Optimal. Leaf size=56 \[ \frac{b n \sin \left (a+b \log \left (c x^n\right )\right )}{x \left (b^2 n^2+1\right )}-\frac{\cos \left (a+b \log \left (c x^n\right )\right )}{x \left (b^2 n^2+1\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.015031, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {4486} \[ \frac{b n \sin \left (a+b \log \left (c x^n\right )\right )}{x \left (b^2 n^2+1\right )}-\frac{\cos \left (a+b \log \left (c x^n\right )\right )}{x \left (b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4486
Rubi steps
\begin{align*} \int \frac{\cos \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=-\frac{\cos \left (a+b \log \left (c x^n\right )\right )}{\left (1+b^2 n^2\right ) x}+\frac{b n \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+b^2 n^2\right ) x}\\ \end{align*}
Mathematica [A] time = 0.0565324, size = 41, normalized size = 0.73 \[ \frac{b n \sin \left (a+b \log \left (c x^n\right )\right )-\cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2 x+x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.09827, size = 281, normalized size = 5.02 \begin{align*} \frac{{\left ({\left (b \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \sin \left (b \log \left (c\right )\right )\right )} n - \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) - \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) - \cos \left (b \log \left (c\right )\right )\right )} \cos \left (b \log \left (x^{n}\right ) + a\right ) +{\left ({\left (b \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \cos \left (b \log \left (c\right )\right )\right )} n + \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + \sin \left (b \log \left (c\right )\right )\right )} \sin \left (b \log \left (x^{n}\right ) + a\right )}{2 \,{\left ({\left (b^{2} \cos \left (b \log \left (c\right )\right )^{2} + b^{2} \sin \left (b \log \left (c\right )\right )^{2}\right )} n^{2} + \cos \left (b \log \left (c\right )\right )^{2} + \sin \left (b \log \left (c\right )\right )^{2}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.487415, size = 120, normalized size = 2.14 \begin{align*} \frac{b n \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{{\left (b^{2} n^{2} + 1\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 25.1753, size = 287, normalized size = 5.12 \begin{align*} \begin{cases} - \frac{i \log{\left (x \right )} \sin{\left (- a + i \log{\left (x \right )} + \frac{i \log{\left (c \right )}}{n} \right )}}{2 x} + \frac{\log{\left (x \right )} \cos{\left (- a + i \log{\left (x \right )} + \frac{i \log{\left (c \right )}}{n} \right )}}{2 x} - \frac{i \sin{\left (- a + i \log{\left (x \right )} + \frac{i \log{\left (c \right )}}{n} \right )}}{2 x} - \frac{i \log{\left (c \right )} \sin{\left (- a + i \log{\left (x \right )} + \frac{i \log{\left (c \right )}}{n} \right )}}{2 n x} + \frac{\log{\left (c \right )} \cos{\left (- a + i \log{\left (x \right )} + \frac{i \log{\left (c \right )}}{n} \right )}}{2 n x} & \text{for}\: b = - \frac{i}{n} \\- \frac{i \log{\left (x \right )} \sin{\left (a + i \log{\left (x \right )} + \frac{i \log{\left (c \right )}}{n} \right )}}{2 x} + \frac{\log{\left (x \right )} \cos{\left (a + i \log{\left (x \right )} + \frac{i \log{\left (c \right )}}{n} \right )}}{2 x} - \frac{i \sin{\left (a + i \log{\left (x \right )} + \frac{i \log{\left (c \right )}}{n} \right )}}{2 x} - \frac{i \log{\left (c \right )} \sin{\left (a + i \log{\left (x \right )} + \frac{i \log{\left (c \right )}}{n} \right )}}{2 n x} + \frac{\log{\left (c \right )} \cos{\left (a + i \log{\left (x \right )} + \frac{i \log{\left (c \right )}}{n} \right )}}{2 n x} & \text{for}\: b = \frac{i}{n} \\\frac{b n \sin{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{b^{2} n^{2} x + x} - \frac{\cos{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{b^{2} n^{2} x + x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b \log \left (c x^{n}\right ) + a\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]