3.1.89 \(\int \frac {\cos (a+b \log (c x^n))}{x} \, dx\) [89]

Optimal. Leaf size=18 \[ \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[Out]

sin(a+b*ln(c*x^n))/b/n

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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2717} \begin {gather*} \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*Log[c*x^n]]/x,x]

[Out]

Sin[a + b*Log[c*x^n]]/(b*n)

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\cos \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \cos (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(18)=36\).
time = 0.04, size = 37, normalized size = 2.06 \begin {gather*} \frac {\cos \left (b \log \left (c x^n\right )\right ) \sin (a)}{b n}+\frac {\cos (a) \sin \left (b \log \left (c x^n\right )\right )}{b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*Log[c*x^n]]/x,x]

[Out]

(Cos[b*Log[c*x^n]]*Sin[a])/(b*n) + (Cos[a]*Sin[b*Log[c*x^n]])/(b*n)

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Maple [A]
time = 0.03, size = 19, normalized size = 1.06

method	result	size
derivativedivides	\(\frac {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\)	\(19\)
default	\(\frac {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a+b*ln(c*x^n))/x,x,method=_RETURNVERBOSE)

[Out]

sin(a+b*ln(c*x^n))/b/n

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Maxima [A]
time = 0.27, size = 18, normalized size = 1.00 \begin {gather*} \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )}{b n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*log(c*x^n))/x,x, algorithm="maxima")

[Out]

sin(b*log(c*x^n) + a)/(b*n)

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Fricas [A]
time = 2.31, size = 19, normalized size = 1.06 \begin {gather*} \frac {\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*log(c*x^n))/x,x, algorithm="fricas")

[Out]

sin(b*n*log(x) + b*log(c) + a)/(b*n)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (14) = 28\).
time = 0.26, size = 34, normalized size = 1.89 \begin {gather*} \begin {cases} \log {\left (x \right )} \cos {\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cos {\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\sin {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b n} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*ln(c*x**n))/x,x)

[Out]

Piecewise((log(x)*cos(a), Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log(x)*cos(a + b*log(c)), Eq(n, 0)), (sin(a + b*
log(c*x**n))/(b*n), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*log(c*x^n))/x,x, algorithm="giac")

[Out]

integrate(cos(b*log(c*x^n) + a)/x, x)

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Mupad [B]
time = 2.28, size = 18, normalized size = 1.00 \begin {gather*} \frac {\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}{b\,n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*log(c*x^n))/x,x)

[Out]

sin(a + b*log(c*x^n))/(b*n)

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