3.217 \(\int x^2 \cot ^2(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=162 \[ -\frac {2 i x^3 \, _2F_1\left (1,-\frac {3 i}{2 b d n};1-\frac {3 i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n}+\frac {i x^3 \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {x^3 (-b d n+3 i)}{3 b d n} \]

[Out]

1/3*(3*I-b*d*n)*x^3/b/d/n+I*x^3*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/b/d/n/(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))-2*
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))/b/d/n

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Rubi [F]  time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^2 \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int x^2 \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int x^2 \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}

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Mathematica [A]  time = 5.32, size = 185, normalized size = 1.14 \[ -\frac {x^3 \left (9 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {3 i}{2 b d n};2-\frac {3 i}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n-3 i) \left (3 i \, _2F_1\left (1,-\frac {3 i}{2 b d n};1-\frac {3 i}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+3 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )\right )}{3 b d n (2 b d n-3 i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(x^3*(9*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)*(b*d*n + 3*Cot[d*(a + b*Log[c*x^n])] + (3*I)*Hypergeometri
c2F1[1, ((-3*I)/2)/(b*d*n), 1 - ((3*I)/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])))/(b*d*n*(-3*I + 2*b*d*n))

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \cot \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 0.81, size = 0, normalized size = 0.00 \[ \int x^{2} \left (\cot ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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