Integrand size = 19, antiderivative size = 401 \[ \int \frac {x^3 \log (c+d x)}{a+b x^4} \, dx=\frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b}+\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b} \] 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)/b+1/4*ln(d*((-a)^(1/4)-b^(1/4)*x)/(b^(1/4)*c+(-a)^(1/4)*d))*ln (d*x+c)/b+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)/b+1/4*ln(-d*((-a)^(1/4)+b^(1/4)*x)/(b^(1/4)*c-(-a)^ (1/4)*d))*ln(d*x+c)/b+1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-(-(-a)^(1/2 ))^(1/2)*d))/b+1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+(-(-a)^(1/2))^(1/2 )*d))/b+1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-(-a)^(1/4)*d))/b+1/4*poly log(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+(-a)^(1/4)*d))/b
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \log (c+d x)}{a+b x^4} \, dx=\frac {\log \left (\frac {d \left (i \sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac {\log \left (-\frac {d \left (i \sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b} \] Input:
Integrate[(x^3*Log[c + d*x])/(a + b*x^4),x]
Output:
(Log[(d*(I*(-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)]*Log[c + d*x])/(4*b) + (Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d) ]*Log[c + d*x])/(4*b) + (Log[-((d*(I*(-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - I*(-a)^(1/4)*d))]*Log[c + d*x])/(4*b) + (Log[-((d*((-a)^(1/4) + b^(1/4)*x ))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*b) + PolyLog[2, (b^(1/4)* (c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(4*b) + PolyLog[2, (b^(1/4)*(c + d* x))/(b^(1/4)*c - I*(-a)^(1/4)*d)]/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/( b^(1/4)*c + I*(-a)^(1/4)*d)]/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/ 4)*c + (-a)^(1/4)*d)]/(4*b)
Time = 1.29 (sec) , antiderivative size = 401, 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.
\(\displaystyle \int \frac {x^3 \log (c+d x)}{a+b x^4} \, dx\) |
\(\Big \downarrow \) 2863 |
\(\displaystyle \int \left (\frac {x \log (c+d x)}{2 \left (b x^2-\sqrt {-a} \sqrt {b}\right )}+\frac {x \log (c+d x)}{2 \left (\sqrt {-a} \sqrt {b}+b x^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b}+\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 b}+\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b}+\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b}+\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b}\) |
Input:
Int[(x^3*Log[c + d*x])/(a + b*x^4),x]
Output:
(Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Lo g[c + d*x])/(4*b) + (Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1 /4)*d)]*Log[c + d*x])/(4*b) + (Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^ (1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*b) + (Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*b) + PolyLog[ 2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)]/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]/(4*b) + PolyLog[2, ( b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(4*b) + PolyLog[2, (b^(1/4) *(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)]/(4*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 = 85, normalized size of antiderivative = 0.21
method | result | size |
derivativedivides | \(\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{4 b}\) | \(85\) |
default | \(\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{4 b}\) | \(85\) |
risch | \(\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{4 b}\) | \(85\) |
parts | \(\frac {\ln \left (d x +c \right ) \ln \left (b \,x^{4}+a \right )}{4 b}-\frac {d \left (\frac {\ln \left (d x +c \right ) \ln \left (b \,x^{4}+a \right )}{d}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{d}\right )}{4 b}\) | \(130\) |
Input:
int(x^3*ln(d*x+c)/(b*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/4/b*sum(ln(d*x+c)*ln((-d*x+_R1-c)/_R1)+dilog((-d*x+_R1-c)/_R1),_R1=RootO f(_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))
\[ \int \frac {x^3 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{3} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \] Input:
integrate(x^3*log(d*x+c)/(b*x^4+a),x, algorithm="fricas")
Output:
integral(x^3*log(d*x + c)/(b*x^4 + a), x)
Timed out. \[ \int \frac {x^3 \log (c+d x)}{a+b x^4} \, dx=\text {Timed out} \] Input:
integrate(x**3*ln(d*x+c)/(b*x**4+a),x)
Output:
Timed out
\[ \int \frac {x^3 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{3} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \] Input:
integrate(x^3*log(d*x+c)/(b*x^4+a),x, algorithm="maxima")
Output:
integrate(x^3*log(d*x + c)/(b*x^4 + a), x)
\[ \int \frac {x^3 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{3} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \] Input:
integrate(x^3*log(d*x+c)/(b*x^4+a),x, algorithm="giac")
Output:
integrate(x^3*log(d*x + c)/(b*x^4 + a), x)
Timed out. \[ \int \frac {x^3 \log (c+d x)}{a+b x^4} \, dx=\int \frac {x^3\,\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \] Input:
int((x^3*log(c + d*x))/(a + b*x^4),x)
Output:
int((x^3*log(c + d*x))/(a + b*x^4), x)
\[ \int \frac {x^3 \log (c+d x)}{a+b x^4} \, dx=\int \frac {\mathrm {log}\left (d x +c \right ) x^{3}}{b \,x^{4}+a}d x \] Input:
int(x^3*log(d*x+c)/(b*x^4+a),x)
Output:
int((log(c + d*x)*x**3)/(a + b*x**4),x)