\(\int \frac {\log (c+d x)}{a+b x^3} \, dx\) [290]

Optimal result

Integrand size = 16, antiderivative size = 359 \[ \int \frac {\log (c+d x)}{a+b x^3} \, dx=\frac {\log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {\sqrt [3]{-1} \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{2/3} \sqrt [3]{b}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\sqrt [3]{-1} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}} \] Output:

1/3*ln(-d*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*c-a^(1/3)*d))*ln(d*x+c)/a^(2/3)/b^( 
1/3)+1/3*(-1)^(2/3)*ln(d*(a^(1/3)-(-1)^(1/3)*b^(1/3)*x)/((-1)^(1/3)*b^(1/3 
)*c+a^(1/3)*d))*ln(d*x+c)/a^(2/3)/b^(1/3)-1/3*(-1)^(1/3)*ln(-d*(a^(1/3)+(- 
1)^(2/3)*b^(1/3)*x)/((-1)^(2/3)*b^(1/3)*c-a^(1/3)*d))*ln(d*x+c)/a^(2/3)/b^ 
(1/3)+1/3*polylog(2,b^(1/3)*(d*x+c)/(b^(1/3)*c-a^(1/3)*d))/a^(2/3)/b^(1/3) 
-1/3*(-1)^(1/3)*polylog(2,(-1)^(2/3)*b^(1/3)*(d*x+c)/((-1)^(2/3)*b^(1/3)*c 
-a^(1/3)*d))/a^(2/3)/b^(1/3)+1/3*(-1)^(2/3)*polylog(2,(-1)^(1/3)*b^(1/3)*( 
d*x+c)/((-1)^(1/3)*b^(1/3)*c+a^(1/3)*d))/a^(2/3)/b^(1/3)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.82 \[ \int \frac {\log (c+d x)}{a+b x^3} \, dx=\frac {\log \left (\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)+(-1)^{2/3} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)-\sqrt [3]{-1} \log \left (\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{-(-1)^{2/3} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)+\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )-\sqrt [3]{-1} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )+(-1)^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}} \] Input:

Integrate[Log[c + d*x]/(a + b*x^3),x]
 

Output:

(Log[(d*(a^(1/3) + b^(1/3)*x))/(-(b^(1/3)*c) + a^(1/3)*d)]*Log[c + d*x] + 
(-1)^(2/3)*Log[(d*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((-1)^(1/3)*b^(1/3)*c 
+ a^(1/3)*d)]*Log[c + d*x] - (-1)^(1/3)*Log[(d*(a^(1/3) + (-1)^(2/3)*b^(1/ 
3)*x))/(-((-1)^(2/3)*b^(1/3)*c) + a^(1/3)*d)]*Log[c + d*x] + PolyLog[2, (b 
^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)] - (-1)^(1/3)*PolyLog[2, ((-1)^( 
2/3)*b^(1/3)*(c + d*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d)] + (-1)^(2/3)*P 
olyLog[2, ((-1)^(1/3)*b^(1/3)*(c + d*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d 
)])/(3*a^(2/3)*b^(1/3))
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 359, 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.

\(\displaystyle \int \frac {\log (c+d x)}{a+b x^3} \, dx\)

\(\Big \downarrow \) 2856

\(\displaystyle \int \left (-\frac {\log (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\log (c+d x)}{3 a^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{b} x-\sqrt [3]{a}\right )}-\frac {\log (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\sqrt [3]{-1} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\sqrt [3]{-1} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}\)

Input:

Int[Log[c + d*x]/(a + b*x^3),x]
 

Output:

(Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/( 
3*a^(2/3)*b^(1/3)) + ((-1)^(2/3)*Log[(d*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/ 
((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)]*Log[c + d*x])/(3*a^(2/3)*b^(1/3)) - (( 
-1)^(1/3)*Log[-((d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/((-1)^(2/3)*b^(1/3)*c 
 - a^(1/3)*d))]*Log[c + d*x])/(3*a^(2/3)*b^(1/3)) + PolyLog[2, (b^(1/3)*(c 
 + d*x))/(b^(1/3)*c - a^(1/3)*d)]/(3*a^(2/3)*b^(1/3)) - ((-1)^(1/3)*PolyLo 
g[2, ((-1)^(2/3)*b^(1/3)*(c + d*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d)])/( 
3*a^(2/3)*b^(1/3)) + ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*b^(1/3)*(c + d*x)) 
/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)])/(3*a^(2/3)*b^(1/3))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2856

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]))
 

Maple [C] (verified)

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

Time = 1.37 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.26

method	result	size
derivativedivides	\(\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b}\)	\(94\)
default	\(\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b}\)	\(94\)
risch	\(\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b}\)	\(94\)

Input:

int(ln(d*x+c)/(b*x^3+a),x,method=_RETURNVERBOSE)
 

Output:

1/3*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^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))
 

Fricas [F]

\[ \int \frac {\log (c+d x)}{a+b x^3} \, dx=\int { \frac {\log \left (d x + c\right )}{b x^{3} + a} \,d x } \] Input:

integrate(log(d*x+c)/(b*x^3+a),x, algorithm="fricas")
 

Output:

integral(log(d*x + c)/(b*x^3 + a), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\log (c+d x)}{a+b x^3} \, dx=\text {Timed out} \] Input:

integrate(ln(d*x+c)/(b*x**3+a),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\log (c+d x)}{a+b x^3} \, dx=\int { \frac {\log \left (d x + c\right )}{b x^{3} + a} \,d x } \] Input:

integrate(log(d*x+c)/(b*x^3+a),x, algorithm="maxima")
 

Output:

integrate(log(d*x + c)/(b*x^3 + a), x)
 

Giac [F]

\[ \int \frac {\log (c+d x)}{a+b x^3} \, dx=\int { \frac {\log \left (d x + c\right )}{b x^{3} + a} \,d x } \] Input:

integrate(log(d*x+c)/(b*x^3+a),x, algorithm="giac")
 

Output:

integrate(log(d*x + c)/(b*x^3 + a), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log (c+d x)}{a+b x^3} \, dx=\int \frac {\ln \left (c+d\,x\right )}{b\,x^3+a} \,d x \] Input:

int(log(c + d*x)/(a + b*x^3),x)
 

Output:

int(log(c + d*x)/(a + b*x^3), x)
 

Reduce [F]

\[ \int \frac {\log (c+d x)}{a+b x^3} \, dx=\int \frac {\mathrm {log}\left (d x +c \right )}{b \,x^{3}+a}d x \] Input:

int(log(d*x+c)/(b*x^3+a),x)
 

Output:

int(log(c + d*x)/(a + b*x**3),x)