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

Optimal. Leaf size=153 \[ -\frac{2 i x \text{Hypergeometric2F1}\left (1,-\frac{i}{2 b d n},1-\frac{i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n}+\frac{i x \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac{x (-b d n+i)}{b d n} \]

[Out]

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

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Rubi [F]  time = 0.0139643, 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 = {} \[ \int \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[d*(a + b*Log[c*x^n])]^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 9.63724, size = 178, normalized size = 1.16 \[ -\frac{x \left (e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,1-\frac{i}{2 b d n},2-\frac{i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n-i) \left (i \text{Hypergeometric2F1}\left (1,-\frac{i}{2 b d n},1-\frac{i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )\right )}{b d n (2 b d n-i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((x*(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*Log[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)))

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Maple [F]  time = 1.293, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{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)

[Out]

int(cot(d*(a+b*ln(c*x^n)))^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, algorithm="maxima")

[Out]

Timed out

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

[Out]

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

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

[Out]

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

[Out]

Exception raised: TypeError