3.1.57 \(\int \frac {a+b \tanh ^{-1}(c x)}{x^3 (d+c d x)^2} \, dx\) [57]

Optimal. Leaf size=212 \[ -\frac {b c}{2 d^2 x}+\frac {b c^2}{2 d^2 (1+c x)}-\frac {a+b \tanh ^{-1}(c x)}{2 d^2 x^2}+\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}+\frac {3 a c^2 \log (x)}{d^2}-\frac {2 b c^2 \log (x)}{d^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{d^2}+\frac {b c^2 \log \left (1-c^2 x^2\right )}{d^2}-\frac {3 b c^2 \text {PolyLog}(2,-c x)}{2 d^2}+\frac {3 b c^2 \text {PolyLog}(2,c x)}{2 d^2}-\frac {3 b c^2 \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 d^2} \]

[Out]

-1/2*b*c/d^2/x+1/2*b*c^2/d^2/(c*x+1)+1/2*(-a-b*arctanh(c*x))/d^2/x^2+2*c*(a+b*arctanh(c*x))/d^2/x+c^2*(a+b*arc
tanh(c*x))/d^2/(c*x+1)+3*a*c^2*ln(x)/d^2-2*b*c^2*ln(x)/d^2+3*c^2*(a+b*arctanh(c*x))*ln(2/(c*x+1))/d^2+b*c^2*ln
(-c^2*x^2+1)/d^2-3/2*b*c^2*polylog(2,-c*x)/d^2+3/2*b*c^2*polylog(2,c*x)/d^2-3/2*b*c^2*polylog(2,1-2/(c*x+1))/d
^2

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Rubi [A]
time = 0.19, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 16, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6087, 6037, 331, 212, 272, 36, 29, 31, 6031, 6063, 641, 46, 213, 6055, 2449, 2352} \begin {gather*} \frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (c x+1)}+\frac {3 c^2 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac {a+b \tanh ^{-1}(c x)}{2 d^2 x^2}+\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 x}+\frac {3 a c^2 \log (x)}{d^2}-\frac {3 b c^2 \text {Li}_2(-c x)}{2 d^2}+\frac {3 b c^2 \text {Li}_2(c x)}{2 d^2}-\frac {3 b c^2 \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 d^2}+\frac {b c^2 \log \left (1-c^2 x^2\right )}{d^2}+\frac {b c^2}{2 d^2 (c x+1)}-\frac {2 b c^2 \log (x)}{d^2}-\frac {b c}{2 d^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 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 331

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

Rule 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 6031

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Simp[(b/2)*PolyLog[2, (-c)*x]
, x] + Simp[(b/2)*PolyLog[2, c*x], x]) /; FreeQ[{a, b, c}, x]

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 6055

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

Rule 6063

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

Rule 6087

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[E
xpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[
p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

Rubi steps

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

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Mathematica [A]
time = 0.73, size = 189, normalized size = 0.89 \begin {gather*} \frac {-\frac {2 a}{x^2}+\frac {8 a c}{x}-\frac {2 b c}{x}+\frac {4 a c^2}{1+c x}+b c^2 \cosh \left (2 \tanh ^{-1}(c x)\right )+12 a c^2 \log (x)-12 a c^2 \log (1+c x)-8 b c^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-6 b c^2 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )-b c^2 \sinh \left (2 \tanh ^{-1}(c x)\right )+2 b \tanh ^{-1}(c x) \left (c^2-\frac {1}{x^2}+\frac {4 c}{x}+c^2 \cosh \left (2 \tanh ^{-1}(c x)\right )+6 c^2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-c^2 \sinh \left (2 \tanh ^{-1}(c x)\right )\right )}{4 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a)/x^2 + (8*a*c)/x - (2*b*c)/x + (4*a*c^2)/(1 + c*x) + b*c^2*Cosh[2*ArcTanh[c*x]] + 12*a*c^2*Log[x] - 12*
a*c^2*Log[1 + c*x] - 8*b*c^2*Log[(c*x)/Sqrt[1 - c^2*x^2]] - 6*b*c^2*PolyLog[2, E^(-2*ArcTanh[c*x])] - b*c^2*Si
nh[2*ArcTanh[c*x]] + 2*b*ArcTanh[c*x]*(c^2 - x^(-2) + (4*c)/x + c^2*Cosh[2*ArcTanh[c*x]] + 6*c^2*Log[1 - E^(-2
*ArcTanh[c*x])] - c^2*Sinh[2*ArcTanh[c*x]]))/(4*d^2)

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Maple [A]
time = 0.34, size = 303, normalized size = 1.43

method	result	size
derivativedivides	\(c^{2} \left (\frac {a}{d^{2} \left (c x +1\right )}-\frac {3 a \ln \left (c x +1\right )}{d^{2}}-\frac {a}{2 d^{2} c^{2} x^{2}}+\frac {2 a}{d^{2} c x}+\frac {3 a \ln \left (c x \right )}{d^{2}}+\frac {b \arctanh \left (c x \right )}{d^{2} \left (c x +1\right )}-\frac {3 b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d^{2}}-\frac {b \arctanh \left (c x \right )}{2 d^{2} c^{2} x^{2}}+\frac {2 b \arctanh \left (c x \right )}{d^{2} c x}+\frac {3 b \arctanh \left (c x \right ) \ln \left (c x \right )}{d^{2}}-\frac {3 b \dilog \left (c x \right )}{2 d^{2}}-\frac {3 b \dilog \left (c x +1\right )}{2 d^{2}}-\frac {3 b \ln \left (c x \right ) \ln \left (c x +1\right )}{2 d^{2}}+\frac {3 b \ln \left (c x +1\right )^{2}}{4 d^{2}}-\frac {3 b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d^{2}}+\frac {3 b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}+\frac {3 b \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}+\frac {b}{2 d^{2} \left (c x +1\right )}+\frac {b \ln \left (c x +1\right )}{d^{2}}-\frac {b}{2 d^{2} c x}-\frac {2 b \ln \left (c x \right )}{d^{2}}+\frac {b \ln \left (c x -1\right )}{d^{2}}\right )\)	\(303\)
default	\(c^{2} \left (\frac {a}{d^{2} \left (c x +1\right )}-\frac {3 a \ln \left (c x +1\right )}{d^{2}}-\frac {a}{2 d^{2} c^{2} x^{2}}+\frac {2 a}{d^{2} c x}+\frac {3 a \ln \left (c x \right )}{d^{2}}+\frac {b \arctanh \left (c x \right )}{d^{2} \left (c x +1\right )}-\frac {3 b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d^{2}}-\frac {b \arctanh \left (c x \right )}{2 d^{2} c^{2} x^{2}}+\frac {2 b \arctanh \left (c x \right )}{d^{2} c x}+\frac {3 b \arctanh \left (c x \right ) \ln \left (c x \right )}{d^{2}}-\frac {3 b \dilog \left (c x \right )}{2 d^{2}}-\frac {3 b \dilog \left (c x +1\right )}{2 d^{2}}-\frac {3 b \ln \left (c x \right ) \ln \left (c x +1\right )}{2 d^{2}}+\frac {3 b \ln \left (c x +1\right )^{2}}{4 d^{2}}-\frac {3 b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d^{2}}+\frac {3 b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}+\frac {3 b \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}+\frac {b}{2 d^{2} \left (c x +1\right )}+\frac {b \ln \left (c x +1\right )}{d^{2}}-\frac {b}{2 d^{2} c x}-\frac {2 b \ln \left (c x \right )}{d^{2}}+\frac {b \ln \left (c x -1\right )}{d^{2}}\right )\)	\(303\)
risch	\(\frac {c^{2} b \ln \left (-c x +1\right )}{4 d^{2} \left (-c x -1\right )}-\frac {3 c^{2} b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}+\frac {3 c^{2} b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{2 d^{2}}-\frac {b c}{2 d^{2} x}+\frac {b \,c^{2}}{2 d^{2} \left (c x +1\right )}+\frac {3 c^{2} \dilog \left (-c x +1\right ) b}{2 d^{2}}-\frac {3 c^{2} b \dilog \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}-\frac {3 c^{2} b \dilog \left (c x +1\right )}{2 d^{2}}-\frac {c b \ln \left (-c x +1\right )}{d^{2} x}-\frac {c^{3} b \ln \left (-c x +1\right ) x}{4 d^{2} \left (-c x -1\right )}+\frac {c b \ln \left (c x +1\right )}{d^{2} x}+\frac {c^{2} b \ln \left (c x +1\right )}{2 d^{2} \left (c x +1\right )}-\frac {a}{2 d^{2} x^{2}}-\frac {5 c^{2} b \ln \left (c x \right )}{4 d^{2}}-\frac {b \ln \left (c x +1\right )}{4 d^{2} x^{2}}-\frac {3 c^{2} b \ln \left (c x +1\right )^{2}}{4 d^{2}}+\frac {2 c a}{d^{2} x}+\frac {3 c^{2} a \ln \left (-c x \right )}{d^{2}}-\frac {c^{2} a}{d^{2} \left (-c x -1\right )}-\frac {3 c^{2} a \ln \left (-c x -1\right )}{d^{2}}-\frac {3 c^{2} b \ln \left (-c x \right )}{4 d^{2}}+\frac {3 c^{2} b \ln \left (-c x +1\right )}{4 d^{2}}-\frac {c^{2} b \ln \left (-c x -1\right )}{4 d^{2}}+\frac {b \ln \left (-c x +1\right )}{4 d^{2} x^{2}}+\frac {5 c^{2} b \ln \left (c x +1\right )}{4 d^{2}}\)	\(412\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*(6*c^2*log(c*x + 1)/d^2 - 6*c^2*log(x)/d^2 - (6*c^2*x^2 + 3*c*x - 1)/(c*d^2*x^3 + d^2*x^2)) + 1/2*b*int
egrate((log(c*x + 1) - log(-c*x + 1))/(c^2*d^2*x^5 + 2*c*d^2*x^4 + d^2*x^3), x)

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Fricas [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))/x^3/(c*d*x+d)^2,x, algorithm="fricas")

[Out]

integral((b*arctanh(c*x) + a)/(c^2*d^2*x^5 + 2*c*d^2*x^4 + d^2*x^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a}{c^{2} x^{5} + 2 c x^{4} + x^{3}}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{5} + 2 c x^{4} + x^{3}}\, dx}{d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a/(c**2*x**5 + 2*c*x**4 + x**3), x) + Integral(b*atanh(c*x)/(c**2*x**5 + 2*c*x**4 + x**3), x))/d**2

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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))/x^3/(c*d*x+d)^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x^3\,{\left (d+c\,d\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x))/(x^3*(d + c*d*x)^2),x)

[Out]

int((a + b*atanh(c*x))/(x^3*(d + c*d*x)^2), x)

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