3.2.22 \(\int \frac {(a+b \tanh ^{-1}(c x^3))^2}{x^7} \, dx\) [122]

Optimal. Leaf size=88 \[ -\frac {b c \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{3 x^3}+\frac {1}{6} c^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2-\frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{6 x^6}+b^2 c^2 \log (x)-\frac {1}{6} b^2 c^2 \log \left (1-c^2 x^6\right ) \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6039, 6037, 6129, 272, 36, 29, 31, 6095} \begin {gather*} \frac {1}{6} c^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2-\frac {b c \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{3 x^3}-\frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{6 x^6}-\frac {1}{6} b^2 c^2 \log \left (1-c^2 x^6\right )+b^2 c^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^3])^2/x^7,x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

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

Rule 272

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

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 6129

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{x^7} \, dx &=\int \left (\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{4 x^7}-\frac {b \left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{2 x^7}+\frac {b^2 \log ^2\left (1+c x^3\right )}{4 x^7}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{x^7} \, dx-\frac {1}{2} b \int \frac {\left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{x^7} \, dx+\frac {1}{4} b^2 \int \frac {\log ^2\left (1+c x^3\right )}{x^7} \, dx\\ &=\frac {1}{12} \text {Subst}\left (\int \frac {(2 a-b \log (1-c x))^2}{x^3} \, dx,x,x^3\right )-\frac {1}{6} b \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log (1+c x)}{x^3} \, dx,x,x^3\right )+\frac {1}{12} b^2 \text {Subst}\left (\int \frac {\log ^2(1+c x)}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}-\frac {b^2 \log ^2\left (1+c x^3\right )}{24 x^6}+\frac {1}{12} (b c) \text {Subst}\left (\int \frac {2 a-b \log (1-c x)}{x^2 (1-c x)} \, dx,x,x^3\right )-\frac {1}{12} (b c) \text {Subst}\left (\int \frac {-2 a+b \log (1-c x)}{x^2 (1+c x)} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^2 (1-c x)} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^2 (1+c x)} \, dx,x,x^3\right )\\ &=-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}-\frac {b^2 \log ^2\left (1+c x^3\right )}{24 x^6}-\frac {1}{12} b \text {Subst}\left (\int \frac {2 a-b \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-c x^3\right )-\frac {1}{12} (b c) \text {Subst}\left (\int \left (\frac {-2 a+b \log (1-c x)}{x^2}-\frac {c (-2 a+b \log (1-c x))}{x}+\frac {c^2 (-2 a+b \log (1-c x))}{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)}{x^2}+\frac {c \log (1+c x)}{x}-\frac {c^2 \log (1+c x)}{-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)}{x^2}-\frac {c \log (1+c x)}{x}+\frac {c^2 \log (1+c x)}{1+c x}\right ) \, dx,x,x^3\right )\\ &=-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}-\frac {b^2 \log ^2\left (1+c x^3\right )}{24 x^6}-\frac {1}{12} b \text {Subst}\left (\int \frac {2 a-b \log (x)}{\left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-c x^3\right )-\frac {1}{12} (b c) \text {Subst}\left (\int \frac {2 a-b \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )} \, dx,x,1-c x^3\right )-\frac {1}{12} (b c) \text {Subst}\left (\int \frac {-2 a+b \log (1-c x)}{x^2} \, dx,x,x^3\right )+2 \left (\frac {1}{12} \left (b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^2} \, dx,x,x^3\right )\right )+\frac {1}{12} \left (b c^2\right ) \text {Subst}\left (\int \frac {-2 a+b \log (1-c x)}{x} \, dx,x,x^3\right )-\frac {1}{12} \left (b c^3\right ) \text {Subst}\left (\int \frac {-2 a+b \log (1-c x)}{1+c x} \, dx,x,x^3\right )-\frac {1}{12} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{-1+c x} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{1+c x} \, dx,x,x^3\right )\\ &=-\frac {1}{2} a b c^2 \log (x)-\frac {b c \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac {b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}+\frac {1}{12} b c^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )-\frac {1}{12} b^2 c^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}-\frac {b^2 \log ^2\left (1+c x^3\right )}{24 x^6}-\frac {1}{12} (b c) \text {Subst}\left (\int \frac {2 a-b \log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x^3\right )-\frac {1}{12} \left (b^2 c\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x^3\right )-\frac {1}{12} \left (b c^2\right ) \text {Subst}\left (\int \frac {2 a-b \log (x)}{x} \, dx,x,1-c x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x (1-c x)} \, dx,x,x^3\right )+2 \left (-\frac {b^2 c \log \left (1+c x^3\right )}{12 x^3}+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x (1+c x)} \, dx,x,x^3\right )\right )+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+c x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^3\right )-\frac {1}{12} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )\\ &=\frac {1}{4} b^2 c^2 \log (x)-\frac {b c \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac {b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}+\frac {1}{24} c^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}+\frac {1}{12} b c^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )-\frac {1}{12} b^2 c^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}+\frac {1}{24} b^2 c^2 \log ^2\left (1+c x^3\right )-\frac {b^2 \log ^2\left (1+c x^3\right )}{24 x^6}-\frac {1}{12} b^2 c^2 \text {Li}_2\left (c x^3\right )+\frac {1}{12} \left (b^2 c\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1-c x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1+c x^3\right )+\frac {1}{12} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{1-c x} \, dx,x,x^3\right )+2 \left (-\frac {b^2 c \log \left (1+c x^3\right )}{12 x^3}+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^3\right )-\frac {1}{12} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{1+c x} \, dx,x,x^3\right )\right )\\ &=\frac {1}{2} b^2 c^2 \log (x)-\frac {1}{12} b^2 c^2 \log \left (1-c x^3\right )-\frac {b c \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac {b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}+\frac {1}{24} c^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}+\frac {1}{12} b c^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )-\frac {1}{12} b^2 c^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}+\frac {1}{24} b^2 c^2 \log ^2\left (1+c x^3\right )-\frac {b^2 \log ^2\left (1+c x^3\right )}{24 x^6}+2 \left (\frac {1}{4} b^2 c^2 \log (x)-\frac {1}{12} b^2 c^2 \log \left (1+c x^3\right )-\frac {b^2 c \log \left (1+c x^3\right )}{12 x^3}\right )-\frac {1}{12} b^2 c^2 \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )-\frac {1}{12} b^2 c^2 \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 111, normalized size = 1.26 \begin {gather*} \frac {1}{6} \left (-\frac {a^2}{x^6}-\frac {2 a b c}{x^3}-\frac {2 b \left (a+b c x^3\right ) \tanh ^{-1}\left (c x^3\right )}{x^6}+\frac {b^2 \left (-1+c^2 x^6\right ) \tanh ^{-1}\left (c x^3\right )^2}{x^6}+6 b^2 c^2 \log (x)-b (a+b) c^2 \log \left (1-c x^3\right )+(a-b) b c^2 \log \left (1+c x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^3])^2/x^7,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(256\) vs. \(2(80)=160\).
time = 0.17, size = 257, normalized size = 2.92

method	result	size
risch	\(\frac {b^{2} \left (c^{2} x^{6}-1\right ) \ln \left (c \,x^{3}+1\right )^{2}}{24 x^{6}}-\frac {b \left (b \,c^{2} \ln \left (-c \,x^{3}+1\right ) x^{6}+2 b c \,x^{3}-b \ln \left (-c \,x^{3}+1\right )+2 a \right ) \ln \left (c \,x^{3}+1\right )}{12 x^{6}}+\frac {b^{2} c^{2} x^{6} \ln \left (-c \,x^{3}+1\right )^{2}+4 b \,c^{2} \ln \left (c \,x^{3}+1\right ) x^{6} a -4 b^{2} c^{2} \ln \left (c \,x^{3}+1\right ) x^{6}-4 b \,c^{2} \ln \left (c \,x^{3}-1\right ) x^{6} a -4 b^{2} c^{2} \ln \left (c \,x^{3}-1\right ) x^{6}+24 b^{2} c^{2} \ln \left (x \right ) x^{6}+4 b^{2} c \,x^{3} \ln \left (-c \,x^{3}+1\right )-8 a b c \,x^{3}-b^{2} \ln \left (-c \,x^{3}+1\right )^{2}+4 b \ln \left (-c \,x^{3}+1\right ) a -4 a^{2}}{24 x^{6}}\)	\(257\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^3))^2/x^7,x,method=_RETURNVERBOSE)

[Out]

1/24*b^2*(c^2*x^6-1)/x^6*ln(c*x^3+1)^2-1/12*b*(b*c^2*ln(-c*x^3+1)*x^6+2*b*c*x^3-b*ln(-c*x^3+1)+2*a)/x^6*ln(c*x
^3+1)+1/24*(b^2*c^2*x^6*ln(-c*x^3+1)^2+4*b*c^2*ln(c*x^3+1)*x^6*a-4*b^2*c^2*ln(c*x^3+1)*x^6-4*b*c^2*ln(c*x^3-1)
*x^6*a-4*b^2*c^2*ln(c*x^3-1)*x^6+24*b^2*c^2*ln(x)*x^6+4*b^2*c*x^3*ln(-c*x^3+1)-8*a*b*c*x^3-b^2*ln(-c*x^3+1)^2+
4*b*ln(-c*x^3+1)*a-4*a^2)/x^6

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (80) = 160\).
time = 0.28, size = 175, normalized size = 1.99 \begin {gather*} \frac {1}{6} \, {\left ({\left (c \log \left (c x^{3} + 1\right ) - c \log \left (c x^{3} - 1\right ) - \frac {2}{x^{3}}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x^{3}\right )}{x^{6}}\right )} a b + \frac {1}{24} \, {\left ({\left (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 ) + 24 \, \log \left (x\right )\right )} c^{2} + 4 \, {\left (c \log \left (c x^{3} + 1\right ) - c \log \left (c x^{3} - 1\right ) - \frac {2}{x^{3}}\right )} c \operatorname {artanh}\left (c x^{3}\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c x^{3}\right )^{2}}{6 \, x^{6}} - \frac {a^{2}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))^2/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.40, size = 151, normalized size = 1.72 \begin {gather*} \frac {24 \, b^{2} c^{2} x^{6} \log \left (x\right ) + 4 \, {\left (a b - b^{2}\right )} c^{2} x^{6} \log \left (c x^{3} + 1\right ) - 4 \, {\left (a b + b^{2}\right )} c^{2} x^{6} \log \left (c x^{3} - 1\right ) - 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 \, a^{2} - 4 \, {\left (b^{2} c x^{3} + a b\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{24 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))^2/x^7,x, algorithm="fricas")

[Out]

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

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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((a+b*atanh(c*x**3))**2/x**7,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))^2/x^7,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^3) + a)^2/x^7, x)

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Mupad [B]
time = 1.54, size = 278, normalized size = 3.16 \begin {gather*} \frac {b^2\,c^2\,{\ln \left (c\,x^3+1\right )}^2}{24}-\frac {b^2\,c^2\,\ln \left (c\,x^3-1\right )}{6}-\frac {b^2\,c^2\,\ln \left (c\,x^3+1\right )}{6}-\frac {a^2}{6\,x^6}+\frac {b^2\,c^2\,{\ln \left (1-c\,x^3\right )}^2}{24}-\frac {b^2\,{\ln \left (c\,x^3+1\right )}^2}{24\,x^6}-\frac {b^2\,{\ln \left (1-c\,x^3\right )}^2}{24\,x^6}+b^2\,c^2\,\ln \left (x\right )-\frac {a\,b\,c^2\,\ln \left (c\,x^3-1\right )}{6}+\frac {a\,b\,c^2\,\ln \left (c\,x^3+1\right )}{6}-\frac {a\,b\,c}{3\,x^3}-\frac {a\,b\,\ln \left (c\,x^3+1\right )}{6\,x^6}+\frac {a\,b\,\ln \left (1-c\,x^3\right )}{6\,x^6}-\frac {b^2\,c^2\,\ln \left (c\,x^3+1\right )\,\ln \left (1-c\,x^3\right )}{12}-\frac {b^2\,c\,\ln \left (c\,x^3+1\right )}{6\,x^3}+\frac {b^2\,c\,\ln \left (1-c\,x^3\right )}{6\,x^3}+\frac {b^2\,\ln \left (c\,x^3+1\right )\,\ln \left (1-c\,x^3\right )}{12\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^3))^2/x^7,x)

[Out]

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

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