Integrand size = 18, antiderivative size = 81 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 b^2 n^2 (d x)^{1+m}}{d (1+m)^3}-\frac {2 b n (d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^2}{d (1+m)} \]
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Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2342, 2341} \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^2}{d (m+1)}-\frac {2 b n (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{d (m+1)^2}+\frac {2 b^2 n^2 (d x)^{m+1}}{d (m+1)^3} \]
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Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = \frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^2}{d (1+m)}-\frac {(2 b n) \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx}{1+m} \\ & = \frac {2 b^2 n^2 (d x)^{1+m}}{d (1+m)^3}-\frac {2 b n (d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^2}{d (1+m)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.94 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {x (d x)^m \left (a^2 (1+m)^2-2 a b (1+m) n+2 b^2 n^2+2 b (1+m) (a+a m-b n) \log \left (c x^n\right )+b^2 (1+m)^2 \log ^2\left (c x^n\right )\right )}{(1+m)^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(242\) vs. \(2(81)=162\).
Time = 0.22 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.00
method | result | size |
parallelrisch | \(-\frac {-x \left (d x \right )^{m} a^{2} m^{2}-2 x \left (d x \right )^{m} b^{2} n^{2}-2 x \left (d x \right )^{m} a^{2} m -x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{2} b^{2}-2 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) a b \,m^{2}+2 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) b^{2} m n -4 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) a b m +2 x \left (d x \right )^{m} a b m n -x \left (d x \right )^{m} a^{2}-x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{2} b^{2} m^{2}-2 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{2} b^{2} m +2 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) b^{2} n -2 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) a b +2 x \left (d x \right )^{m} a b n}{m^{3}+3 m^{2}+3 m +1}\) | \(243\) |
risch | \(\text {Expression too large to display}\) | \(2027\) |
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Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (81) = 162\).
Time = 0.28 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.57 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {{\left ({\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} x \log \left (x\right )^{2} + {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} x \log \left (c\right )^{2} + 2 \, {\left (a b m^{2} + 2 \, a b m + a b - {\left (b^{2} m + b^{2}\right )} n\right )} x \log \left (c\right ) + {\left (a^{2} m^{2} + 2 \, b^{2} n^{2} + 2 \, a^{2} m + a^{2} - 2 \, {\left (a b m + a b\right )} n\right )} x + 2 \, {\left ({\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n x \log \left (c\right ) - {\left ({\left (b^{2} m + b^{2}\right )} n^{2} - {\left (a b m^{2} + 2 \, a b m + a b\right )} n\right )} x\right )} \log \left (x\right )\right )} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (73) = 146\).
Time = 6.38 (sec) , antiderivative size = 502, normalized size of antiderivative = 6.20 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\begin {cases} \frac {a^{2} m^{2} x \left (d x\right )^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 a^{2} m x \left (d x\right )^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {a^{2} x \left (d x\right )^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 a b m^{2} x \left (d x\right )^{m} \log {\left (c x^{n} \right )}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 a b m n x \left (d x\right )^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {4 a b m x \left (d x\right )^{m} \log {\left (c x^{n} \right )}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 a b n x \left (d x\right )^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 a b x \left (d x\right )^{m} \log {\left (c x^{n} \right )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {b^{2} m^{2} x \left (d x\right )^{m} \log {\left (c x^{n} \right )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 b^{2} m n x \left (d x\right )^{m} \log {\left (c x^{n} \right )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 b^{2} m x \left (d x\right )^{m} \log {\left (c x^{n} \right )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 b^{2} n^{2} x \left (d x\right )^{m}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 b^{2} n x \left (d x\right )^{m} \log {\left (c x^{n} \right )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {b^{2} x \left (d x\right )^{m} \log {\left (c x^{n} \right )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} & \text {for}\: m \neq -1 \\\frac {\begin {cases} \frac {a^{2} \log {\left (c x^{n} \right )} + a b \log {\left (c x^{n} \right )}^{2} + \frac {b^{2} \log {\left (c x^{n} \right )}^{3}}{3}}{n} & \text {for}\: n \neq 0 \\\left (a^{2} + 2 a b \log {\left (c \right )} + b^{2} \log {\left (c \right )}^{2}\right ) \log {\left (x \right )} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.63 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^2 \, dx=-\frac {2 \, a b d^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} - 2 \, {\left (\frac {d^{m} n x x^{m} \log \left (c x^{n}\right )}{{\left (m + 1\right )}^{2}} - \frac {d^{m} n^{2} x x^{m}}{{\left (m + 1\right )}^{3}}\right )} b^{2} + \frac {\left (d x\right )^{m + 1} b^{2} \log \left (c x^{n}\right )^{2}}{d {\left (m + 1\right )}} + \frac {2 \, \left (d x\right )^{m + 1} a b \log \left (c x^{n}\right )}{d {\left (m + 1\right )}} + \frac {\left (d x\right )^{m + 1} a^{2}}{d {\left (m + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (81) = 162\).
Time = 0.36 (sec) , antiderivative size = 402, normalized size of antiderivative = 4.96 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {b^{2} d^{m} m^{2} n^{2} x x^{m} \log \left (x\right )^{2}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac {2 \, b^{2} d^{m} m n^{2} x x^{m} \log \left (x\right )^{2}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} - \frac {2 \, b^{2} d^{m} m n^{2} x x^{m} \log \left (x\right )}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac {2 \, b^{2} d^{m} m n x x^{m} \log \left (c\right ) \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac {b^{2} d^{m} n^{2} x x^{m} \log \left (x\right )^{2}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac {2 \, a b d^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac {2 \, b^{2} d^{m} n^{2} x x^{m} \log \left (x\right )}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac {2 \, b^{2} d^{m} n x x^{m} \log \left (c\right ) \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac {2 \, b^{2} d^{m} n^{2} x x^{m}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} - \frac {2 \, b^{2} d^{m} n x x^{m} \log \left (c\right )}{m^{2} + 2 \, m + 1} + \frac {2 \, a b d^{m} n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac {2 \, a b d^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {\left (d x\right )^{m} b^{2} x \log \left (c\right )^{2}}{m + 1} + \frac {2 \, \left (d x\right )^{m} a b x \log \left (c\right )}{m + 1} + \frac {\left (d x\right )^{m} a^{2} x}{m + 1} \]
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Timed out. \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int {\left (d\,x\right )}^m\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
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