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

Optimal. Leaf size=74 \[ \frac{i x^3}{3}-\frac{2}{3} i x^3 \text{Hypergeometric2F1}\left (1,-\frac{3 i}{2 b d n},1-\frac{3 i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]

[Out]

(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)]

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Rubi [F]  time = 0.0290022, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [B]  time = 5.74146, size = 229, normalized size = 3.09 \[ -\frac{x^3 \left (3 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,1-\frac{3 i}{2 b d n},2-\frac{3 i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n-3 i) \left (i \text{Hypergeometric2F1}\left (1,-\frac{3 i}{2 b d n},1-\frac{3 i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\cot \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+\sin (b d n \log (x)) \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )\right )}{6 b d n-9 i} \]

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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Maple [F]  time = 1.595, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\cot \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int x^{2} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \cot \left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int x^{2} \cot{\left (a d + b d \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError