Optimal. Leaf size=359 \[ \frac{\text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac{\sqrt [3]{-1} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac{(-1)^{2/3} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac{\log (c+d x) \log \left (-\frac{d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac{(-1)^{2/3} \log (c+d x) \log \left (\frac{d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac{\sqrt [3]{-1} \log (c+d x) \log \left (-\frac{d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{2/3} \sqrt [3]{b}} \]
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Rubi [A] time = 0.238828, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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.
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Rule 2409
Rule 2394
Rule 2393
Rule 2391
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 = 0.0696822, size = 294, normalized size = 0.82 \[ \frac{\text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )-\sqrt [3]{-1} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )+(-1)^{2/3} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )+\log (c+d x) \log \left (\frac{d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} c}\right )+(-1)^{2/3} \log (c+d x) \log \left (\frac{d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )-\sqrt [3]{-1} \log (c+d x) \log \left (\frac{d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} c}\right )}{3 a^{2/3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.377, size = 94, normalized size = 0.3 \begin{align*}{\frac{{d}^{2}}{3\,b}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{3}-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }{\frac{1}{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}} \left ( \ln \left ( dx+c \right ) \ln \left ({\frac{-dx+{\it \_R1}-c}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{-dx+{\it \_R1}-c}{{\it \_R1}}} \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (d x + c\right )}{b x^{3} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (d x + c\right )}{b x^{3} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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