3.292 \(\int \frac{\csc (a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=20 \[ -\frac{\tanh ^{-1}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.015723, antiderivative size = 20, normalized size of antiderivative = 1., 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 = {3770} \[ -\frac{\tanh ^{-1}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*Log[c*x^n]]/x,x]

[Out]

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

Rule 3770

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

Rubi steps

\begin{align*} \int \frac{\csc \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \csc (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\tanh ^{-1}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ \end{align*}

Mathematica [B]  time = 0.0571344, size = 54, normalized size = 2.7 \[ \frac{\log \left (\sin \left (\frac{a}{2}+\frac{1}{2} b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\log \left (\cos \left (\frac{a}{2}+\frac{1}{2} b \log \left (c x^n\right )\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*Log[c*x^n]]/x,x]

[Out]

-(Log[Cos[a/2 + (b*Log[c*x^n])/2]]/(b*n)) + Log[Sin[a/2 + (b*Log[c*x^n])/2]]/(b*n)

________________________________________________________________________________________

Maple [A]  time = 0.027, size = 33, normalized size = 1.7 \begin{align*} -{\frac{\ln \left ( \csc \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +\cot \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.995549, size = 43, normalized size = 2.15 \begin{align*} -\frac{\log \left (\cot \left (b \log \left (c x^{n}\right ) + a\right ) + \csc \left (b \log \left (c x^{n}\right ) + a\right )\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(cot(b*log(c*x^n) + a) + csc(b*log(c*x^n) + a))/(b*n)

________________________________________________________________________________________

Fricas [B]  time = 0.497139, size = 147, normalized size = 7.35 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \frac{1}{2}\right ) - \log \left (-\frac{1}{2} \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \frac{1}{2}\right )}{2 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 2.88101, size = 49, normalized size = 2.45 \begin{align*} - \begin{cases} - \log{\left (x \right )} \csc{\left (a \right )} & \text{for}\: b = 0 \\- \log{\left (x \right )} \csc{\left (a + b \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\\frac{\log{\left (\cot{\left (a + b \log{\left (c x^{n} \right )} \right )} + \csc{\left (a + b \log{\left (c x^{n} \right )} \right )} \right )}}{b n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Piecewise((-log(x)*csc(a), Eq(b, 0)), (-log(x)*csc(a + b*log(c)), Eq(n, 0)), (log(cot(a + b*log(c*x**n)) + cs
c(a + b*log(c*x**n)))/(b*n), True))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b \log \left (c x^{n}\right ) + a\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csc(b*log(c*x^n) + a)/x, x)