Integrand size = 12, antiderivative size = 133 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-3 p x-\frac {\sqrt {3} \sqrt [3]{a} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt [3]{a} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right ) \]
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Time = 0.06 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2498, 327, 206, 31, 648, 631, 210, 642} \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-\frac {\sqrt [3]{a} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \sqrt [3]{a} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right )+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-3 p x \]
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Rule 31
Rule 206
Rule 210
Rule 327
Rule 631
Rule 642
Rule 648
Rule 2498
Rubi steps \begin{align*} \text {integral}& = x \log \left (c \left (a+b x^3\right )^p\right )-(3 b p) \int \frac {x^3}{a+b x^3} \, dx \\ & = -3 p x+x \log \left (c \left (a+b x^3\right )^p\right )+(3 a p) \int \frac {1}{a+b x^3} \, dx \\ & = -3 p x+x \log \left (c \left (a+b x^3\right )^p\right )+\left (\sqrt [3]{a} p\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx+\left (\sqrt [3]{a} p\right ) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx \\ & = -3 p x+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right )+\frac {1}{2} \left (3 a^{2/3} p\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx-\frac {\left (\sqrt [3]{a} p\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{b}} \\ & = -3 p x+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt [3]{a} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right )+\frac {\left (3 \sqrt [3]{a} p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}} \\ & = -3 p x-\frac {\sqrt {3} \sqrt [3]{a} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt [3]{a} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.97 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-3 p x-\frac {\sqrt {3} \sqrt [3]{a} p \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt [3]{a} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right ) \]
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Time = 0.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92
method | result | size |
default | \(x \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )-3 p b \left (\frac {x}{b}-\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a}{b}\right )\) | \(122\) |
parts | \(x \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )-3 p b \left (\frac {x}{b}-\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a}{b}\right )\) | \(122\) |
risch | \(x \ln \left (\left (b \,x^{3}+a \right )^{p}\right )+\frac {i {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) x \pi }{2}-\frac {i \pi x \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi x {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{2}+\frac {i \operatorname {csgn}\left (i c \right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} x \pi }{2}+\frac {a p \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{b}+x \ln \left (c \right )-3 p x\) | \(167\) |
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Time = 0.34 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=p x \log \left (b x^{3} + a\right ) + \sqrt {3} p \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - \frac {1}{2} \, p \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) + p \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) - 3 \, p x + x \log \left (c\right ) \]
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Time = 24.71 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.24 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\begin {cases} x \log {\left (0^{p} c \right )} & \text {for}\: a = 0 \wedge b = 0 \\- 3 p x + x \log {\left (c \left (b x^{3}\right )^{p} \right )} & \text {for}\: a = 0 \\x \log {\left (a^{p} c \right )} & \text {for}\: b = 0 \\- 3 p x + x \log {\left (c \left (a + b x^{3}\right )^{p} \right )} - \frac {3 b p \left (- \frac {a}{b}\right )^{\frac {4}{3}} \log {\left (4 x^{2} + 4 x \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{2 a} - \frac {\sqrt {3} b p \left (- \frac {a}{b}\right )^{\frac {4}{3}} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{a} + \frac {b \left (- \frac {a}{b}\right )^{\frac {4}{3}} \log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{a} & \text {otherwise} \end {cases} \]
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Time = 0.37 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.94 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-\frac {1}{2} \, b p {\left (\frac {6 \, x}{b} - \frac {2 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {a \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, a \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )} + x \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \]
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Time = 0.31 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-\frac {1}{2} \, a b p {\left (\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a b} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b^{2}}\right )} + p x \log \left (b x^{3} + a\right ) - {\left (3 \, p - \log \left (c\right )\right )} x \]
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Time = 0.45 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.01 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=x\,\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )-3\,p\,x-\frac {{\left (-a\right )}^{1/3}\,p\,\ln \left ({\left (-a\right )}^{4/3}+a\,b^{1/3}\,x\right )}{b^{1/3}}+\frac {{\left (-a\right )}^{1/3}\,p\,\ln \left (2\,a\,b^{1/3}\,x-{\left (-a\right )}^{4/3}-\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{1/3}}-\frac {{\left (-a\right )}^{1/3}\,p\,\ln \left (2\,a\,b^{1/3}\,x-{\left (-a\right )}^{4/3}+\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{1/3}} \]
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