3.3.65 \(\int \frac {(a+b \coth ^{-1}(c x^n)) (d+e \log (f x^m))}{x} \, dx\) [265]

Optimal. Leaf size=160 \[ a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}+\frac {b d \text {PolyLog}\left (2,-\frac {x^{-n}}{c}\right )}{2 n}+\frac {b e \log \left (f x^m\right ) \text {PolyLog}\left (2,-\frac {x^{-n}}{c}\right )}{2 n}-\frac {b d \text {PolyLog}\left (2,\frac {x^{-n}}{c}\right )}{2 n}-\frac {b e \log \left (f x^m\right ) \text {PolyLog}\left (2,\frac {x^{-n}}{c}\right )}{2 n}+\frac {b e m \text {PolyLog}\left (3,-\frac {x^{-n}}{c}\right )}{2 n^2}-\frac {b e m \text {PolyLog}\left (3,\frac {x^{-n}}{c}\right )}{2 n^2} \]

[Out]

a*d*ln(x)+1/2*a*e*ln(f*x^m)^2/m+1/2*b*d*polylog(2,-1/c/(x^n))/n+1/2*b*e*ln(f*x^m)*polylog(2,-1/c/(x^n))/n-1/2*
b*d*polylog(2,1/c/(x^n))/n-1/2*b*e*ln(f*x^m)*polylog(2,1/c/(x^n))/n+1/2*b*e*m*polylog(3,-1/c/(x^n))/n^2-1/2*b*
e*m*polylog(3,1/c/(x^n))/n^2

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Rubi [A]
time = 0.41, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2338, 6874, 6036, 6032, 6219, 6217, 2421, 6724} \begin {gather*} a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}+\frac {b d \text {Li}_2\left (-\frac {x^{-n}}{c}\right )}{2 n}-\frac {b d \text {Li}_2\left (\frac {x^{-n}}{c}\right )}{2 n}+\frac {b e \text {Li}_2\left (-\frac {x^{-n}}{c}\right ) \log \left (f x^m\right )}{2 n}-\frac {b e \text {Li}_2\left (\frac {x^{-n}}{c}\right ) \log \left (f x^m\right )}{2 n}+\frac {b e m \text {Li}_3\left (-\frac {x^{-n}}{c}\right )}{2 n^2}-\frac {b e m \text {Li}_3\left (\frac {x^{-n}}{c}\right )}{2 n^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*ArcCoth[c*x^n])*(d + e*Log[f*x^m]))/x,x]

[Out]

a*d*Log[x] + (a*e*Log[f*x^m]^2)/(2*m) + (b*d*PolyLog[2, -(1/(c*x^n))])/(2*n) + (b*e*Log[f*x^m]*PolyLog[2, -(1/
(c*x^n))])/(2*n) - (b*d*PolyLog[2, 1/(c*x^n)])/(2*n) - (b*e*Log[f*x^m]*PolyLog[2, 1/(c*x^n)])/(2*n) + (b*e*m*P
olyLog[3, -(1/(c*x^n))])/(2*n^2) - (b*e*m*PolyLog[3, 1/(c*x^n)])/(2*n^2)

Rule 2338

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

Rule 2421

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

Rule 6032

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Simp[(b/2)*PolyLog[2, -(c*x)^(
-1)], x] - Simp[(b/2)*PolyLog[2, 1/(c*x)], x]) /; FreeQ[{a, b, c}, x]

Rule 6036

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

Rule 6217

Int[(ArcCoth[(c_.)*(x_)^(n_.)]*Log[(d_.)*(x_)^(m_.)])/(x_), x_Symbol] :> Dist[1/2, Int[Log[d*x^m]*(Log[1 + 1/(
c*x^n)]/x), x], x] - Dist[1/2, Int[Log[d*x^m]*(Log[1 - 1/(c*x^n)]/x), x], x] /; FreeQ[{c, d, m, n}, x]

Rule 6219

Int[(Log[(d_.)*(x_)^(m_.)]*(ArcCoth[(c_.)*(x_)^(n_.)]*(b_.) + (a_)))/(x_), x_Symbol] :> Dist[a, Int[Log[d*x^m]
/x, x], x] + Dist[b, Int[(Log[d*x^m]*ArcCoth[c*x^n])/x, x], x] /; FreeQ[{a, b, c, d, m, n}, x]

Rule 6724

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

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (a+b \coth ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx &=\int \left (\frac {d \left (a+b \coth ^{-1}\left (c x^n\right )\right )}{x}+\frac {e \left (a+b \coth ^{-1}\left (c x^n\right )\right ) \log \left (f x^m\right )}{x}\right ) \, dx\\ &=d \int \frac {a+b \coth ^{-1}\left (c x^n\right )}{x} \, dx+e \int \frac {\left (a+b \coth ^{-1}\left (c x^n\right )\right ) \log \left (f x^m\right )}{x} \, dx\\ &=(a e) \int \frac {\log \left (f x^m\right )}{x} \, dx+(b e) \int \frac {\coth ^{-1}\left (c x^n\right ) \log \left (f x^m\right )}{x} \, dx+\frac {d \text {Subst}\left (\int \frac {a+b \coth ^{-1}(c x)}{x} \, dx,x,x^n\right )}{n}\\ &=a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}+\frac {b d \text {Li}_2\left (-\frac {x^{-n}}{c}\right )}{2 n}-\frac {b d \text {Li}_2\left (\frac {x^{-n}}{c}\right )}{2 n}-\frac {1}{2} (b e) \int \frac {\log \left (f x^m\right ) \log \left (1-\frac {x^{-n}}{c}\right )}{x} \, dx+\frac {1}{2} (b e) \int \frac {\log \left (f x^m\right ) \log \left (1+\frac {x^{-n}}{c}\right )}{x} \, dx\\ &=a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}+\frac {b d \text {Li}_2\left (-\frac {x^{-n}}{c}\right )}{2 n}+\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (-\frac {x^{-n}}{c}\right )}{2 n}-\frac {b d \text {Li}_2\left (\frac {x^{-n}}{c}\right )}{2 n}-\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (\frac {x^{-n}}{c}\right )}{2 n}-\frac {(b e m) \int \frac {\text {Li}_2\left (-\frac {x^{-n}}{c}\right )}{x} \, dx}{2 n}+\frac {(b e m) \int \frac {\text {Li}_2\left (\frac {x^{-n}}{c}\right )}{x} \, dx}{2 n}\\ &=a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}+\frac {b d \text {Li}_2\left (-\frac {x^{-n}}{c}\right )}{2 n}+\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (-\frac {x^{-n}}{c}\right )}{2 n}-\frac {b d \text {Li}_2\left (\frac {x^{-n}}{c}\right )}{2 n}-\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (\frac {x^{-n}}{c}\right )}{2 n}+\frac {b e m \text {Li}_3\left (-\frac {x^{-n}}{c}\right )}{2 n^2}-\frac {b e m \text {Li}_3\left (\frac {x^{-n}}{c}\right )}{2 n^2}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.21, size = 131, normalized size = 0.82 \begin {gather*} -\frac {b c e m x^n \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2},\frac {3}{2};c^2 x^{2 n}\right )}{n^2}+\frac {b c x^n \, _3F_2\left (\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2};c^2 x^{2 n}\right ) \left (d+e \log \left (f x^m\right )\right )}{n}-\frac {1}{2} \left (a+b \coth ^{-1}\left (c x^n\right )-b \tanh ^{-1}\left (c x^n\right )\right ) \log (x) \left (e m \log (x)-2 \left (d+e \log \left (f x^m\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*ArcCoth[c*x^n])*(d + e*Log[f*x^m]))/x,x]

[Out]

-((b*c*e*m*x^n*HypergeometricPFQ[{1/2, 1/2, 1/2, 1}, {3/2, 3/2, 3/2}, c^2*x^(2*n)])/n^2) + (b*c*x^n*Hypergeome
tricPFQ[{1/2, 1/2, 1}, {3/2, 3/2}, c^2*x^(2*n)]*(d + e*Log[f*x^m]))/n - ((a + b*ArcCoth[c*x^n] - b*ArcTanh[c*x
^n])*Log[x]*(e*m*Log[x] - 2*(d + e*Log[f*x^m])))/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.21, size = 920, normalized size = 5.75

method	result	size
risch	\(\text {Expression too large to display}\)	\(920\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccoth(c*x^n))*(d+e*ln(f*x^m))/x,x,method=_RETURNVERBOSE)

[Out]

1/2*e*b*m/n*ln(x)*polylog(2,c*x^n)+1/4*I/n*Pi*dilog(c*x^n+1)*b*e*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/2*I/n*P
i*ln(x^n)*a*e*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/4*I/n*ln(c*x^n-1)*ln(c*x^n)*Pi*b*e*csgn(I*x^m)*csgn(I*f*x^
m)^2+1/4*I/n*dilog(c*x^n)*Pi*b*e*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/4*I/n*ln(c*x^n-1)*ln(c*x^n)*Pi*b*e*csgn
(I*f)*csgn(I*f*x^m)^2+1/n*ln(x^n)*a*d+1/2*e*b*ln(1-c*x^n)*ln(x^m)*ln(x)-1/2/n*ln(c*x^n-1)*ln(c*x^n)*b*d+1/4*e*
b*ln(c*x^n-1)*m*ln(x)^2-1/2*e*b*ln(c*x^n-1)*ln(x^m)*ln(x)-1/2/n*dilog(c*x^n)*b*d-1/2/n*dilog(c*x^n+1)*b*d-1/2*
e*b*m/n^2*polylog(3,c*x^n)-1/2*e*b/n*dilog(c*x^n)*ln(x^m)+1/n*ln(f)*ln(x^n)*a*e+1/2*e*a/m*ln(x^m)^2+1/2*e*b*m/
n^2*polylog(3,-c*x^n)-1/2*e*b/n*dilog(c*x^n+1)*ln(x^m)-1/2/n*ln(f)*dilog(c*x^n+1)*b*e-1/4*e*b*m*ln(x)^2*ln(1-c
*x^n)+1/2*I/n*Pi*ln(x^n)*a*e*csgn(I*x^m)*csgn(I*f*x^m)^2-1/4*I/n*Pi*dilog(c*x^n+1)*b*e*csgn(I*x^m)*csgn(I*f*x^
m)^2+1/2*I/n*Pi*ln(x^n)*a*e*csgn(I*f)*csgn(I*f*x^m)^2-1/2*e*b*m/n*ln(x)*polylog(2,-c*x^n)+1/2*e*b/n*dilog(c*x^
n+1)*m*ln(x)-1/4*I/n*Pi*dilog(c*x^n+1)*b*e*csgn(I*f)*csgn(I*f*x^m)^2-1/4*I/n*dilog(c*x^n)*Pi*b*e*csgn(I*x^m)*c
sgn(I*f*x^m)^2+1/2*e*b/n*ln(1-c*x^n)*ln(c*x^n)*m*ln(x)+1/4*I/n*ln(c*x^n-1)*ln(c*x^n)*Pi*b*e*csgn(I*f*x^m)^3-1/
4*I/n*dilog(c*x^n)*Pi*b*e*csgn(I*f)*csgn(I*f*x^m)^2-1/2*I/n*Pi*ln(x^n)*a*e*csgn(I*f*x^m)^3-1/2/n*ln(c*x^n-1)*l
n(c*x^n)*ln(f)*b*e+1/4*I/n*Pi*dilog(c*x^n+1)*b*e*csgn(I*f*x^m)^3-1/2*e*b/n*ln(1-c*x^n)*ln(c*x^n)*ln(x^m)+1/2*e
*b/n*dilog(c*x^n)*m*ln(x)+1/4*I/n*dilog(c*x^n)*Pi*b*e*csgn(I*f*x^m)^3-1/2/n*dilog(c*x^n)*ln(f)*b*e+1/4*I/n*ln(
c*x^n-1)*ln(c*x^n)*Pi*b*e*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)

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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*arccoth(c*x^n))*(d+e*log(f*x^m))/x,x, algorithm="maxima")

[Out]

1/2*a*e*log(f*x^m)^2/m + a*d*log(x) - 1/4*(b*m*e*log(x)^2 - 2*b*e*log(x)*log(x^m) - 2*(b*e*log(f) + b*d)*log(x
))*log(c*x^n + 1) + 1/4*(b*m*e*log(x)^2 - 2*b*e*log(x)*log(x^m) - 2*(b*e*log(f) + b*d)*log(x))*log(c*x^n - 1)
+ integrate(1/2*(2*b*c*n*e^(n*log(x) + 1)*log(x)*log(x^m) - (b*c*m*n*e*log(x)^2 - 2*(b*c*n*e*log(f) + b*c*d*n)
*log(x))*x^n)/(c^2*x*x^(2*n) - x), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (147) = 294\).
time = 0.41, size = 460, normalized size = 2.88 \begin {gather*} \frac {2 \, {\left (a m n^{2} \cosh \left (1\right ) + a m n^{2} \sinh \left (1\right )\right )} \log \left (x\right )^{2} + 2 \, {\left (b d n + {\left (b n \cosh \left (1\right ) + b n \sinh \left (1\right )\right )} \log \left (f\right ) + {\left (b m n \cosh \left (1\right ) + b m n \sinh \left (1\right )\right )} \log \left (x\right )\right )} {\rm Li}_2\left (c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right )\right ) - 2 \, {\left (b d n + {\left (b n \cosh \left (1\right ) + b n \sinh \left (1\right )\right )} \log \left (f\right ) + {\left (b m n \cosh \left (1\right ) + b m n \sinh \left (1\right )\right )} \log \left (x\right )\right )} {\rm Li}_2\left (-c \cosh \left (n \log \left (x\right )\right ) - c \sinh \left (n \log \left (x\right )\right )\right ) - {\left ({\left (b m n^{2} \cosh \left (1\right ) + b m n^{2} \sinh \left (1\right )\right )} \log \left (x\right )^{2} + 2 \, {\left (b d n^{2} + {\left (b n^{2} \cosh \left (1\right ) + b n^{2} \sinh \left (1\right )\right )} \log \left (f\right )\right )} \log \left (x\right )\right )} \log \left (c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right ) + 1\right ) + {\left ({\left (b m n^{2} \cosh \left (1\right ) + b m n^{2} \sinh \left (1\right )\right )} \log \left (x\right )^{2} + 2 \, {\left (b d n^{2} + {\left (b n^{2} \cosh \left (1\right ) + b n^{2} \sinh \left (1\right )\right )} \log \left (f\right )\right )} \log \left (x\right )\right )} \log \left (-c \cosh \left (n \log \left (x\right )\right ) - c \sinh \left (n \log \left (x\right )\right ) + 1\right ) + 4 \, {\left (a d n^{2} + {\left (a n^{2} \cosh \left (1\right ) + a n^{2} \sinh \left (1\right )\right )} \log \left (f\right )\right )} \log \left (x\right ) + {\left ({\left (b m n^{2} \cosh \left (1\right ) + b m n^{2} \sinh \left (1\right )\right )} \log \left (x\right )^{2} + 2 \, {\left (b d n^{2} + {\left (b n^{2} \cosh \left (1\right ) + b n^{2} \sinh \left (1\right )\right )} \log \left (f\right )\right )} \log \left (x\right )\right )} \log \left (\frac {c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right ) + 1}{c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right ) - 1}\right ) - 2 \, {\left (b m \cosh \left (1\right ) + b m \sinh \left (1\right )\right )} {\rm polylog}\left (3, c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right )\right ) + 2 \, {\left (b m \cosh \left (1\right ) + b m \sinh \left (1\right )\right )} {\rm polylog}\left (3, -c \cosh \left (n \log \left (x\right )\right ) - c \sinh \left (n \log \left (x\right )\right )\right )}{4 \, n^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccoth(c*x^n))*(d+e*log(f*x^m))/x,x, algorithm="fricas")

[Out]

1/4*(2*(a*m*n^2*cosh(1) + a*m*n^2*sinh(1))*log(x)^2 + 2*(b*d*n + (b*n*cosh(1) + b*n*sinh(1))*log(f) + (b*m*n*c
osh(1) + b*m*n*sinh(1))*log(x))*dilog(c*cosh(n*log(x)) + c*sinh(n*log(x))) - 2*(b*d*n + (b*n*cosh(1) + b*n*sin
h(1))*log(f) + (b*m*n*cosh(1) + b*m*n*sinh(1))*log(x))*dilog(-c*cosh(n*log(x)) - c*sinh(n*log(x))) - ((b*m*n^2
*cosh(1) + b*m*n^2*sinh(1))*log(x)^2 + 2*(b*d*n^2 + (b*n^2*cosh(1) + b*n^2*sinh(1))*log(f))*log(x))*log(c*cosh
(n*log(x)) + c*sinh(n*log(x)) + 1) + ((b*m*n^2*cosh(1) + b*m*n^2*sinh(1))*log(x)^2 + 2*(b*d*n^2 + (b*n^2*cosh(
1) + b*n^2*sinh(1))*log(f))*log(x))*log(-c*cosh(n*log(x)) - c*sinh(n*log(x)) + 1) + 4*(a*d*n^2 + (a*n^2*cosh(1
) + a*n^2*sinh(1))*log(f))*log(x) + ((b*m*n^2*cosh(1) + b*m*n^2*sinh(1))*log(x)^2 + 2*(b*d*n^2 + (b*n^2*cosh(1
) + b*n^2*sinh(1))*log(f))*log(x))*log((c*cosh(n*log(x)) + c*sinh(n*log(x)) + 1)/(c*cosh(n*log(x)) + c*sinh(n*
log(x)) - 1)) - 2*(b*m*cosh(1) + b*m*sinh(1))*polylog(3, c*cosh(n*log(x)) + c*sinh(n*log(x))) + 2*(b*m*cosh(1)
+ b*m*sinh(1))*polylog(3, -c*cosh(n*log(x)) - c*sinh(n*log(x))))/n^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((a+b*acoth(c*x**n))*(d+e*ln(f*x**m))/x,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*arccoth(c*x^n))*(d+e*log(f*x^m))/x,x, algorithm="giac")

[Out]

integrate((b*arccoth(c*x^n) + a)*(e*log(f*x^m) + d)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x^n\right )\right )\,\left (d+e\,\ln \left (f\,x^m\right )\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*acoth(c*x^n))*(d + e*log(f*x^m)))/x,x)

[Out]

int(((a + b*acoth(c*x^n))*(d + e*log(f*x^m)))/x, x)

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