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*ln(c*x^n)))/b/n

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Rubi [A]  time = 0.02, antiderivative size = 20, 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 = {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*}

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Mathematica [B]  time = 0.06, size = 54, normalized size = 2.70 \[ \frac {\log \left (\sin \left (\frac {a}{2}+\frac {1}{2} b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\log \left (\cos \left (\frac {a}{2}+\frac {1}{2} b \log \left (c x^n\right )\right )\right )}{b n} \]

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)

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fricas [B]  time = 0.43, size = 45, normalized size = 2.25 \[ -\frac {\log \left (\frac {1}{2} \, \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + \frac {1}{2}\right ) - \log \left (-\frac {1}{2} \, \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + \frac {1}{2}\right )}{2 \, b n} \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b \log \left (c x^{n}\right ) + a\right )}{x}\,{d x} \]

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)

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maple [A]  time = 0.03, size = 33, normalized size = 1.65 \[ -\frac {\ln \left (\csc \left (a +b \ln \left (c \,x^{n}\right )\right )+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{n b} \]

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)))

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maxima [A]  time = 0.31, size = 32, normalized size = 1.60 \[ -\frac {\log \left (\cot \left (b \log \left (c x^{n}\right ) + a\right ) + \csc \left (b \log \left (c x^{n}\right ) + a\right )\right )}{b n} \]

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)

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mupad [B]  time = 4.00, size = 68, normalized size = 3.40 \[ \frac {\ln \left (\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,2{}\mathrm {i}-2{}\mathrm {i}}{x}\right )}{b\,n}-\frac {\ln \left (\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,2{}\mathrm {i}+2{}\mathrm {i}}{x}\right )}{b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*sin(a + b*log(c*x^n))),x)

[Out]

log((exp(a*1i)*(c*x^n)^(b*1i)*2i - 2i)/x)/(b*n) - log((exp(a*1i)*(c*x^n)^(b*1i)*2i + 2i)/x)/(b*n)

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sympy [A]  time = 2.29, size = 49, normalized size = 2.45 \[ - \begin {cases} - \log {\relax (x )} \csc {\relax (a )} & \text {for}\: b = 0 \\- \log {\relax (x )} \csc {\left (a + b \log {\relax (c )} \right )} & \text {for}\: n = 0 \\\frac {\log {\left (\cot {\left (a + b \log {\left (c x^{n} \right )} \right )} + \csc {\left (a + b \log {\left (c x^{n} \right )} \right )} \right )}}{b n} & \text {otherwise} \end {cases} \]

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))

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