Integrand size = 16, antiderivative size = 497 \[ \int \frac {\log (c+d x)}{a+b x^4} \, dx=\frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/4} \sqrt [4]{b}}-\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/4} \sqrt [4]{b}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/4} \sqrt [4]{b}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/4} \sqrt [4]{b}} \] 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)/(-(-a)^(1/2))^(3/2)/b^(1/4)+1/4*ln(d*((-a)^(1/4)-b^(1/4)*x)/(b ^(1/4)*c+(-a)^(1/4)*d))*ln(d*x+c)/(-a)^(3/4)/b^(1/4)-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))^(3/2)/b^(1/4)-1/4*ln(-d*((-a)^(1/4)+b^(1/4)*x)/(b^(1/4)*c-(-a)^(1/4) *d))*ln(d*x+c)/(-a)^(3/4)/b^(1/4)-1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c -(-(-a)^(1/2))^(1/2)*d))/(-(-a)^(1/2))^(3/2)/b^(1/4)+1/4*polylog(2,b^(1/4) *(d*x+c)/(b^(1/4)*c+(-(-a)^(1/2))^(1/2)*d))/(-(-a)^(1/2))^(3/2)/b^(1/4)-1/ 4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-(-a)^(1/4)*d))/(-a)^(3/4)/b^(1/4)+1 /4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+(-a)^(1/4)*d))/(-a)^(3/4)/b^(1/4)
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.72 \[ \int \frac {\log (c+d x)}{a+b x^4} \, dx=\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-i \log \left (\frac {d \left (\sqrt [4]{-a}-i \sqrt [4]{b} x\right )}{i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)+i \log \left (\frac {d \left (\sqrt [4]{-a}+i \sqrt [4]{b} x\right )}{-i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\log \left (\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )+i \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/4} \sqrt [4]{b}} \] Input:
Integrate[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] - I*Log[(d*((-a)^(1/4) - I*b^(1/4)*x))/(I*b^(1/4)*c + (-a)^(1/4)*d)]*Log[ c + d*x] + I*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)] - I*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/4)*d )] + I*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)] + Poly Log[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*(-a)^(3/4)*b^(1 /4))
Time = 1.30 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2856, 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.
| \(\displaystyle \int \frac {\log (c+d x)}{a+b x^4} \, dx\) |
| \(\Big \downarrow \) 2856 |
| \(\displaystyle \int \left (\frac {\sqrt {-a} \log (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x^2\right )}+\frac {\sqrt {-a} \log (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/4} \sqrt [4]{b}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/4} \sqrt [4]{b}}+\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/4} \sqrt [4]{b}}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/4} \sqrt [4]{b}}\) |
Input:
Int[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*(-Sqrt[-a])^(3/2)*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)^(3/4)*b^(1/4)) - (Lo g[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Lo g[c + d*x])/(4*(-Sqrt[-a])^(3/2)*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)^(3/4)*b^(1/4)) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)]/(4*(-Sqrt[ -a])^(3/2)*b^(1/4)) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sq rt[-a]]*d)]/(4*(-Sqrt[-a])^(3/2)*b^(1/4)) - PolyLog[2, (b^(1/4)*(c + d*x)) /(b^(1/4)*c - (-a)^(1/4)*d)]/(4*(-a)^(3/4)*b^(1/4)) + PolyLog[2, (b^(1/4)* (c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)]/(4*(-a)^(3/4)*b^(1/4))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. )*(x_)^(r_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x) ^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && I GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.42 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.23
| method | result | size |
| derivativedivides | \(-\frac {d^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1}^{3}+3 c \,\textit {\_R1}^{2}-3 c^{2} \textit {\_R1} +c^{3}}\right )}{4 b}\) | \(112\) |
| default | \(-\frac {d^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1}^{3}+3 c \,\textit {\_R1}^{2}-3 c^{2} \textit {\_R1} +c^{3}}\right )}{4 b}\) | \(112\) |
| risch | \(-\frac {d^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1}^{3}+3 c \,\textit {\_R1}^{2}-3 c^{2} \textit {\_R1} +c^{3}}\right )}{4 b}\) | \(112\) |
Input:
int(ln(d*x+c)/(b*x^4+a),x,method=_RETURNVERBOSE)
Output:
-1/4*d^3/b*sum(1/(-_R1^3+3*_R1^2*c-3*_R1*c^2+c^3)*(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))
\[ \int \frac {\log (c+d x)}{a+b x^4} \, dx=\int { \frac {\log \left (d x + c\right )}{b x^{4} + a} \,d x } \] Input:
integrate(log(d*x+c)/(b*x^4+a),x, algorithm="fricas")
Output:
integral(log(d*x + c)/(b*x^4 + a), x)
Timed out. \[ \int \frac {\log (c+d x)}{a+b x^4} \, dx=\text {Timed out} \] Input:
integrate(ln(d*x+c)/(b*x**4+a),x)
Output:
Timed out
\[ \int \frac {\log (c+d x)}{a+b x^4} \, dx=\int { \frac {\log \left (d x + c\right )}{b x^{4} + a} \,d x } \] Input:
integrate(log(d*x+c)/(b*x^4+a),x, algorithm="maxima")
Output:
integrate(log(d*x + c)/(b*x^4 + a), x)
\[ \int \frac {\log (c+d x)}{a+b x^4} \, dx=\int { \frac {\log \left (d x + c\right )}{b x^{4} + a} \,d x } \] Input:
integrate(log(d*x+c)/(b*x^4+a),x, algorithm="giac")
Output:
integrate(log(d*x + c)/(b*x^4 + a), x)
Timed out. \[ \int \frac {\log (c+d x)}{a+b x^4} \, dx=\int \frac {\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \] Input:
int(log(c + d*x)/(a + b*x^4),x)
Output:
int(log(c + d*x)/(a + b*x^4), x)
\[ \int \frac {\log (c+d x)}{a+b x^4} \, dx=\int \frac {\mathrm {log}\left (d x +c \right )}{b \,x^{4}+a}d x \] Input:
int(log(d*x+c)/(b*x^4+a),x)
Output:
int(log(c + d*x)/(a + b*x**4),x)