Optimal. Leaf size=52 \[ \frac{1}{2} x^2 \log ^2\left (c \left (b x^n\right )^p\right )-\frac{1}{2} n p x^2 \log \left (c \left (b x^n\right )^p\right )+\frac{1}{4} n^2 p^2 x^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0446348, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2305, 2304, 2445} \[ \frac{1}{2} x^2 \log ^2\left (c \left (b x^n\right )^p\right )-\frac{1}{2} n p x^2 \log \left (c \left (b x^n\right )^p\right )+\frac{1}{4} n^2 p^2 x^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2305
Rule 2304
Rule 2445
Rubi steps
\begin{align*} \int x \log ^2\left (c \left (b x^n\right )^p\right ) \, dx &=\operatorname{Subst}\left (\int x \log ^2\left (b^p c x^{n p}\right ) \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=\frac{1}{2} x^2 \log ^2\left (c \left (b x^n\right )^p\right )-\operatorname{Subst}\left ((n p) \int x \log \left (b^p c x^{n p}\right ) \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=\frac{1}{4} n^2 p^2 x^2-\frac{1}{2} n p x^2 \log \left (c \left (b x^n\right )^p\right )+\frac{1}{2} x^2 \log ^2\left (c \left (b x^n\right )^p\right )\\ \end{align*}
Mathematica [A] time = 0.0058748, size = 43, normalized size = 0.83 \[ \frac{1}{4} x^2 \left (2 \log ^2\left (c \left (b x^n\right )^p\right )-2 n p \log \left (c \left (b x^n\right )^p\right )+n^2 p^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int x \left ( \ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.18958, size = 62, normalized size = 1.19 \begin{align*} \frac{1}{4} \, n^{2} p^{2} x^{2} - \frac{1}{2} \, n p x^{2} \log \left (\left (b x^{n}\right )^{p} c\right ) + \frac{1}{2} \, x^{2} \log \left (\left (b x^{n}\right )^{p} c\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.94347, size = 292, normalized size = 5.62 \begin{align*} \frac{1}{2} \, n^{2} p^{2} x^{2} \log \left (x\right )^{2} + \frac{1}{4} \, n^{2} p^{2} x^{2} - \frac{1}{2} \, n p^{2} x^{2} \log \left (b\right ) + \frac{1}{2} \, p^{2} x^{2} \log \left (b\right )^{2} + \frac{1}{2} \, x^{2} \log \left (c\right )^{2} - \frac{1}{2} \,{\left (n p x^{2} - 2 \, p x^{2} \log \left (b\right )\right )} \log \left (c\right ) - \frac{1}{2} \,{\left (n^{2} p^{2} x^{2} - 2 \, n p^{2} x^{2} \log \left (b\right ) - 2 \, n p x^{2} \log \left (c\right )\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 2.81901, size = 133, normalized size = 2.56 \begin{align*} \frac{n^{2} p^{2} x^{2} \log{\left (x \right )}^{2}}{2} - \frac{n^{2} p^{2} x^{2} \log{\left (x \right )}}{2} + \frac{n^{2} p^{2} x^{2}}{4} + n p^{2} x^{2} \log{\left (b \right )} \log{\left (x \right )} - \frac{n p^{2} x^{2} \log{\left (b \right )}}{2} + n p x^{2} \log{\left (c \right )} \log{\left (x \right )} - \frac{n p x^{2} \log{\left (c \right )}}{2} + \frac{p^{2} x^{2} \log{\left (b \right )}^{2}}{2} + p x^{2} \log{\left (b \right )} \log{\left (c \right )} + \frac{x^{2} \log{\left (c \right )}^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.26043, size = 151, normalized size = 2.9 \begin{align*} \frac{1}{2} \, n^{2} p^{2} x^{2} \log \left (x\right )^{2} - \frac{1}{2} \, n^{2} p^{2} x^{2} \log \left (x\right ) + n p^{2} x^{2} \log \left (b\right ) \log \left (x\right ) + \frac{1}{4} \, n^{2} p^{2} x^{2} - \frac{1}{2} \, n p^{2} x^{2} \log \left (b\right ) + \frac{1}{2} \, p^{2} x^{2} \log \left (b\right )^{2} + n p x^{2} \log \left (c\right ) \log \left (x\right ) - \frac{1}{2} \, n p x^{2} \log \left (c\right ) + p x^{2} \log \left (b\right ) \log \left (c\right ) + \frac{1}{2} \, x^{2} \log \left (c\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]