Optimal. Leaf size=73 \[ \frac{\sin \left (a+b \log \left (c x^n\right )\right ) \cos ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{3 \log (x)}{8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0441463, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2635, 8} \[ \frac{\sin \left (a+b \log \left (c x^n\right )\right ) \cos ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{3 \log (x)}{8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \cos ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \operatorname{Subst}\left (\int \cos ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n}\\ &=\frac{3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{\cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \operatorname{Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{8 n}\\ &=\frac{3 \log (x)}{8}+\frac{3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{\cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end{align*}
Mathematica [A] time = 0.10127, size = 51, normalized size = 0.7 \[ \frac{12 \left (a+b \log \left (c x^n\right )\right )+8 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.027, size = 84, normalized size = 1.2 \begin{align*}{\frac{ \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}\sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{4\,bn}}+{\frac{3\,\cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{8\,bn}}+{\frac{3\,\ln \left ( c{x}^{n} \right ) }{8\,n}}+{\frac{3\,a}{8\,bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.14422, size = 126, normalized size = 1.73 \begin{align*} \frac{12 \, b n \log \left (x\right ) + \cos \left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right ) \sin \left (4 \, b \log \left (c\right )\right ) + 8 \, \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) \sin \left (2 \, b \log \left (c\right )\right ) + \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right ) + 8 \, \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}{32 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.500121, size = 177, normalized size = 2.42 \begin{align*} \frac{3 \, b n \log \left (x\right ) +{\left (2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{8 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 121.549, size = 110, normalized size = 1.51 \begin{align*} \frac{\begin{cases} \log{\left (x \right )} \cos{\left (2 a \right )} & \text{for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log{\left (x \right )} \cos{\left (2 a + 2 b \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\\frac{\sin{\left (2 a + 2 b n \log{\left (x \right )} + 2 b \log{\left (c \right )} \right )}}{2 b n} & \text{otherwise} \end{cases}}{2} + \frac{\begin{cases} \log{\left (x \right )} \cos{\left (4 a \right )} & \text{for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log{\left (x \right )} \cos{\left (4 a + 4 b \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\\frac{\sin{\left (4 a + 4 b n \log{\left (x \right )} + 4 b \log{\left (c \right )} \right )}}{4 b n} & \text{otherwise} \end{cases}}{8} + \frac{3 \log{\left (x \right )}}{8} \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 )^{4}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]