3.1.23 \(\int \frac {(a+b \tanh ^{-1}(c x))^2}{x^5} \, dx\) [23]

Optimal. Leaf size=117 \[ -\frac {b^2 c^2}{12 x^2}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac {b c^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}+\frac {1}{4} c^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {2}{3} b^2 c^4 \log (x)-\frac {1}{3} b^2 c^4 \log \left (1-c^2 x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6037, 6129, 272, 46, 36, 29, 31, 6095} \begin {gather*} \frac {1}{4} c^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}+\frac {2}{3} b^2 c^4 \log (x)-\frac {b^2 c^2}{12 x^2}-\frac {1}{3} b^2 c^4 \log \left (1-c^2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x])^2/x^5,x]

[Out]

-1/12*(b^2*c^2)/x^2 - (b*c*(a + b*ArcTanh[c*x]))/(6*x^3) - (b*c^3*(a + b*ArcTanh[c*x]))/(2*x) + (c^4*(a + b*Ar
cTanh[c*x])^2)/4 - (a + b*ArcTanh[c*x])^2/(4*x^4) + (2*b^2*c^4*Log[x])/3 - (b^2*c^4*Log[1 - c^2*x^2])/3

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 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 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 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}(c x)\right )^2}{x^5} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{2} (b c) \int \frac {a+b \tanh ^{-1}(c x)}{x^4 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{2} (b c) \int \frac {a+b \tanh ^{-1}(c x)}{x^4} \, dx+\frac {1}{2} \left (b c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{6} \left (b^2 c^2\right ) \int \frac {1}{x^3 \left (1-c^2 x^2\right )} \, dx+\frac {1}{2} \left (b c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\frac {1}{2} \left (b c^5\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac {b c^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}+\frac {1}{4} c^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{2} \left (b^2 c^4\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac {b c^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}+\frac {1}{4} c^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}+\frac {c^2}{x}-\frac {c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 c^4\right ) \text {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2}{12 x^2}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac {b c^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}+\frac {1}{4} c^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{6} b^2 c^4 \log (x)-\frac {1}{12} b^2 c^4 \log \left (1-c^2 x^2\right )+\frac {1}{4} \left (b^2 c^4\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 c^6\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2}{12 x^2}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac {b c^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}+\frac {1}{4} c^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {2}{3} b^2 c^4 \log (x)-\frac {1}{3} b^2 c^4 \log \left (1-c^2 x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 164, normalized size = 1.40 \begin {gather*} -\frac {3 a^2+2 a b c x+b^2 c^2 x^2+6 a b c^3 x^3+2 b \left (3 a+b c x+3 b c^3 x^3\right ) \tanh ^{-1}(c x)-3 b^2 \left (-1+c^4 x^4\right ) \tanh ^{-1}(c x)^2-8 b^2 c^4 x^4 \log (x)+3 a b c^4 x^4 \log (1-c x)+4 b^2 c^4 x^4 \log (1-c x)-3 a b c^4 x^4 \log (1+c x)+4 b^2 c^4 x^4 \log (1+c x)}{12 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(3*a^2 + 2*a*b*c*x + b^2*c^2*x^2 + 6*a*b*c^3*x^3 + 2*b*(3*a + b*c*x + 3*b*c^3*x^3)*ArcTanh[c*x] - 3*b^2*
(-1 + c^4*x^4)*ArcTanh[c*x]^2 - 8*b^2*c^4*x^4*Log[x] + 3*a*b*c^4*x^4*Log[1 - c*x] + 4*b^2*c^4*x^4*Log[1 - c*x]
- 3*a*b*c^4*x^4*Log[1 + c*x] + 4*b^2*c^4*x^4*Log[1 + c*x])/x^4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(270\) vs. \(2(103)=206\).
time = 0.05, size = 271, normalized size = 2.32

method	result	size
derivativedivides	\(c^{4} \left (-\frac {a^{2}}{4 c^{4} x^{4}}-\frac {b^{2} \arctanh \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{4}-\frac {b^{2} \arctanh \left (c x \right )}{6 c^{3} x^{3}}-\frac {b^{2} \arctanh \left (c x \right )}{2 c x}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{4}+\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{8}-\frac {b^{2} \ln \left (c x -1\right )^{2}}{16}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{8}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{8}-\frac {b^{2} \ln \left (c x +1\right )^{2}}{16}-\frac {b^{2} \ln \left (c x -1\right )}{3}-\frac {b^{2}}{12 c^{2} x^{2}}+\frac {2 b^{2} \ln \left (c x \right )}{3}-\frac {b^{2} \ln \left (c x +1\right )}{3}-\frac {a b \arctanh \left (c x \right )}{2 c^{4} x^{4}}-\frac {a b \ln \left (c x -1\right )}{4}-\frac {a b}{6 c^{3} x^{3}}-\frac {a b}{2 c x}+\frac {a b \ln \left (c x +1\right )}{4}\right )\)	\(271\)
default	\(c^{4} \left (-\frac {a^{2}}{4 c^{4} x^{4}}-\frac {b^{2} \arctanh \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{4}-\frac {b^{2} \arctanh \left (c x \right )}{6 c^{3} x^{3}}-\frac {b^{2} \arctanh \left (c x \right )}{2 c x}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{4}+\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{8}-\frac {b^{2} \ln \left (c x -1\right )^{2}}{16}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{8}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{8}-\frac {b^{2} \ln \left (c x +1\right )^{2}}{16}-\frac {b^{2} \ln \left (c x -1\right )}{3}-\frac {b^{2}}{12 c^{2} x^{2}}+\frac {2 b^{2} \ln \left (c x \right )}{3}-\frac {b^{2} \ln \left (c x +1\right )}{3}-\frac {a b \arctanh \left (c x \right )}{2 c^{4} x^{4}}-\frac {a b \ln \left (c x -1\right )}{4}-\frac {a b}{6 c^{3} x^{3}}-\frac {a b}{2 c x}+\frac {a b \ln \left (c x +1\right )}{4}\right )\)	\(271\)
risch	\(\frac {b^{2} \left (c^{4} x^{4}-1\right ) \ln \left (c x +1\right )^{2}}{16 x^{4}}-\frac {b \left (3 x^{4} b \ln \left (-c x +1\right ) c^{4}+6 b \,c^{3} x^{3}+2 b c x -3 b \ln \left (-c x +1\right )+6 a \right ) \ln \left (c x +1\right )}{24 x^{4}}+\frac {3 b^{2} c^{4} x^{4} \ln \left (-c x +1\right )^{2}+12 c^{4} b \ln \left (-c x -1\right ) x^{4} a -16 c^{4} b^{2} \ln \left (-c x -1\right ) x^{4}-12 a b \,c^{4} x^{4} \ln \left (-c x +1\right )-16 c^{4} b^{2} \ln \left (-c x +1\right ) x^{4}+32 b^{2} c^{4} \ln \left (x \right ) x^{4}+12 b^{2} c^{3} x^{3} \ln \left (-c x +1\right )-24 a b \,c^{3} x^{3}-4 b^{2} c^{2} x^{2}+4 b^{2} c x \ln \left (-c x +1\right )-8 a b c x -3 b^{2} \ln \left (-c x +1\right )^{2}+12 b \ln \left (-c x +1\right ) a -12 a^{2}}{48 x^{4}}\)	\(281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*(-1/4*a^2/c^4/x^4-1/4*b^2/c^4/x^4*arctanh(c*x)^2-1/4*b^2*arctanh(c*x)*ln(c*x-1)-1/6*b^2*arctanh(c*x)/c^3/x
^3-1/2*b^2*arctanh(c*x)/c/x+1/4*b^2*arctanh(c*x)*ln(c*x+1)+1/8*b^2*ln(c*x-1)*ln(1/2*c*x+1/2)-1/16*b^2*ln(c*x-1
)^2+1/8*b^2*ln(-1/2*c*x+1/2)*ln(c*x+1)-1/8*b^2*ln(-1/2*c*x+1/2)*ln(1/2*c*x+1/2)-1/16*b^2*ln(c*x+1)^2-1/3*b^2*l
n(c*x-1)-1/12*b^2/c^2/x^2+2/3*b^2*ln(c*x)-1/3*b^2*ln(c*x+1)-1/2*a*b/c^4/x^4*arctanh(c*x)-1/4*a*b*ln(c*x-1)-1/6
*a*b/c^3/x^3-1/2*a*b/c/x+1/4*a*b*ln(c*x+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (103) = 206\).
time = 0.26, size = 224, normalized size = 1.91 \begin {gather*} \frac {1}{12} \, {\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c x\right )}{x^{4}}\right )} a b + \frac {1}{48} \, {\left ({\left (32 \, c^{2} \log \left (x\right ) - \frac {3 \, c^{2} x^{2} \log \left (c x + 1\right )^{2} + 3 \, c^{2} x^{2} \log \left (c x - 1\right )^{2} + 16 \, c^{2} x^{2} \log \left (c x - 1\right ) - 2 \, {\left (3 \, c^{2} x^{2} \log \left (c x - 1\right ) - 8 \, c^{2} x^{2}\right )} \log \left (c x + 1\right ) + 4}{x^{2}}\right )} c^{2} + 4 \, {\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c \operatorname {artanh}\left (c x\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c x\right )^{2}}{4 \, x^{4}} - \frac {a^{2}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*((3*c^3*log(c*x + 1) - 3*c^3*log(c*x - 1) - 2*(3*c^2*x^2 + 1)/x^3)*c - 6*arctanh(c*x)/x^4)*a*b + 1/48*((3
2*c^2*log(x) - (3*c^2*x^2*log(c*x + 1)^2 + 3*c^2*x^2*log(c*x - 1)^2 + 16*c^2*x^2*log(c*x - 1) - 2*(3*c^2*x^2*l
og(c*x - 1) - 8*c^2*x^2)*log(c*x + 1) + 4)/x^2)*c^2 + 4*(3*c^3*log(c*x + 1) - 3*c^3*log(c*x - 1) - 2*(3*c^2*x^
2 + 1)/x^3)*c*arctanh(c*x))*b^2 - 1/4*b^2*arctanh(c*x)^2/x^4 - 1/4*a^2/x^4

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Fricas [A]
time = 0.36, size = 173, normalized size = 1.48 \begin {gather*} \frac {32 \, b^{2} c^{4} x^{4} \log \left (x\right ) + 4 \, {\left (3 \, a b - 4 \, b^{2}\right )} c^{4} x^{4} \log \left (c x + 1\right ) - 4 \, {\left (3 \, a b + 4 \, b^{2}\right )} c^{4} x^{4} \log \left (c x - 1\right ) - 24 \, a b c^{3} x^{3} - 4 \, b^{2} c^{2} x^{2} - 8 \, a b c x + 3 \, {\left (b^{2} c^{4} x^{4} - b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} - 12 \, a^{2} - 4 \, {\left (3 \, b^{2} c^{3} x^{3} + b^{2} c x + 3 \, a b\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{48 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(32*b^2*c^4*x^4*log(x) + 4*(3*a*b - 4*b^2)*c^4*x^4*log(c*x + 1) - 4*(3*a*b + 4*b^2)*c^4*x^4*log(c*x - 1)
- 24*a*b*c^3*x^3 - 4*b^2*c^2*x^2 - 8*a*b*c*x + 3*(b^2*c^4*x^4 - b^2)*log(-(c*x + 1)/(c*x - 1))^2 - 12*a^2 - 4*
(3*b^2*c^3*x^3 + b^2*c*x + 3*a*b)*log(-(c*x + 1)/(c*x - 1)))/x^4

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Sympy [A]
time = 0.57, size = 184, normalized size = 1.57 \begin {gather*} \begin {cases} - \frac {a^{2}}{4 x^{4}} + \frac {a b c^{4} \operatorname {atanh}{\left (c x \right )}}{2} - \frac {a b c^{3}}{2 x} - \frac {a b c}{6 x^{3}} - \frac {a b \operatorname {atanh}{\left (c x \right )}}{2 x^{4}} + \frac {2 b^{2} c^{4} \log {\left (x \right )}}{3} - \frac {2 b^{2} c^{4} \log {\left (x - \frac {1}{c} \right )}}{3} + \frac {b^{2} c^{4} \operatorname {atanh}^{2}{\left (c x \right )}}{4} - \frac {2 b^{2} c^{4} \operatorname {atanh}{\left (c x \right )}}{3} - \frac {b^{2} c^{3} \operatorname {atanh}{\left (c x \right )}}{2 x} - \frac {b^{2} c^{2}}{12 x^{2}} - \frac {b^{2} c \operatorname {atanh}{\left (c x \right )}}{6 x^{3}} - \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{4 x^{4}} & \text {for}\: c \neq 0 \\- \frac {a^{2}}{4 x^{4}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x))**2/x**5,x)

[Out]

Piecewise((-a**2/(4*x**4) + a*b*c**4*atanh(c*x)/2 - a*b*c**3/(2*x) - a*b*c/(6*x**3) - a*b*atanh(c*x)/(2*x**4)
+ 2*b**2*c**4*log(x)/3 - 2*b**2*c**4*log(x - 1/c)/3 + b**2*c**4*atanh(c*x)**2/4 - 2*b**2*c**4*atanh(c*x)/3 - b
**2*c**3*atanh(c*x)/(2*x) - b**2*c**2/(12*x**2) - b**2*c*atanh(c*x)/(6*x**3) - b**2*atanh(c*x)**2/(4*x**4), Ne
(c, 0)), (-a**2/(4*x**4), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (103) = 206\).
time = 0.42, size = 612, normalized size = 5.23 \begin {gather*} \frac {1}{6} \, {\left (4 \, b^{2} c^{3} \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - 4 \, b^{2} c^{3} \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {3 \, {\left (\frac {{\left (c x + 1\right )}^{3} b^{2} c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {{\left (c x + 1\right )} b^{2} c^{3}}{c x - 1}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{\frac {{\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {4 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {4 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {2 \, {\left (\frac {6 \, {\left (c x + 1\right )}^{3} a b c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )} a b c^{3}}{c x - 1} + \frac {3 \, {\left (c x + 1\right )}^{3} b^{2} c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2} b^{2} c^{3}}{{\left (c x - 1\right )}^{2}} + \frac {5 \, {\left (c x + 1\right )} b^{2} c^{3}}{c x - 1} + 2 \, b^{2} c^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {4 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {4 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {2 \, {\left (\frac {6 \, {\left (c x + 1\right )}^{3} a^{2} c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )} a^{2} c^{3}}{c x - 1} + \frac {6 \, {\left (c x + 1\right )}^{3} a b c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {12 \, {\left (c x + 1\right )}^{2} a b c^{3}}{{\left (c x - 1\right )}^{2}} + \frac {10 \, {\left (c x + 1\right )} a b c^{3}}{c x - 1} + 4 \, a b c^{3} + \frac {{\left (c x + 1\right )}^{3} b^{2} c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {2 \, {\left (c x + 1\right )}^{2} b^{2} c^{3}}{{\left (c x - 1\right )}^{2}} + \frac {{\left (c x + 1\right )} b^{2} c^{3}}{c x - 1}\right )}}{\frac {{\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {4 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {4 \, {\left (c x + 1\right )}}{c x - 1} + 1}\right )} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(4*b^2*c^3*log(-(c*x + 1)/(c*x - 1) - 1) - 4*b^2*c^3*log(-(c*x + 1)/(c*x - 1)) + 3*((c*x + 1)^3*b^2*c^3/(c
*x - 1)^3 + (c*x + 1)*b^2*c^3/(c*x - 1))*log(-(c*x + 1)/(c*x - 1))^2/((c*x + 1)^4/(c*x - 1)^4 + 4*(c*x + 1)^3/
(c*x - 1)^3 + 6*(c*x + 1)^2/(c*x - 1)^2 + 4*(c*x + 1)/(c*x - 1) + 1) + 2*(6*(c*x + 1)^3*a*b*c^3/(c*x - 1)^3 +
6*(c*x + 1)*a*b*c^3/(c*x - 1) + 3*(c*x + 1)^3*b^2*c^3/(c*x - 1)^3 + 6*(c*x + 1)^2*b^2*c^3/(c*x - 1)^2 + 5*(c*x
+ 1)*b^2*c^3/(c*x - 1) + 2*b^2*c^3)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^4/(c*x - 1)^4 + 4*(c*x + 1)^3/(c*x -
1)^3 + 6*(c*x + 1)^2/(c*x - 1)^2 + 4*(c*x + 1)/(c*x - 1) + 1) + 2*(6*(c*x + 1)^3*a^2*c^3/(c*x - 1)^3 + 6*(c*x
+ 1)*a^2*c^3/(c*x - 1) + 6*(c*x + 1)^3*a*b*c^3/(c*x - 1)^3 + 12*(c*x + 1)^2*a*b*c^3/(c*x - 1)^2 + 10*(c*x + 1
)*a*b*c^3/(c*x - 1) + 4*a*b*c^3 + (c*x + 1)^3*b^2*c^3/(c*x - 1)^3 + 2*(c*x + 1)^2*b^2*c^3/(c*x - 1)^2 + (c*x +
1)*b^2*c^3/(c*x - 1))/((c*x + 1)^4/(c*x - 1)^4 + 4*(c*x + 1)^3/(c*x - 1)^3 + 6*(c*x + 1)^2/(c*x - 1)^2 + 4*(c
*x + 1)/(c*x - 1) + 1))*c

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Mupad [B]
time = 1.90, size = 303, normalized size = 2.59 \begin {gather*} \frac {b^2\,c^4\,{\ln \left (c\,x+1\right )}^2}{16}-\frac {a^2}{4\,x^4}+\frac {b^2\,c^4\,{\ln \left (1-c\,x\right )}^2}{16}-\frac {b^2\,{\ln \left (c\,x+1\right )}^2}{16\,x^4}-\frac {b^2\,{\ln \left (1-c\,x\right )}^2}{16\,x^4}-\frac {b^2\,c^2}{12\,x^2}+\frac {2\,b^2\,c^4\,\ln \left (x\right )}{3}-\frac {b^2\,c^4\,\ln \left (c\,x-1\right )}{3}-\frac {b^2\,c^4\,\ln \left (c\,x+1\right )}{3}-\frac {a\,b\,\ln \left (c\,x+1\right )}{4\,x^4}+\frac {a\,b\,\ln \left (1-c\,x\right )}{4\,x^4}+\frac {b^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )}{8\,x^4}-\frac {a\,b\,c}{6\,x^3}-\frac {b^2\,c\,\ln \left (c\,x+1\right )}{12\,x^3}+\frac {b^2\,c\,\ln \left (1-c\,x\right )}{12\,x^3}-\frac {a\,b\,c^3}{2\,x}-\frac {b^2\,c^3\,\ln \left (c\,x+1\right )}{4\,x}+\frac {b^2\,c^3\,\ln \left (1-c\,x\right )}{4\,x}-\frac {a\,b\,c^4\,\ln \left (c\,x-1\right )}{4}+\frac {a\,b\,c^4\,\ln \left (c\,x+1\right )}{4}-\frac {b^2\,c^4\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x))^2/x^5,x)

[Out]

(b^2*c^4*log(c*x + 1)^2)/16 - a^2/(4*x^4) + (b^2*c^4*log(1 - c*x)^2)/16 - (b^2*log(c*x + 1)^2)/(16*x^4) - (b^2
*log(1 - c*x)^2)/(16*x^4) - (b^2*c^2)/(12*x^2) + (2*b^2*c^4*log(x))/3 - (b^2*c^4*log(c*x - 1))/3 - (b^2*c^4*lo
g(c*x + 1))/3 - (a*b*log(c*x + 1))/(4*x^4) + (a*b*log(1 - c*x))/(4*x^4) + (b^2*log(c*x + 1)*log(1 - c*x))/(8*x
^4) - (a*b*c)/(6*x^3) - (b^2*c*log(c*x + 1))/(12*x^3) + (b^2*c*log(1 - c*x))/(12*x^3) - (a*b*c^3)/(2*x) - (b^2
*c^3*log(c*x + 1))/(4*x) + (b^2*c^3*log(1 - c*x))/(4*x) - (a*b*c^4*log(c*x - 1))/4 + (a*b*c^4*log(c*x + 1))/4
- (b^2*c^4*log(c*x + 1)*log(1 - c*x))/8

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