Integrand size = 19, antiderivative size = 423 \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^3\right )} \, dx=-\frac {d}{2 a c x}-\frac {d^2 \log (x)}{2 a c^2}+\frac {d^2 \log (c+d x)}{2 a c^2}-\frac {\log (c+d x)}{2 a x^2}-\frac {b^{2/3} \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{5/3}}-\frac {b^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{5/3}} \]
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Time = 0.31 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {331, 206, 31, 648, 631, 210, 642, 2463, 2442, 46, 2456, 2441, 2440, 2438} \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^3\right )} \, dx=-\frac {b^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{5/3}}-\frac {b^{2/3} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 a^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{5/3}}-\frac {d^2 \log (x)}{2 a c^2}+\frac {d^2 \log (c+d x)}{2 a c^2}-\frac {\log (c+d x)}{2 a x^2}-\frac {d}{2 a c x} \]
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Rule 31
Rule 46
Rule 206
Rule 210
Rule 331
Rule 631
Rule 642
Rule 648
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2456
Rule 2463
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\log (c+d x)}{a x^3}-\frac {b \log (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx \\ & = \frac {\int \frac {\log (c+d x)}{x^3} \, dx}{a}-\frac {b \int \frac {\log (c+d x)}{a+b x^3} \, dx}{a} \\ & = -\frac {\log (c+d x)}{2 a x^2}-\frac {b \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}{a}+\frac {d \int \frac {1}{x^2 (c+d x)} \, dx}{2 a} \\ & = -\frac {\log (c+d x)}{2 a x^2}+\frac {b \int \frac {\log (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 a^{5/3}}+\frac {b \int \frac {\log (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{5/3}}+\frac {b \int \frac {\log (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{5/3}}+\frac {d \int \left (\frac {1}{c x^2}-\frac {d}{c^2 x}+\frac {d^2}{c^2 (c+d x)}\right ) \, dx}{2 a} \\ & = -\frac {d}{2 a c x}-\frac {d^2 \log (x)}{2 a c^2}+\frac {d^2 \log (c+d x)}{2 a c^2}-\frac {\log (c+d x)}{2 a x^2}-\frac {b^{2/3} \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^{5/3}}-\frac {(-1)^{2/3} b^{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^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \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^{5/3}}+\frac {\left (b^{2/3} d\right ) \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^{5/3}}-\frac {\left (\sqrt [3]{-1} b^{2/3} 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^{5/3}}+\frac {\left ((-1)^{2/3} b^{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^{5/3}} \\ & = -\frac {d}{2 a c x}-\frac {d^2 \log (x)}{2 a c^2}+\frac {d^2 \log (c+d x)}{2 a c^2}-\frac {\log (c+d x)}{2 a x^2}-\frac {b^{2/3} \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^{5/3}}-\frac {(-1)^{2/3} b^{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^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \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^{5/3}}+\frac {b^{2/3} \text {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^{5/3}}-\frac {\left (\sqrt [3]{-1} b^{2/3}\right ) \text {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^{5/3}}+\frac {\left ((-1)^{2/3} b^{2/3}\right ) \text {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^{5/3}} \\ & = -\frac {d}{2 a c x}-\frac {d^2 \log (x)}{2 a c^2}+\frac {d^2 \log (c+d x)}{2 a c^2}-\frac {\log (c+d x)}{2 a x^2}-\frac {b^{2/3} \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^{5/3}}-\frac {(-1)^{2/3} b^{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^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \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^{5/3}}-\frac {b^{2/3} \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \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^{5/3}}-\frac {(-1)^{2/3} b^{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^{5/3}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.88 \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^3\right )} \, dx=\frac {-\frac {3 a^{2/3} \log (c+d x)}{x^2}-2 b^{2/3} \log \left (\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)-2 (-1)^{2/3} b^{2/3} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)+2 \sqrt [3]{-1} b^{2/3} \log \left (\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{-(-1)^{2/3} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)-\frac {3 a^{2/3} d (c+d x \log (x)-d x \log (c+d x))}{c^2 x}-2 b^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )+2 \sqrt [3]{-1} b^{2/3} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )-2 (-1)^{2/3} b^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{6 a^{5/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.63 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.35
method | result | size |
derivativedivides | \(d^{2} \left (\frac {-\frac {\ln \left (-d x \right )}{2 c^{2}}-\frac {1}{2 c d x}-\frac {\ln \left (d x +c \right ) \left (d x +c \right ) \left (-d x +c \right )}{2 c^{2} d^{2} x^{2}}}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{3 a}\right )\) | \(150\) |
default | \(d^{2} \left (\frac {-\frac {\ln \left (-d x \right )}{2 c^{2}}-\frac {1}{2 c d x}-\frac {\ln \left (d x +c \right ) \left (d x +c \right ) \left (-d x +c \right )}{2 c^{2} d^{2} x^{2}}}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{3 a}\right )\) | \(150\) |
risch | \(-\frac {d^{2} \ln \left (-d x \right )}{2 a \,c^{2}}-\frac {d}{2 a c x}+\frac {d^{2} \ln \left (d x +c \right )}{2 a \,c^{2}}-\frac {\ln \left (d x +c \right )}{2 a \,x^{2}}-\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 a}\) | \(154\) |
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\[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{3}} \,d x } \]
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\[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^3\right )} \, dx=\int \frac {\ln \left (c+d\,x\right )}{x^3\,\left (b\,x^3+a\right )} \,d x \]
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