3.221 \(\int \frac{\cot ^2(d (a+b \log (c x^n)))}{x^2} \, dx\)

Optimal. Leaf size=156 \[ -\frac{2 i \text{Hypergeometric2F1}\left (1,\frac{i}{2 b d n},1+\frac{i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x}+\frac{i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac{1+\frac{i}{b d n}}{x} \]

[Out]

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

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Rubi [F]  time = 0.0569329, 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 = {} \[ \int \frac{\cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 4.36142, size = 181, normalized size = 1.16 \[ \frac{e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,1+\frac{i}{2 b d n},2+\frac{i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n+i) \left (-i \text{Hypergeometric2F1}\left (1,\frac{i}{2 b d n},1+\frac{i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )}{b d n x (2 b d n+i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.628, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cot \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{2}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out