\(\int \frac {\cot ^3(a+b \log (c x^n))}{x} \, dx\) [223]

Optimal result

Integrand size = 17, antiderivative size = 44 \[ \int \frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\log \left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3554, 3556} \[ \int \frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\log \left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[In]

Int[Cot[a + b*Log[c*x^n]]^3/x,x]

[Out]

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

Rule 3554

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 3556

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*} \text {integral}& = \frac {\text {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 {\text {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*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.18 \[ \int \frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\cot ^2\left (a+b \log \left (c x^n\right )\right )+2 \log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )+2 \log \left (\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n} \]

[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)

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {-\frac {{\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}+\frac {\ln \left ({\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}+1\right )}{2}}{n b}\)	\(42\)
default	\(\frac {-\frac {{\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}+\frac {\ln \left ({\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}+1\right )}{2}}{n b}\)	\(42\)
parallelrisch	\(\frac {-2 \ln \left (\tan \left (a +b \ln \left (c \,x^{n}\right )\right )\right )+\ln \left ({\sec \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}\right )-{\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2 b n}\)	\(53\)
risch	\(-i \ln \left (x \right )+\frac {2 i a}{n b}+\frac {2 i \ln \left (c \right )}{n}+\frac {2 i \ln \left (x^{n}\right )}{n}+\frac {\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{n}-\frac {\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{n}-\frac {\pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{n}+\frac {\pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{n}+\frac {2 \left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}}{b n {\left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}-1\right )}^{2}}-\frac {\ln \left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}-1\right )}{b n}\)	\(453\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.59 \[ \int \frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {{\left (\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + \frac {1}{2}\right ) - 2}{2 \, {\left (b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) - b n\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (36) = 72\).

Time = 4.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.20 \[ \int \frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\log {\left (x \right )} \cot ^{3}{\left (a \right )} & \text {for}\: b = 0 \\\log {\left (x \right )} \cot ^{3}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\tilde {\infty } \log {\left (x \right )} & \text {for}\: a = - b \log {\left (c x^{n} \right )} \\\frac {\log {\left (\tan ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} + 1 \right )}}{2 b n} - \frac {\log {\left (\tan {\left (a + b \log {\left (c x^{n} \right )} \right )} \right )}}{b n} - \frac {1}{2 b n \tan ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (log(x)*cot(a)**3, Eq(b, 0)), (log(x)*cot(a + b*log(c)
)**3, Eq(n, 0)), (zoo*log(x), Eq(a, -b*log(c*x**n))), (log(tan(a + b*log(c*x**n))**2 + 1)/(2*b*n) - log(tan(a
+ b*log(c*x**n)))/(b*n) - 1/(2*b*n*tan(a + b*log(c*x**n))**2), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1713 vs. \(2 (42) = 84\).

Time = 0.26 (sec) , antiderivative size = 1713, normalized size of antiderivative = 38.93 \[ \int \frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]

[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))

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 29.41 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.41 \[ \int \frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\ln \left (x\right )\,1{}\mathrm {i}+\frac {2}{b\,n\,\left (1+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\right )}+\frac {2}{b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )}-\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )}{b\,n} \]

[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)