\(\int \frac {\cot ^4(a+b \log (c x^n))}{x} \, dx\) [224]

Optimal result

Integrand size = 17, antiderivative size = 44 \[ \int \frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\log (x) \]

[Out]

cot(a+b*ln(c*x^n))/b/n-1/3*cot(a+b*ln(c*x^n))^3/b/n+ln(x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3554, 8} \[ \int \frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n}+\log (x) \]

[In]

Int[Cot[a + b*Log[c*x^n]]^4/x,x]

[Out]

Cot[a + b*Log[c*x^n]]/(b*n) - Cot[a + b*Log[c*x^n]]^3/(3*b*n) + Log[x]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \cot ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\text {Subst}\left (\int \cot ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\text {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\log (x) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.05 \[ \int \frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\cot ^3\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2\left (a+b \log \left (c x^n\right )\right )\right )}{3 b n} \]

[In]

Integrate[Cot[a + b*Log[c*x^n]]^4/x,x]

[Out]

-1/3*(Cot[a + b*Log[c*x^n]]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[a + b*Log[c*x^n]]^2])/(b*n)

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(\frac {-{\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}+3 \ln \left (x \right ) b n +3 \cot \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 b n}\)	\(44\)
derivativedivides	\(\frac {-\frac {{\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{3}+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\frac {\pi }{2}+\operatorname {arccot}\left (\cot \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{n b}\)	\(50\)
default	\(\frac {-\frac {{\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{3}+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\frac {\pi }{2}+\operatorname {arccot}\left (\cot \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{n b}\)	\(50\)
risch	\(\ln \left (x \right )+\frac {4 i \left (3 \left (x^{n}\right )^{4 i b} c^{4 i b} {\mathrm e}^{2 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-2 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-2 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{2 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{4 i a}-3 \left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+2\right )}{3 b n {\left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}-1\right )}^{3}}\)	\(335\)

[In]

int(cot(a+b*ln(c*x^n))^4/x,x,method=_RETURNVERBOSE)

[Out]

1/3*(-cot(a+b*ln(c*x^n))^3+3*ln(x)*b*n+3*cot(a+b*ln(c*x^n)))/b/n

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (42) = 84\).

Time = 0.25 (sec) , antiderivative size = 132, normalized size of antiderivative = 3.00 \[ \int \frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {4 \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )^{2} + 3 \, {\left (b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) \log \left (x\right ) - b n \log \left (x\right )\right )} \sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 2 \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) - 2}{3 \, {\left (b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) - b n\right )} \sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )} \]

[In]

integrate(cot(a+b*log(c*x^n))^4/x,x, algorithm="fricas")

[Out]

1/3*(4*cos(2*b*n*log(x) + 2*b*log(c) + 2*a)^2 + 3*(b*n*cos(2*b*n*log(x) + 2*b*log(c) + 2*a)*log(x) - b*n*log(x
))*sin(2*b*n*log(x) + 2*b*log(c) + 2*a) + 2*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) - 2)/((b*n*cos(2*b*n*log(x) +
2*b*log(c) + 2*a) - b*n)*sin(2*b*n*log(x) + 2*b*log(c) + 2*a))

Sympy [A] (verification not implemented)

Time = 1.39 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.48 \[ \int \frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \log {\left (x \right )} \cot ^{4}{\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cot ^{4}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\log {\left (c x^{n} \right )}}{n} - \frac {\cot ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{3 b n} + \frac {\cot {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b n} & \text {otherwise} \end {cases} \]

[In]

integrate(cot(a+b*ln(c*x**n))**4/x,x)

[Out]

Piecewise((log(x)*cot(a)**4, Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log(x)*cot(a + b*log(c))**4, Eq(n, 0)), (log(
c*x**n)/n - cot(a + b*log(c*x**n))**3/(3*b*n) + cot(a + b*log(c*x**n))/(b*n), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2172 vs. \(2 (42) = 84\).

Time = 0.28 (sec) , antiderivative size = 2172, normalized size of antiderivative = 49.36 \[ \int \frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]

[In]

integrate(cot(a+b*log(c*x^n))^4/x,x, algorithm="maxima")

[Out]

1/3*(3*(b*cos(6*b*log(c))^2 + b*sin(6*b*log(c))^2)*n*cos(6*b*log(x^n) + 6*a)^2*log(x) + 27*(b*cos(4*b*log(c))^
2 + b*sin(4*b*log(c))^2)*n*cos(4*b*log(x^n) + 4*a)^2*log(x) + 27*(b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n
*cos(2*b*log(x^n) + 2*a)^2*log(x) + 3*(b*cos(6*b*log(c))^2 + b*sin(6*b*log(c))^2)*n*log(x)*sin(6*b*log(x^n) +
6*a)^2 + 27*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*log(x)*sin(4*b*log(x^n) + 4*a)^2 + 27*(b*cos(2*b*log
(c))^2 + b*sin(2*b*log(c))^2)*n*log(x)*sin(2*b*log(x^n) + 2*a)^2 + 3*b*n*log(x) - 2*(3*b*n*cos(6*b*log(c))*log
(x) + 3*(3*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n*log(x) - 2*cos(4*b*log(c)
)*sin(6*b*log(c)) + 2*cos(6*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 3*(3*(b*cos(6*b*log(c))*cos(2
*b*log(c)) + b*sin(6*b*log(c))*sin(2*b*log(c)))*n*log(x) - 2*cos(2*b*log(c))*sin(6*b*log(c)) + 2*cos(6*b*log(c
))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 3*(3*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(
4*b*log(c)))*n*log(x) + 2*cos(6*b*log(c))*cos(4*b*log(c)) + 2*sin(6*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n
) + 4*a) - 3*(3*(b*cos(2*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n*log(x) + 2*cos(6*b*l
og(c))*cos(2*b*log(c)) + 2*sin(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) - 4*sin(6*b*log(c)))*cos(6
*b*log(x^n) + 6*a) + 6*(3*b*n*cos(4*b*log(c))*log(x) - 9*(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)*log(x) - 9*(b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))
*sin(2*b*log(c)))*n*log(x)*sin(2*b*log(x^n) + 2*a) - 2*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 6*(3*b*n*cos
(2*b*log(c))*log(x) - 2*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 2*(3*b*n*log(x)*sin(6*b*log(c)) + 3*(3*(b*c
os(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n*log(x) + 2*cos(6*b*log(c))*cos(4*b*log(c
)) + 2*sin(6*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 3*(3*(b*cos(2*b*log(c))*sin(6*b*log(c)) - b*
cos(6*b*log(c))*sin(2*b*log(c)))*n*log(x) + 2*cos(6*b*log(c))*cos(2*b*log(c)) + 2*sin(6*b*log(c))*sin(2*b*log(
c)))*cos(2*b*log(x^n) + 2*a) - 3*(3*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n*
log(x) - 2*cos(4*b*log(c))*sin(6*b*log(c)) + 2*cos(6*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) + 3*(3
*(b*cos(6*b*log(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(2*b*log(c)))*n*log(x) - 2*cos(2*b*log(c))*sin(6*b*
log(c)) + 2*cos(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + 4*cos(6*b*log(c)))*sin(6*b*log(x^n) + 6
*a) + 6*(9*(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)*l
og(x) - 3*b*n*log(x)*sin(4*b*log(c)) - 9*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)
))*n*log(x)*sin(2*b*log(x^n) + 2*a) - 2*cos(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) + 6*(3*b*n*log(x)*sin(2*b*log
(c)) + 2*cos(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))/((b*cos(6*b*log(c))^2 + b*sin(6*b*log(c))^2)*n*cos(6*b*log(
x^n) + 6*a)^2 + 9*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*cos(4*b*log(x^n) + 4*a)^2 - 6*b*n*cos(2*b*log(
c))*cos(2*b*log(x^n) + 2*a) + 9*(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*c
os(6*b*log(c))^2 + b*sin(6*b*log(c))^2)*n*sin(6*b*log(x^n) + 6*a)^2 + 9*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c
))^2)*n*sin(4*b*log(x^n) + 4*a)^2 + 6*b*n*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + 9*(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(6*b*log(c)) + 3*(b*cos(6*b*log(c))*cos(4*b*
log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n*cos(4*b*log(x^n) + 4*a) - 3*(b*cos(6*b*log(c))*cos(2*b*log(c))
+ b*sin(6*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) + 3*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(
6*b*log(c))*sin(4*b*log(c)))*n*sin(4*b*log(x^n) + 4*a) - 3*(b*cos(2*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(
c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) + 6*(b*n*cos(4*b*log(c)) - 3*(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) - 3*(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*
(3*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n*cos(4*b*log(x^n) + 4*a) - 3*(b*co
s(2*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) + b*n*sin(6*b*log
(c)) - 3*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n*sin(4*b*log(x^n) + 4*a) + 3
*(b*cos(6*b*log(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*sin(6*b*lo
g(x^n) + 6*a) + 6*(3*(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)) - 3*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*n*s
in(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a))

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]

[In]

integrate(cot(a+b*log(c*x^n))^4/x,x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 35.99 (sec) , antiderivative size = 182, normalized size of antiderivative = 4.14 \[ \int \frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\ln \left (x\right )+\frac {\frac {4{}\mathrm {i}}{3\,b\,n}+\frac {{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}\,4{}\mathrm {i}}{3\,b\,n}}{3\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-3\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+{\mathrm {e}}^{a\,6{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,6{}\mathrm {i}}-1}+\frac {4{}\mathrm {i}}{3\,b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )}+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\,4{}\mathrm {i}}{3\,b\,n\,\left (1+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\right )} \]

[In]

int(cot(a + b*log(c*x^n))^4/x,x)

[Out]

log(x) + (4i/(3*b*n) + (exp(a*4i)*(c*x^n)^(b*4i)*4i)/(3*b*n))/(3*exp(a*2i)*(c*x^n)^(b*2i) - 3*exp(a*4i)*(c*x^n
)^(b*4i) + exp(a*6i)*(c*x^n)^(b*6i) - 1) + 4i/(3*b*n*(exp(a*2i)*(c*x^n)^(b*2i) - 1)) + (exp(a*2i)*(c*x^n)^(b*2
i)*4i)/(3*b*n*(exp(a*4i)*(c*x^n)^(b*4i) - 2*exp(a*2i)*(c*x^n)^(b*2i) + 1))