\(\int \cot ^p(d (a+b \log (c x^n))) \, dx\) [229]

Optimal result

Integrand size = 15, antiderivative size = 190 \[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b d n},p,-p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]

[Out]

x*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*(-I*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d
)))^p*AppellF1(-1/2*I/b/d/n,p,-p,1-1/2*I/b/d/n,exp(2*I*a*d)*(c*x^n)^(2*I*b*d),-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))
/((1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4590, 4592, 1986, 525, 524} \[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b d n},p,-p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]

[In]

Int[Cot[d*(a + b*Log[c*x^n])]^p,x]

[Out]

(x*(1 - E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^p*(((-I)*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(1 - E^((2*I)*a*d
)*(c*x^n)^((2*I)*b*d)))^p*AppellF1[(-1/2*I)/(b*d*n), p, -p, 1 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*
d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])/(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^p

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 1986

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^(r_.))^(p_), x_Symbol] :> Dist[Simp
[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))], Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)
^(p*r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]

Rule 4590

Int[Cot[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Cot[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n,
1])

Rule 4592

Int[Cot[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*((-I - I*E^(2*I*a*d)
*x^(2*I*b*d))/(1 - E^(2*I*a*d)*x^(2*I*b*d)))^p, x] /; FreeQ[{a, b, d, e, m, p}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \cot ^p(d (a+b \log (x))) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \left (\frac {-i-i e^{2 i a d} x^{2 i b d}}{1-e^{2 i a d} x^{2 i b d}}\right )^p \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-1/n} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (-i-i e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (\frac {-i-i e^{2 i a d} \left (c x^n\right )^{2 i b d}}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \left (1-e^{2 i a d} x^{2 i b d}\right )^{-p} \left (-i-i e^{2 i a d} x^{2 i b d}\right )^p \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-1/n} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (\frac {-i-i e^{2 i a d} \left (c x^n\right )^{2 i b d}}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \left (1-e^{2 i a d} x^{2 i b d}\right )^{-p} \left (1+e^{2 i a d} x^{2 i b d}\right )^p \, dx,x,c x^n\right )}{n} \\ & = x \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b d n},p,-p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(458\) vs. \(2(190)=380\).

Time = 1.03 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.41 \[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(-i+2 b d n) x \left (\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b d n},p,-p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{2 b d e^{2 i a d} n p \left (c x^n\right )^{2 i b d} \operatorname {AppellF1}\left (1-\frac {i}{2 b d n},p,1-p,2-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+2 b d e^{2 i a d} n p \left (c x^n\right )^{2 i b d} \operatorname {AppellF1}\left (1-\frac {i}{2 b d n},1+p,-p,2-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+(-i+2 b d n) \operatorname {AppellF1}\left (-\frac {i}{2 b d n},p,-p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} \]

[In]

Integrate[Cot[d*(a + b*Log[c*x^n])]^p,x]

[Out]

((-I + 2*b*d*n)*x*((I*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))^p*App
ellF1[(-1/2*I)/(b*d*n), p, -p, 1 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^(
(2*I)*b*d))])/(2*b*d*E^((2*I)*a*d)*n*p*(c*x^n)^((2*I)*b*d)*AppellF1[1 - (I/2)/(b*d*n), p, 1 - p, 2 - (I/2)/(b*
d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))] + 2*b*d*E^((2*I)*a*d)*n*p*(c*x^
n)^((2*I)*b*d)*AppellF1[1 - (I/2)/(b*d*n), 1 + p, -p, 2 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(
E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))] + (-I + 2*b*d*n)*AppellF1[(-1/2*I)/(b*d*n), p, -p, 1 - (I/2)/(b*d*n), E^((
2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])

Maple [F]

\[\int {\cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{p}d x\]

[In]

int(cot(d*(a+b*ln(c*x^n)))^p,x)

[Out]

int(cot(d*(a+b*ln(c*x^n)))^p,x)

Fricas [F]

\[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p} \,d x } \]

[In]

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

[Out]

integral(cot(b*d*log(c*x^n) + a*d)^p, x)

Sympy [F]

\[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \cot ^{p}{\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \]

[In]

integrate(cot(d*(a+b*ln(c*x**n)))**p,x)

[Out]

Integral(cot(d*(a + b*log(c*x**n)))**p, x)

Maxima [F]

\[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p} \,d x } \]

[In]

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

[Out]

integrate(cot((b*log(c*x^n) + a)*d)^p, x)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int {\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^p \,d x \]

[In]

int(cot(d*(a + b*log(c*x^n)))^p,x)

[Out]

int(cot(d*(a + b*log(c*x^n)))^p, x)