Integrand size = 19, antiderivative size = 536 \[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^4\right )} \, dx=\frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x}+\frac {\sqrt [4]{b} \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt {-a}\right )^{5/2}}+\frac {\sqrt [4]{b} \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{5/4}}-\frac {\sqrt [4]{b} \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt {-a}\right )^{5/2}}-\frac {\sqrt [4]{b} \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{5/4}}-\frac {\sqrt [4]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{5/2}}+\frac {\sqrt [4]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{5/2}}-\frac {\sqrt [4]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{5/4}}+\frac {\sqrt [4]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{5/4}} \]
d*ln(x)/a/c-d*ln(d*x+c)/a/c-ln(d*x+c)/a/x+1/4*b^(1/4)*ln(d*((-a)^(1/4)-b^( 1/4)*x)/(b^(1/4)*c+(-a)^(1/4)*d))*ln(d*x+c)/(-a)^(5/4)-1/4*b^(1/4)*ln(-d*( (-a)^(1/4)+b^(1/4)*x)/(b^(1/4)*c-(-a)^(1/4)*d))*ln(d*x+c)/(-a)^(5/4)-1/4*b ^(1/4)*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-(-a)^(1/4)*d))/(-a)^(5/4)+1/4* b^(1/4)*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+(-a)^(1/4)*d))/(-a)^(5/4)-1/4 *b^(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)^(1/2))^(5/2)+1/4*b^(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)^(1/2))^( 5/2)-1/4*b^(1/4)*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-d*(-(-a)^(1/2))^(1/2 )))/(-(-a)^(1/2))^(5/2)+1/4*b^(1/4)*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+d *(-(-a)^(1/2))^(1/2)))/(-(-a)^(1/2))^(5/2)
Time = 0.58 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.00 \[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^4\right )} \, dx=\frac {\frac {a \sqrt [4]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{\sqrt {-\sqrt {-a}}}-\frac {-4 a d x \log (x)+4 a c \log (c+d x)+4 a d x \log (c+d x)+\frac {a \sqrt [4]{b} c x \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{\sqrt {-\sqrt {-a}}}-(-a)^{3/4} \sqrt [4]{b} c x \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\frac {a \sqrt [4]{b} c x \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{\sqrt {-\sqrt {-a}}}+(-a)^{3/4} \sqrt [4]{b} c x \log \left (\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)+\frac {a \sqrt [4]{b} c x \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{\sqrt {-\sqrt {-a}}}+(-a)^{3/4} \sqrt [4]{b} c x \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )-(-a)^{3/4} \sqrt [4]{b} c x \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{c x}}{4 a^2} \]
((a*b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d) ])/Sqrt[-Sqrt[-a]] - (-4*a*d*x*Log[x] + 4*a*c*Log[c + d*x] + 4*a*d*x*Log[c + d*x] + (a*b^(1/4)*c*x*Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/Sqrt[-Sqrt[-a]] - (-a)^(3/4)*b^(1/4)*c *x*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d* x] - (a*b^(1/4)*c*x*Log[(d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(-(b^(1/4)*c) + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/Sqrt[-Sqrt[-a]] + (-a)^(3/4)*b^(1/4)*c*x *Log[(d*((-a)^(1/4) + b^(1/4)*x))/(-(b^(1/4)*c) + (-a)^(1/4)*d)]*Log[c + d *x] + (a*b^(1/4)*c*x*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqr t[-a]]*d)])/Sqrt[-Sqrt[-a]] + (-a)^(3/4)*b^(1/4)*c*x*PolyLog[2, (b^(1/4)*( c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)] - (-a)^(3/4)*b^(1/4)*c*x*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(c*x))/(4*a^2)
Time = 1.01 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2863, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\log (c+d x)}{x^2 \left (a+b x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {\log (c+d x)}{a x^2}-\frac {b x^2 \log (c+d x)}{a \left (a+b x^4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [4]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{5/2}}+\frac {\sqrt [4]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{5/2}}-\frac {\sqrt [4]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{5/4}}+\frac {\sqrt [4]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{5/4}}+\frac {\sqrt [4]{b} \log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 \left (-\sqrt {-a}\right )^{5/2}}+\frac {\sqrt [4]{b} \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{5/4}}-\frac {\sqrt [4]{b} \log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{5/2}}-\frac {\sqrt [4]{b} \log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{5/4}}+\frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x}\) |
(d*Log[x])/(a*c) - (d*Log[c + d*x])/(a*c) - Log[c + d*x]/(a*x) + (b^(1/4)* Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log [c + d*x])/(4*(-Sqrt[-a])^(5/2)) + (b^(1/4)*Log[(d*((-a)^(1/4) - b^(1/4)*x ))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*(-a)^(5/4)) - (b^(1/4)*Log [-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log [c + d*x])/(4*(-Sqrt[-a])^(5/2)) - (b^(1/4)*Log[-((d*((-a)^(1/4) + b^(1/4) *x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)^(5/4)) - (b^(1/4)* PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*(-Sqrt [-a])^(5/2)) + (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[- Sqrt[-a]]*d)])/(4*(-Sqrt[-a])^(5/2)) - (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d *x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(4*(-a)^(5/4)) + (b^(1/4)*PolyLog[2, (b^ (1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*(-a)^(5/4))
3.4.2.3.1 Defintions of rubi rules used
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.65 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.25
| method | result | size |
| derivativedivides | \(d \left (\frac {\frac {\ln \left (-d x \right )}{c}-\frac {\ln \left (d x +c \right ) \left (d x +c \right )}{c d x}}{a}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}}{4 a}\right )\) | \(132\) |
| default | \(d \left (\frac {\frac {\ln \left (-d x \right )}{c}-\frac {\ln \left (d x +c \right ) \left (d x +c \right )}{c d x}}{a}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}}{4 a}\right )\) | \(132\) |
| risch | \(\frac {d \ln \left (-d x \right )}{a c}-\frac {d \ln \left (d x +c \right )}{a c}-\frac {\ln \left (d x +c \right )}{a x}+\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}\right )}{4 a}\) | \(137\) |
d*(1/a*(1/c*ln(-d*x)-ln(d*x+c)*(d*x+c)/c/d/x)+1/4/a*sum(1/(-_R1+c)*(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)))
\[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^4\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^4\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x^{2}} \,d x } \]
\[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^4\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^4\right )} \, dx=\int \frac {\ln \left (c+d\,x\right )}{x^2\,\left (b\,x^4+a\right )} \,d x \]