3.213 \(\int \frac{\cot (d (a+b \log (c x^n)))}{x} \, dx\)

Optimal. Leaf size=25 \[ \frac{\log \left (\sin \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]

[Out]

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

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Rubi [A]  time = 0.0184988, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {3475} \[ \frac{\log \left (\sin \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

Int[Cot[d*(a + b*Log[c*x^n])]/x,x]

[Out]

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

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0591389, size = 40, normalized size = 1.6 \[ \frac{\log \left (\tan \left (a d+b d \log \left (c x^n\right )\right )\right )+\log \left (\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[d*(a + b*Log[c*x^n])]/x,x]

[Out]

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

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Maple [A]  time = 0.019, size = 30, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ( \left ( \cot \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{2}+1 \right ) }{2\,bdn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*(a+b*ln(c*x^n)))/x,x)

[Out]

-1/2/n/d/b*ln(cot(d*(a+b*ln(c*x^n)))^2+1)

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Maxima [A]  time = 0.971066, size = 32, normalized size = 1.28 \begin{align*} \frac{\log \left (\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\right )}{b d n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.502593, size = 97, normalized size = 3.88 \begin{align*} \frac{\log \left (-\frac{1}{2} \, \cos \left (2 \, b d n \log \left (x\right ) + 2 \, b d \log \left (c\right ) + 2 \, a d\right ) + \frac{1}{2}\right )}{2 \, b d n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(-1/2*cos(2*b*d*n*log(x) + 2*b*d*log(c) + 2*a*d) + 1/2)/(b*d*n)

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Sympy [A]  time = 5.00643, size = 46, normalized size = 1.84 \begin{align*} \begin{cases} \log{\left (x \right )} \cot{\left (a d \right )} & \text{for}\: b = 0 \\\tilde{\infty } \log{\left (x \right )} & \text{for}\: d = 0 \\\log{\left (x \right )} \cot{\left (a d + b d \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\\frac{\log{\left (\sin{\left (a d + b d \log{\left (c x^{n} \right )} \right )} \right )}}{b d n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((log(x)*cot(a*d), Eq(b, 0)), (zoo*log(x), Eq(d, 0)), (log(x)*cot(a*d + b*d*log(c)), Eq(n, 0)), (log(
sin(a*d + b*d*log(c*x**n)))/(b*d*n), True))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError