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

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 [F]  time = 0.03, 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 = {} \[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*Cot[d*(a + b*Log[c*x^n])],x]

[Out]

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

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]  time = 5.58, size = 229, normalized size = 3.09 \[ -\frac {x^3 \left (3 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {3 i}{2 b d n};2-\frac {3 i}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n-3 i) \left (i \, _2F_1\left (1,-\frac {3 i}{2 b d n};1-\frac {3 i}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\cot \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+\sin (b d n \log (x)) \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )\right )}{6 b d n-9 i} \]

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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fricas [F]  time = 2.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \cot \left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]

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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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))),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 1.42, 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 \[ \int x^{2} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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