\(\int x \cot (d (a+b \log (c x^n))) \, dx\) [211]

Optimal result

Integrand size = 15, antiderivative size = 68 \[ \int x \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {i x^2}{2}-i x^2 \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{b d n},1-\frac {i}{b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]

[Out]

1/2*I*x^2-I*x^2*hypergeom([1, -I/b/d/n],[1-I/b/d/n],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4594, 4592, 470, 371} \[ \int x \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {i x^2}{2}-i x^2 \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{b d n},1-\frac {i}{b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]

[In]

Int[x*Cot[d*(a + b*Log[c*x^n])],x]

[Out]

(I/2)*x^2 - I*x^2*Hypergeometric2F1[1, (-I)/(b*d*n), 1 - I/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 470

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

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]

Rule 4594

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int x^{-1+\frac {2}{n}} \cot (d (a+b \log (x))) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {2}{n}} \left (-i-i e^{2 i a d} x^{2 i b d}\right )}{1-e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{n} \\ & = \frac {i x^2}{2}-\frac {\left (2 i x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {2}{n}}}{1-e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{n} \\ & = \frac {i x^2}{2}-i x^2 \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{b d n},1-\frac {i}{b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(219\) vs. \(2(68)=136\).

Time = 4.31 (sec) , antiderivative size = 219, normalized size of antiderivative = 3.22 \[ \int x \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {x^2 \left (e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i}{b d n},2-\frac {i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(-i+b d n) \left (\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\cot \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+i \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{b d n},1-\frac {i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right ) \sin (b d n \log (x))\right )\right )}{-2 i+2 b d n} \]

[In]

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

[Out]

-((x^2*(E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - I/(b*d*n), 2 - I/(b*d*n), E^((2*I)*d*(a + b*Lo
g[c*x^n]))] + (-I + b*d*n)*(Cot[d*(a + b*Log[c*x^n])] - Cot[d*(a - b*n*Log[x] + b*Log[c*x^n])] + I*Hypergeomet
ric2F1[1, (-I)/(b*d*n), 1 - I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + Csc[d*(a + b*Log[c*x^n])]*Csc[d*(a -
b*n*Log[x] + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])))/(-2*I + 2*b*d*n))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x*cot(d*(a + b*log(c*x^n))),x)

[Out]

int(x*cot(d*(a + b*log(c*x^n))), x)