Integrand size = 16, antiderivative size = 73 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x^4} \, dx=\frac {p}{9 x^3}-\frac {a p}{6 b x^2}+\frac {a^2 p}{3 b^2 x}-\frac {a^3 p \log \left (a+\frac {b}{x}\right )}{3 b^3}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 x^3} \]
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Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2442, 45} \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x^4} \, dx=-\frac {a^3 p \log \left (a+\frac {b}{x}\right )}{3 b^3}+\frac {a^2 p}{3 b^2 x}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 x^3}-\frac {a p}{6 b x^2}+\frac {p}{9 x^3} \]
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Rule 45
Rule 2442
Rule 2504
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int x^2 \log \left (c (a+b x)^p\right ) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 x^3}+\frac {1}{3} (b p) \text {Subst}\left (\int \frac {x^3}{a+b x} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 x^3}+\frac {1}{3} (b p) \text {Subst}\left (\int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = \frac {p}{9 x^3}-\frac {a p}{6 b x^2}+\frac {a^2 p}{3 b^2 x}-\frac {a^3 p \log \left (a+\frac {b}{x}\right )}{3 b^3}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 x^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x^4} \, dx=\frac {p}{9 x^3}-\frac {a p}{6 b x^2}+\frac {a^2 p}{3 b^2 x}-\frac {a^3 p \log \left (a+\frac {b}{x}\right )}{3 b^3}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 x^3} \]
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Time = 0.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03
method | result | size |
parts | \(-\frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{3 x^{3}}-\frac {p b \left (-\frac {1}{3 b \,x^{3}}-\frac {a^{2}}{b^{3} x}+\frac {a}{2 b^{2} x^{2}}-\frac {a^{3} \ln \left (x \right )}{b^{4}}+\frac {a^{3} \ln \left (a x +b \right )}{b^{4}}\right )}{3}\) | \(75\) |
parallelrisch | \(-\frac {6 x^{3} \ln \left (c \left (\frac {a x +b}{x}\right )^{p}\right ) a^{3} p +6 x^{3} a^{3} p^{2}-6 x^{2} a^{2} b \,p^{2}+3 x a \,b^{2} p^{2}+6 \ln \left (c \left (\frac {a x +b}{x}\right )^{p}\right ) b^{3} p -2 b^{3} p^{2}}{18 x^{3} b^{3} p}\) | \(97\) |
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Time = 0.34 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x^4} \, dx=\frac {6 \, a^{2} b p x^{2} - 3 \, a b^{2} p x + 2 \, b^{3} p - 6 \, b^{3} \log \left (c\right ) - 6 \, {\left (a^{3} p x^{3} + b^{3} p\right )} \log \left (\frac {a x + b}{x}\right )}{18 \, b^{3} x^{3}} \]
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Time = 1.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x^4} \, dx=\begin {cases} - \frac {a^{3} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{3 b^{3}} + \frac {a^{2} p}{3 b^{2} x} - \frac {a p}{6 b x^{2}} + \frac {p}{9 x^{3}} - \frac {\log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{3 x^{3}} & \text {for}\: b \neq 0 \\- \frac {\log {\left (a^{p} c \right )}}{3 x^{3}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x^4} \, dx=-\frac {1}{18} \, b p {\left (\frac {6 \, a^{3} \log \left (a x + b\right )}{b^{4}} - \frac {6 \, a^{3} \log \left (x\right )}{b^{4}} - \frac {6 \, a^{2} x^{2} - 3 \, a b x + 2 \, b^{2}}{b^{3} x^{3}}\right )} - \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{3 \, x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (63) = 126\).
Time = 0.32 (sec) , antiderivative size = 234, normalized size of antiderivative = 3.21 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x^4} \, dx=-\frac {\frac {18 \, {\left (a x + b\right )} a^{2} p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b^{2} x} - \frac {18 \, {\left (a x + b\right )} a^{2} p}{b^{2} x} - \frac {18 \, {\left (a x + b\right )}^{2} a p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b^{2} x^{2}} + \frac {18 \, {\left (a x + b\right )} a^{2} \log \left (c\right )}{b^{2} x} + \frac {9 \, {\left (a x + b\right )}^{2} a p}{b^{2} x^{2}} + \frac {6 \, {\left (a x + b\right )}^{3} p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b^{2} x^{3}} - \frac {18 \, {\left (a x + b\right )}^{2} a \log \left (c\right )}{b^{2} x^{2}} - \frac {2 \, {\left (a x + b\right )}^{3} p}{b^{2} x^{3}} + \frac {6 \, {\left (a x + b\right )}^{3} \log \left (c\right )}{b^{2} x^{3}}}{18 \, b} \]
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Time = 1.47 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x^4} \, dx=\frac {\frac {p}{3}+\frac {a^2\,p\,x^2}{b^2}-\frac {a\,p\,x}{2\,b}}{3\,x^3}-\frac {\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{3\,x^3}-\frac {2\,a^3\,p\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{3\,b^3} \]
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