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 = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, 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 8
Rule 2635
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 = 0.10, size = 51, normalized size = 0.70 \[ \frac {12 \left (a+b \log \left (c x^n\right )\right )+8 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 59, normalized size = 0.81 \[ \frac {3 \, b n \log \relax (x) + {\left (2 \, \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + 3 \, \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{8 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 84, normalized size = 1.15 \[ \frac {\left (\cos ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{4 b n}+\frac {3 \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{8 b n}+\frac {3 \ln \left (c \,x^{n}\right )}{8 n}+\frac {3 a}{8 b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.37, size = 93, normalized size = 1.27 \[ \frac {12 \, b n \log \relax (x) + \cos \left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right ) \sin \left (4 \, b \log \relax (c)\right ) + 8 \, \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) \sin \left (2 \, b \log \relax (c)\right ) + \cos \left (4 \, b \log \relax (c)\right ) \sin \left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right ) + 8 \, \cos \left (2 \, b \log \relax (c)\right ) \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}{32 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.55, size = 50, normalized size = 0.68 \[ \frac {3\,\ln \left (x^n\right )}{8\,n}+\frac {\frac {\sin \left (2\,a+2\,b\,\ln \left (c\,x^n\right )\right )}{4}+\frac {\sin \left (4\,a+4\,b\,\ln \left (c\,x^n\right )\right )}{32}}{b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 15.35, size = 110, normalized size = 1.51 \[ \frac {\begin {cases} \log {\relax (x )} \cos {\left (2 a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\relax (x )} \cos {\left (2 a + 2 b \log {\relax (c )} \right )} & \text {for}\: n = 0 \\\frac {\sin {\left (2 a + 2 b n \log {\relax (x )} + 2 b \log {\relax (c )} \right )}}{2 b n} & \text {otherwise} \end {cases}}{2} + \frac {\begin {cases} \log {\relax (x )} \cos {\left (4 a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\relax (x )} \cos {\left (4 a + 4 b \log {\relax (c )} \right )} & \text {for}\: n = 0 \\\frac {\sin {\left (4 a + 4 b n \log {\relax (x )} + 4 b \log {\relax (c )} \right )}}{4 b n} & \text {otherwise} \end {cases}}{8} + \frac {3 \log {\relax (x )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________