x^6\,\ln \left (c\,x^3+1\right )}{6}-\frac {a\,b\,x^6\,\ln \left (1-c\,x^3\right )}{6}+\frac {b^2\,\ln \left (c\,x^3+1\right )\,\ln \left (1-c\,x^3\right )}{12\,c^2}+\frac {a\,b\,x^3}{3\,c}-\frac {b^2\,x^6\,\ln \left (c\,x^3+1\right )\,\ln \left (1-c\,x^3\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*atanh(c*x^3))^2,x)

[Out]

(a^2*x^6)/6 + (b^2*log(c*x^3 - 1))/(6*c^2) + (b^2*log(c*x^3 + 1))/(6*c^2) - (b^2*log(c*x^3 + 1)^2)/(24*c^2) -
(b^2*log(1 - c*x^3)^2)/(24*c^2) + (b^2*x^6*log(c*x^3 + 1)^2)/24 + (b^2*x^6*log(1 - c*x^3)^2)/24 + (b^2*x^3*log
(c*x^3 + 1))/(6*c) - (b^2*x^3*log(1 - c*x^3))/(6*c) + (a*b*log(c*x^3 - 1))/(6*c^2) - (a*b*log(c*x^3 + 1))/(6*c
^2) + (a*b*x^6*log(c*x^3 + 1))/6 - (a*b*x^6*log(1 - c*x^3))/6 + (b^2*log(c*x^3 + 1)*log(1 - c*x^3))/(12*c^2) +
(a*b*x^3)/(3*c) - (b^2*x^6*log(c*x^3 + 1)*log(1 - c*x^3))/12

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