Integrand size = 17, antiderivative size = 473 \[ \int \frac {x \log (c+d x)}{a+b x^4} \, dx=-\frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \sqrt {-a} \sqrt {b}}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 \sqrt {-a} \sqrt {b}}-\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \sqrt {-a} \sqrt {b}}+\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 \sqrt {-a} \sqrt {b}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 \sqrt {-a} \sqrt {b}} \] Output:
-1/4*ln(d*((-(-a)^(1/2))^(1/2)-b^(1/4)*x)/(b^(1/4)*c+(-(-a)^(1/2))^(1/2)*d ))*ln(d*x+c)/(-a)^(1/2)/b^(1/2)+1/4*ln(d*((-a)^(1/4)-b^(1/4)*x)/(b^(1/4)*c +(-a)^(1/4)*d))*ln(d*x+c)/(-a)^(1/2)/b^(1/2)-1/4*ln(-d*((-(-a)^(1/2))^(1/2 )+b^(1/4)*x)/(b^(1/4)*c-(-(-a)^(1/2))^(1/2)*d))*ln(d*x+c)/(-a)^(1/2)/b^(1/ 2)+1/4*ln(-d*((-a)^(1/4)+b^(1/4)*x)/(b^(1/4)*c-(-a)^(1/4)*d))*ln(d*x+c)/(- a)^(1/2)/b^(1/2)-1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-(-(-a)^(1/2))^(1 /2)*d))/(-a)^(1/2)/b^(1/2)-1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+(-(-a) ^(1/2))^(1/2)*d))/(-a)^(1/2)/b^(1/2)+1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4 )*c-(-a)^(1/4)*d))/(-a)^(1/2)/b^(1/2)+1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/ 4)*c+(-a)^(1/4)*d))/(-a)^(1/2)/b^(1/2)
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.74 \[ \int \frac {x \log (c+d x)}{a+b x^4} \, dx=\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\log \left (\frac {d \left (\sqrt [4]{-a}-i \sqrt [4]{b} x\right )}{i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\log \left (\frac {d \left (\sqrt [4]{-a}+i \sqrt [4]{b} x\right )}{-i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)+\log \left (\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)+\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 \sqrt {-a} \sqrt {b}} \] Input:
Integrate[(x*Log[c + d*x])/(a + b*x^4),x]
Output:
(Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x] - Log[(d*((-a)^(1/4) - I*b^(1/4)*x))/(I*b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x] - Log[(d*((-a)^(1/4) + I*b^(1/4)*x))/((-I)*b^(1/4)*c + (-a)^(1/4)*d )]*Log[c + d*x] + Log[(d*((-a)^(1/4) + b^(1/4)*x))/(-(b^(1/4)*c) + (-a)^(1 /4)*d)]*Log[c + d*x] + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1 /4)*d)] - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/4)*d)] - P olyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)] + PolyLog[2, ( b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*Sqrt[-a]*Sqrt[b])
Time = 1.34 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.
| \(\displaystyle \int \frac {x \log (c+d x)}{a+b x^4} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (-\frac {\sqrt {b} x \log (c+d x)}{2 \sqrt {-a} \left (\sqrt {-a} \sqrt {b}-b x^2\right )}-\frac {\sqrt {b} x \log (c+d x)}{2 \sqrt {-a} \left (\sqrt {-a} \sqrt {b}+b x^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 \sqrt {-a} \sqrt {b}}\) |
Input:
Int[(x*Log[c + d*x])/(a + b*x^4),x]
Output:
-1/4*(Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d )]*Log[c + d*x])/(Sqrt[-a]*Sqrt[b]) + (Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b ^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*Sqrt[-a]*Sqrt[b]) - (Log[-((d*( Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d* x])/(4*Sqrt[-a]*Sqrt[b]) + (Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*Sqrt[-a]*Sqrt[b]) - PolyLog[2, (b^(1/4) *(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)]/(4*Sqrt[-a]*Sqrt[b]) - PolyLo g[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]/(4*Sqrt[-a]*Sqrt [b]) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(4*Sqrt[ -a]*Sqrt[b]) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)]/ (4*Sqrt[-a]*Sqrt[b])
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 = 1.45 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.22
| method | result | size |
| derivativedivides | \(\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{4 b}\) | \(102\) |
| default | \(\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{4 b}\) | \(102\) |
| risch | \(\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{4 b}\) | \(102\) |
Input:
int(x*ln(d*x+c)/(b*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/4*d^2/b*sum(1/(_R1^2-2*_R1*c+c^2)*(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))
\[ \int \frac {x \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x \log \left (d x + c\right )}{b x^{4} + a} \,d x } \] Input:
integrate(x*log(d*x+c)/(b*x^4+a),x, algorithm="fricas")
Output:
integral(x*log(d*x + c)/(b*x^4 + a), x)
Timed out. \[ \int \frac {x \log (c+d x)}{a+b x^4} \, dx=\text {Timed out} \] Input:
integrate(x*ln(d*x+c)/(b*x**4+a),x)
Output:
Timed out
\[ \int \frac {x \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x \log \left (d x + c\right )}{b x^{4} + a} \,d x } \] Input:
integrate(x*log(d*x+c)/(b*x^4+a),x, algorithm="maxima")
Output:
integrate(x*log(d*x + c)/(b*x^4 + a), x)
\[ \int \frac {x \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x \log \left (d x + c\right )}{b x^{4} + a} \,d x } \] Input:
integrate(x*log(d*x+c)/(b*x^4+a),x, algorithm="giac")
Output:
integrate(x*log(d*x + c)/(b*x^4 + a), x)
Timed out. \[ \int \frac {x \log (c+d x)}{a+b x^4} \, dx=\int \frac {x\,\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \] Input:
int((x*log(c + d*x))/(a + b*x^4),x)
Output:
int((x*log(c + d*x))/(a + b*x^4), x)
\[ \int \frac {x \log (c+d x)}{a+b x^4} \, dx=\int \frac {\mathrm {log}\left (d x +c \right ) x}{b \,x^{4}+a}d x \] Input:
int(x*log(d*x+c)/(b*x^4+a),x)
Output:
int((log(c + d*x)*x)/(a + b*x**4),x)