3.3.10 \(\int x^2 \cot (d (a+b \log (c x^n))) \, dx\) [210]

Optimal. Leaf size=74 \[ \frac {i x^3}{3}-\frac {2}{3} i x^3 \, _2F_1\left (1,-\frac {3 i}{2 b d n};1-\frac {3 i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]

[Out]

1/3*I*x^3-2/3*I*x^3*hypergeom([1, -3/2*I/b/d/n],[1-3/2*I/b/d/n],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))

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Rubi [A]
time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {4594, 4592, 470, 371} \begin {gather*} \frac {i x^3}{3}-\frac {2}{3} i x^3 \, _2F_1\left (1,-\frac {3 i}{2 b d n};1-\frac {3 i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Cot[d*(a + b*Log[c*x^n])],x]

[Out]

(I/3)*x^3 - ((2*I)/3)*x^3*Hypergeometric2F1[1, ((-3*I)/2)/(b*d*n), 1 - ((3*I)/2)/(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*} \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(229\) vs. \(2(74)=148\).
time = 6.18, size = 229, normalized size = 3.09 \begin {gather*} -\frac {x^3 \left (3 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {3 i}{2 b d n};2-\frac {3 i}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(-3 i+2 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 \, _2F_1\left (1,-\frac {3 i}{2 b d n};1-\frac {3 i}{2 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 )}{-9 i+6 b d n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cot[d*(a + b*Log[c*x^n])],x]

[Out]

-((x^3*(3*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - ((3*I)/2)/(b*d*n), 2 - ((3*I)/2)/(b*d*n), E^
((2*I)*d*(a + b*Log[c*x^n]))] + (-3*I + 2*b*d*n)*(Cot[d*(a + b*Log[c*x^n])] - Cot[d*(a - b*n*Log[x] + b*Log[c*
x^n])] + I*Hypergeometric2F1[1, ((-3*I)/2)/(b*d*n), 1 - ((3*I)/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + C
sc[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]])))/(-9*I + 6*b*d*n))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cot(d*(a+b*ln(c*x^n))),x)

[Out]

int(x^2*cot(d*(a+b*ln(c*x^n))),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cot(d*(a+b*log(c*x^n))),x, algorithm="maxima")

[Out]

integrate(x^2*cot((b*log(c*x^n) + a)*d), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cot(d*(a+b*log(c*x^n))),x, algorithm="fricas")

[Out]

integral(x^2*cot(b*d*log(c*x^n) + a*d), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \cot {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cot(d*(a+b*ln(c*x**n))),x)

[Out]

Integral(x**2*cot(a*d + b*d*log(c*x**n)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cot(d*(a+b*log(c*x^n))),x, algorithm="giac")

[Out]

integrate(x^2*cot((b*log(c*x^n) + a)*d), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cot(d*(a + b*log(c*x^n))),x)

[Out]

int(x^2*cot(d*(a + b*log(c*x^n))), x)

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