Integrand size = 16, antiderivative size = 45 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^4} \, dx=\frac {b p \log (x)}{a}-\frac {b p \log \left (a+b x^3\right )}{3 a}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{3 x^3} \]
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Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2504, 2442, 36, 29, 31} \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^4} \, dx=-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{3 x^3}-\frac {b p \log \left (a+b x^3\right )}{3 a}+\frac {b p \log (x)}{a} \]
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Rule 29
Rule 31
Rule 36
Rule 2442
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x^2} \, dx,x,x^3\right ) \\ & = -\frac {\log \left (c \left (a+b x^3\right )^p\right )}{3 x^3}+\frac {1}{3} (b p) \text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,x^3\right ) \\ & = -\frac {\log \left (c \left (a+b x^3\right )^p\right )}{3 x^3}+\frac {(b p) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^3\right )}{3 a}-\frac {\left (b^2 p\right ) \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,x^3\right )}{3 a} \\ & = \frac {b p \log (x)}{a}-\frac {b p \log \left (a+b x^3\right )}{3 a}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{3 x^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^4} \, dx=\frac {b p \log (x)}{a}-\frac {b p \log \left (a+b x^3\right )}{3 a}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{3 x^3} \]
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Time = 0.40 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93
method | result | size |
parts | \(-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{3 x^{3}}+p b \left (\frac {\ln \left (x \right )}{a}-\frac {\ln \left (b \,x^{3}+a \right )}{3 a}\right )\) | \(42\) |
parallelrisch | \(\frac {3 p^{2} b \ln \left (x \right ) x^{3}-x^{3} \ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) b p -\ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) a p}{3 x^{3} a p}\) | \(59\) |
risch | \(-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right )}{3 x^{3}}-\frac {i \pi a \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2}-i \pi a \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi a {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}+i \pi a {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-6 b p \ln \left (x \right ) x^{3}+2 b p \ln \left (b \,x^{3}+a \right ) x^{3}+2 \ln \left (c \right ) a}{6 x^{3} a}\) | \(173\) |
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Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^4} \, dx=\frac {3 \, b p x^{3} \log \left (x\right ) - {\left (b p x^{3} + a p\right )} \log \left (b x^{3} + a\right ) - a \log \left (c\right )}{3 \, a x^{3}} \]
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Time = 1.80 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.44 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^4} \, dx=\begin {cases} - \frac {\log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{3 x^{3}} + \frac {b p \log {\left (x \right )}}{a} - \frac {b \log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{3 a} & \text {for}\: a \neq 0 \\- \frac {p}{3 x^{3}} - \frac {\log {\left (c \left (b x^{3}\right )^{p} \right )}}{3 x^{3}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^4} \, dx=-\frac {1}{3} \, b p {\left (\frac {\log \left (b x^{3} + a\right )}{a} - \frac {\log \left (x^{3}\right )}{a}\right )} - \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{3 \, x^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.29 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^4} \, dx=-\frac {\frac {b^{2} p \log \left (b x^{3} + a\right )}{a} - \frac {b^{2} p \log \left (b x^{3}\right )}{a} + \frac {b p \log \left (b x^{3} + a\right )}{x^{3}} + \frac {b \log \left (c\right )}{x^{3}}}{3 \, b} \]
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Time = 1.34 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^4} \, dx=\frac {b\,p\,\ln \left (x\right )}{a}-\frac {b\,p\,\ln \left (b\,x^3+a\right )}{3\,a}-\frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{3\,x^3} \]
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