3.223 \(\int \frac {\cot ^3(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=44 \[ -\frac {\log \left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3473, 3475} \[ -\frac {\log \left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*Log[c*x^n]]^3/x,x]

[Out]

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

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

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 ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \cot ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\operatorname {Subst}\left (\int \cot (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\log \left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 52, normalized size = 1.18 \[ -\frac {2 \log \left (\tan \left (a+b \log \left (c x^n\right )\right )\right )+2 \log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )+\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*Log[c*x^n]]^3/x,x]

[Out]

-1/2*(Cot[a + b*Log[c*x^n]]^2 + 2*Log[Cos[a + b*Log[c*x^n]]] + 2*Log[Tan[a + b*Log[c*x^n]]])/(b*n)

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fricas [A]  time = 0.74, size = 70, normalized size = 1.59 \[ -\frac {{\left (\cos \left (2 \, b n \log \relax (x) + 2 \, b \log \relax (c) + 2 \, a\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b n \log \relax (x) + 2 \, b \log \relax (c) + 2 \, a\right ) + \frac {1}{2}\right ) - 2}{2 \, {\left (b n \cos \left (2 \, b n \log \relax (x) + 2 \, b \log \relax (c) + 2 \, a\right ) - b n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+b*log(c*x^n))^3/x,x, algorithm="fricas")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+b*log(c*x^n))^3/x,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.01, size = 47, normalized size = 1.07 \[ -\frac {\cot ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n}+\frac {\ln \left (\cot ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2 n b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a+b*ln(c*x^n))^3/x,x)

[Out]

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

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maxima [B]  time = 1.89, size = 1713, normalized size = 38.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+b*log(c*x^n))^3/x,x, algorithm="maxima")

[Out]

-1/2*(8*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*cos(2*b*log(x^n) + 2*a)^2 + 8*(cos(2*b*log(c))^2 + sin(2*b*log
(c))^2)*sin(2*b*log(x^n) + 2*a)^2 - 4*((cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*cos
(2*b*log(x^n) + 2*a) + (cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) +
2*a))*cos(4*b*log(x^n) + 4*a) - 4*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + ((cos(4*b*log(c))^2 + sin(4*b*log(
c))^2)*cos(4*b*log(x^n) + 4*a)^2 + 4*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*cos(2*b*log(x^n) + 2*a)^2 + (cos(
4*b*log(c))^2 + sin(4*b*log(c))^2)*sin(4*b*log(x^n) + 4*a)^2 + 4*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*sin(2
*b*log(x^n) + 2*a)^2 - 2*(2*(cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^
n) + 2*a) + 2*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) - co
s(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 4*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + 2*(2*(cos(2*b*log(c))*sin
(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 2*(cos(4*b*log(c))*cos(2*b*log(c)) +
sin(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) - sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) + 4*sin(2
*b*log(c))*sin(2*b*log(x^n) + 2*a) + 1)*log((cos(a)^2 + sin(a)^2)*cos(b*log(c))^2 + (cos(a)^2 + sin(a)^2)*sin(
b*log(c))^2 + 2*(cos(b*log(c))*cos(a) - sin(b*log(c))*sin(a))*cos(b*log(x^n)) + cos(b*log(x^n))^2 - 2*(cos(a)*
sin(b*log(c)) + cos(b*log(c))*sin(a))*sin(b*log(x^n)) + sin(b*log(x^n))^2) + ((cos(4*b*log(c))^2 + sin(4*b*log
(c))^2)*cos(4*b*log(x^n) + 4*a)^2 + 4*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*cos(2*b*log(x^n) + 2*a)^2 + (cos
(4*b*log(c))^2 + sin(4*b*log(c))^2)*sin(4*b*log(x^n) + 4*a)^2 + 4*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*sin(
2*b*log(x^n) + 2*a)^2 - 2*(2*(cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x
^n) + 2*a) + 2*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) - c
os(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 4*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + 2*(2*(cos(2*b*log(c))*si
n(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 2*(cos(4*b*log(c))*cos(2*b*log(c))
+ sin(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) - sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) + 4*sin(
2*b*log(c))*sin(2*b*log(x^n) + 2*a) + 1)*log((cos(a)^2 + sin(a)^2)*cos(b*log(c))^2 + (cos(a)^2 + sin(a)^2)*sin
(b*log(c))^2 - 2*(cos(b*log(c))*cos(a) - sin(b*log(c))*sin(a))*cos(b*log(x^n)) + cos(b*log(x^n))^2 + 2*(cos(a)
*sin(b*log(c)) + cos(b*log(c))*sin(a))*sin(b*log(x^n)) + sin(b*log(x^n))^2) + 4*((cos(2*b*log(c))*sin(4*b*log(
c)) - cos(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - (cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*lo
g(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a) + 4*sin(2*b*log(c))*sin(2*b*log(x^n) +
2*a))/((b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*cos(4*b*log(x^n) + 4*a)^2 - 4*b*n*cos(2*b*log(c))*cos(2*
b*log(x^n) + 2*a) + 4*(b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*cos(2*b*log(x^n) + 2*a)^2 + (b*cos(4*b*log
(c))^2 + b*sin(4*b*log(c))^2)*n*sin(4*b*log(x^n) + 4*a)^2 + 4*b*n*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + 4*
(b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*sin(2*b*log(x^n) + 2*a)^2 + b*n + 2*(b*n*cos(4*b*log(c)) - 2*(b*
cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) - 2*(b*cos(2*b*
log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a
) + 2*(2*(b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) - b
*n*sin(4*b*log(c)) - 2*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x
^n) + 2*a))*sin(4*b*log(x^n) + 4*a))

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mupad [B]  time = 4.69, size = 106, normalized size = 2.41 \[ \ln \relax (x)\,1{}\mathrm {i}+\frac {2}{b\,n\,\left (1+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\right )}+\frac {2}{b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )}-\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )}{b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + b*log(c*x^n))^3/x,x)

[Out]

log(x)*1i + 2/(b*n*(exp(a*4i)*(c*x^n)^(b*4i) - 2*exp(a*2i)*(c*x^n)^(b*2i) + 1)) + 2/(b*n*(exp(a*2i)*(c*x^n)^(b
*2i) - 1)) - log(exp(a*2i)*(c*x^n)^(b*2i) - 1)/(b*n)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+b*ln(c*x**n))**3/x,x)

[Out]

Timed out

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