\(\int \frac {\log (c+d x)}{x (a+b x^4)} \, dx\) [295]

Optimal result

Integrand size = 19, antiderivative size = 433 \[ \int \frac {\log (c+d x)}{x \left (a+b x^4\right )} \, dx=\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a}-\frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 a}-\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 a}-\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 a}-\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 a}+\frac {\operatorname {PolyLog}\left (2,1+\frac {d x}{c}\right )}{a} \]

[Out]

ln(-d*x/c)*ln(d*x+c)/a-1/4*ln(d*((-a)^(1/4)-b^(1/4)*x)/(b^(1/4)*c+(-a)^(1/4)*d))*ln(d*x+c)/a-1/4*ln(-d*((-a)^(
1/4)+b^(1/4)*x)/(b^(1/4)*c-(-a)^(1/4)*d))*ln(d*x+c)/a-1/4*ln(d*x+c)*ln(-d*(b^(1/4)*x+(-(-a)^(1/2))^(1/2))/(b^(
1/4)*c-d*(-(-a)^(1/2))^(1/2)))/a-1/4*ln(d*x+c)*ln(d*(-b^(1/4)*x+(-(-a)^(1/2))^(1/2))/(b^(1/4)*c+d*(-(-a)^(1/2)
)^(1/2)))/a-1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-(-a)^(1/4)*d))/a-1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c
+(-a)^(1/4)*d))/a+polylog(2,1+d*x/c)/a-1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-d*(-(-a)^(1/2))^(1/2)))/a-1/4*
polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+d*(-(-a)^(1/2))^(1/2)))/a

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {272, 36, 29, 31, 2463, 2441, 2352, 266, 2440, 2438} \[ \int \frac {\log (c+d x)}{x \left (a+b x^4\right )} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 a}-\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 a}-\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 a}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 a}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 a}+\frac {\operatorname {PolyLog}\left (2,\frac {d x}{c}+1\right )}{a}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a} \]

[In]

Int[Log[c + d*x]/(x*(a + b*x^4)),x]

[Out]

(Log[-((d*x)/c)]*Log[c + d*x])/a - (Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log
[c + d*x])/(4*a) - (Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*a) - (Log[-(
(d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*a) - (Log[-((d*((-a)^(1/4
) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*a) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c -
Sqrt[-Sqrt[-a]]*d)]/(4*a) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]/(4*a) - PolyLog[2
, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(4*a) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/
4)*d)]/(4*a) + PolyLog[2, 1 + (d*x)/c]/a

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 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 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 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 2438

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

Rule 2440

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

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.06 (sec) , antiderivative size = 416, normalized size of antiderivative = 0.96 \[ \int \frac {\log (c+d x)}{x \left (a+b x^4\right )} \, dx=\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a}-\frac {\log \left (\frac {d \left (i \sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 a}-\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 a}-\frac {\log \left (-\frac {d \left (i \sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 a}-\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 a}+\frac {\operatorname {PolyLog}\left (2,\frac {c+d x}{c}\right )}{a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{4 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{4 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 a} \]

[In]

Integrate[Log[c + d*x]/(x*(a + b*x^4)),x]

[Out]

(Log[-((d*x)/c)]*Log[c + d*x])/a - (Log[(d*(I*(-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)]*Log[c + d
*x])/(4*a) - (Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*a) - (Log[-((d*(I*
(-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - I*(-a)^(1/4)*d))]*Log[c + d*x])/(4*a) - (Log[-((d*((-a)^(1/4) + b^(1/4)*
x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*a) + PolyLog[2, (c + d*x)/c]/a - PolyLog[2, (b^(1/4)*(c + d*
x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(4*a) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/4)*d)]/(4*a) - P
olyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)]/(4*a) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c
+ (-a)^(1/4)*d)]/(4*a)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.65 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.26

method	result	size
derivativedivides	\(\frac {\operatorname {dilog}\left (-\frac {x d}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {x d}{c}\right )}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{4 a}\)	\(114\)
default	\(\frac {\operatorname {dilog}\left (-\frac {x d}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {x d}{c}\right )}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{4 a}\)	\(114\)
risch	\(\frac {\ln \left (-\frac {x d}{c}\right ) \ln \left (d x +c \right )}{a}+\frac {\operatorname {dilog}\left (-\frac {x d}{c}\right )}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{4 a}\)	\(116\)
parts	\(\frac {\ln \left (d x +c \right ) \ln \left (x \right )}{a}-\frac {\ln \left (d x +c \right ) \ln \left (b \,x^{4}+a \right )}{4 a}-\frac {d \left (\frac {4 \operatorname {dilog}\left (\frac {d x +c}{c}\right )}{a d}+\frac {4 \ln \left (x \right ) \ln \left (\frac {d x +c}{c}\right )}{a d}-\frac {\ln \left (d x +c \right ) \ln \left (b \,x^{4}+a \right )}{a d}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{a d}\right )}{4}\)	\(183\)

[In]

int(ln(d*x+c)/x/(b*x^4+a),x,method=_RETURNVERBOSE)

[Out]

1/a*(dilog(-x*d/c)+ln(d*x+c)*ln(-x*d/c))-1/4/a*sum(ln(d*x+c)*ln((-d*x+_R1-c)/_R1)+dilog((-d*x+_R1-c)/_R1),_R1=
RootOf(_Z^4*b-4*_Z^3*b*c+6*_Z^2*b*c^2-4*_Z*b*c^3+a*d^4+b*c^4))

Fricas [F]

\[ \int \frac {\log (c+d x)}{x \left (a+b x^4\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x} \,d x } \]

[In]

integrate(log(d*x+c)/x/(b*x^4+a),x, algorithm="fricas")

[Out]

integral(log(d*x + c)/(b*x^5 + a*x), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log (c+d x)}{x \left (a+b x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(ln(d*x+c)/x/(b*x**4+a),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\log (c+d x)}{x \left (a+b x^4\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x} \,d x } \]

[In]

integrate(log(d*x+c)/x/(b*x^4+a),x, algorithm="maxima")

[Out]

integrate(log(d*x + c)/((b*x^4 + a)*x), x)

Giac [F]

\[ \int \frac {\log (c+d x)}{x \left (a+b x^4\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x} \,d x } \]

[In]

integrate(log(d*x+c)/x/(b*x^4+a),x, algorithm="giac")

[Out]

integrate(log(d*x + c)/((b*x^4 + a)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log (c+d x)}{x \left (a+b x^4\right )} \, dx=\int \frac {\ln \left (c+d\,x\right )}{x\,\left (b\,x^4+a\right )} \,d x \]

[In]

int(log(c + d*x)/(x*(a + b*x^4)),x)

[Out]

int(log(c + d*x)/(x*(a + b*x^4)), x)