Integrand size = 16, antiderivative size = 77 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=-\frac {3 b^3 n^3}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2342, 2341} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b^3 n^3}{8 x^2} \]
[In]
[Out]
Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac {1}{2} (3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx \\ & = -\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac {1}{2} \left (3 b^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx \\ & = -\frac {3 b^3 n^3}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=-\frac {4 \left (a+b \log \left (c x^n\right )\right )^3+3 b n \left (2 \left (a+b \log \left (c x^n\right )\right )^2+b n \left (2 a+b n+2 b \log \left (c x^n\right )\right )\right )}{8 x^2} \]
[In]
[Out]
Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.51
method | result | size |
parallelrisch | \(-\frac {4 b^{3} \ln \left (c \,x^{n}\right )^{3}+6 \ln \left (c \,x^{n}\right )^{2} b^{3} n +6 \ln \left (c \,x^{n}\right ) b^{3} n^{2}+3 b^{3} n^{3}+12 a \,b^{2} \ln \left (c \,x^{n}\right )^{2}+12 \ln \left (c \,x^{n}\right ) a \,b^{2} n +6 a \,b^{2} n^{2}+12 a^{2} b \ln \left (c \,x^{n}\right )+6 a^{2} b n +4 a^{3}}{8 x^{2}}\) | \(116\) |
risch | \(\text {Expression too large to display}\) | \(2673\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (69) = 138\).
Time = 0.31 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.45 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=-\frac {4 \, b^{3} n^{3} \log \left (x\right )^{3} + 3 \, b^{3} n^{3} + 4 \, b^{3} \log \left (c\right )^{3} + 6 \, a b^{2} n^{2} + 6 \, a^{2} b n + 4 \, a^{3} + 6 \, {\left (b^{3} n + 2 \, a b^{2}\right )} \log \left (c\right )^{2} + 6 \, {\left (b^{3} n^{3} + 2 \, b^{3} n^{2} \log \left (c\right ) + 2 \, a b^{2} n^{2}\right )} \log \left (x\right )^{2} + 6 \, {\left (b^{3} n^{2} + 2 \, a b^{2} n + 2 \, a^{2} b\right )} \log \left (c\right ) + 6 \, {\left (b^{3} n^{3} + 2 \, b^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} n^{2} + 2 \, a^{2} b n + 2 \, {\left (b^{3} n^{2} + 2 \, a b^{2} n\right )} \log \left (c\right )\right )} \log \left (x\right )}{8 \, x^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (76) = 152\).
Time = 0.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.18 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=- \frac {a^{3}}{2 x^{2}} - \frac {3 a^{2} b n}{4 x^{2}} - \frac {3 a^{2} b \log {\left (c x^{n} \right )}}{2 x^{2}} - \frac {3 a b^{2} n^{2}}{4 x^{2}} - \frac {3 a b^{2} n \log {\left (c x^{n} \right )}}{2 x^{2}} - \frac {3 a b^{2} \log {\left (c x^{n} \right )}^{2}}{2 x^{2}} - \frac {3 b^{3} n^{3}}{8 x^{2}} - \frac {3 b^{3} n^{2} \log {\left (c x^{n} \right )}}{4 x^{2}} - \frac {3 b^{3} n \log {\left (c x^{n} \right )}^{2}}{4 x^{2}} - \frac {b^{3} \log {\left (c x^{n} \right )}^{3}}{2 x^{2}} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=-\frac {3}{8} \, {\left (n {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} + \frac {2 \, n \log \left (c x^{n}\right )^{2}}{x^{2}}\right )} b^{3} - \frac {3}{4} \, a b^{2} {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac {b^{3} \log \left (c x^{n}\right )^{3}}{2 \, x^{2}} - \frac {3 \, a b^{2} \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac {3 \, a^{2} b n}{4 \, x^{2}} - \frac {3 \, a^{2} b \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {a^{3}}{2 \, x^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (69) = 138\).
Time = 0.35 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.64 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=-\frac {b^{3} n^{3} \log \left (x\right )^{3}}{2 \, x^{2}} - \frac {3 \, {\left (b^{3} n^{3} + 2 \, b^{3} n^{2} \log \left (c\right ) + 2 \, a b^{2} n^{2}\right )} \log \left (x\right )^{2}}{4 \, x^{2}} - \frac {3 \, {\left (b^{3} n^{3} + 2 \, b^{3} n^{2} \log \left (c\right ) + 2 \, b^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} n^{2} + 4 \, a b^{2} n \log \left (c\right ) + 2 \, a^{2} b n\right )} \log \left (x\right )}{4 \, x^{2}} - \frac {3 \, b^{3} n^{3} + 6 \, b^{3} n^{2} \log \left (c\right ) + 6 \, b^{3} n \log \left (c\right )^{2} + 4 \, b^{3} \log \left (c\right )^{3} + 6 \, a b^{2} n^{2} + 12 \, a b^{2} n \log \left (c\right ) + 12 \, a b^{2} \log \left (c\right )^{2} + 6 \, a^{2} b n + 12 \, a^{2} b \log \left (c\right ) + 4 \, a^{3}}{8 \, x^{2}} \]
[In]
[Out]
Time = 0.41 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=-\frac {\frac {a^3}{2}+\frac {3\,a^2\,b\,n}{4}+\frac {3\,a\,b^2\,n^2}{4}+\frac {3\,b^3\,n^3}{8}}{x^2}-\frac {\ln \left (c\,x^n\right )\,\left (3\,a^2\,b+3\,a\,b^2\,n+\frac {3\,b^3\,n^2}{2}\right )}{2\,x^2}-\frac {{\ln \left (c\,x^n\right )}^2\,\left (\frac {3\,n\,b^3}{2}+3\,a\,b^2\right )}{2\,x^2}-\frac {b^3\,{\ln \left (c\,x^n\right )}^3}{2\,x^2} \]
[In]
[Out]