Optimal. Leaf size=359 \[ \frac {\text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\sqrt [3]{-1} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\sqrt [3]{-1} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2409, 2394, 2393, 2391} \[ \frac {\text {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\sqrt [3]{-1} \text {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \text {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\sqrt [3]{-1} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rubi steps
\begin {align*} \int \frac {\log (c+d x)}{a+b x^3} \, dx &=\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]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\log (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\log (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 a^{2/3}}-\frac {\int \frac {\log (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{2/3}}-\frac {\int \frac {\log (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{2/3}}\\ &=\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 {d \int \frac {\log \left (\frac {d \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (\sqrt [3]{-1} d\right ) \int \frac {\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 )}{c+d x} \, dx}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left ((-1)^{2/3} d\right ) \int \frac {\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 )}{c+d x} \, dx}{3 a^{2/3} \sqrt [3]{b}}\\ &=\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 {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\sqrt [3]{-1} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {(-1)^{2/3} \sqrt [3]{b} x}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {(-1)^{2/3} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{-1} \sqrt [3]{b} x}{-\sqrt [3]{-1} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 a^{2/3} \sqrt [3]{b}}\\ &=\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 {\text {Li}_2\left (\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} \text {Li}_2\left (\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} \text {Li}_2\left (\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}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 294, normalized size = 0.82 \[ \frac {\text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )-\sqrt [3]{-1} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )+(-1)^{2/3} \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )+\log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} c}\right )+(-1)^{2/3} \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )-\sqrt [3]{-1} \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} c}\right )}{3 a^{2/3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (d x + c\right )}{b x^{3} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (d x + c\right )}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.26, size = 94, normalized size = 0.26 \[ \frac {d^{2} \left (\ln \left (\frac {-d x +\RootOf \left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )-c}{\RootOf \left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}\right ) \ln \left (d x +c \right )+\dilog \left (\frac {-d x +\RootOf \left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )-c}{\RootOf \left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}\right )\right )}{3 b \left (\RootOf \left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )^{2}-2 \RootOf \left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right ) c +c^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (d x + c\right )}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (c+d\,x\right )}{b\,x^3+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________