3.215 \(\int \frac{\cot (d (a+b \log (c x^n)))}{x^3} \, dx\)

Optimal. Leaf size=68 \[ \frac{i \text{Hypergeometric2F1}\left (1,\frac{i}{b d n},1+\frac{i}{b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{x^2}-\frac{i}{2 x^2} \]

[Out]

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

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Rubi [F]  time = 0.029018, 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 = {} \[ \int \frac{\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[d*(a + b*Log[c*x^n])]/x^3,x]

[Out]

Defer[Int][Cot[d*(a + b*Log[c*x^n])]/x^3, x]

Rubi steps

\begin{align*} \int \frac{\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=\int \frac{\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\\ \end{align*}

Mathematica [B]  time = 4.3129, size = 211, normalized size = 3.1 \[ \frac{-\frac{e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,1+\frac{i}{b d n},2+\frac{i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{b d n+i}+i \text{Hypergeometric2F1}\left (1,\frac{i}{b d n},1+\frac{i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\cot \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+\sin (b d n \log (x)) \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[d*(a + b*Log[c*x^n])]/x^3,x]

[Out]

(Cot[d*(a + b*Log[c*x^n])] - Cot[d*(a - b*n*Log[x] + b*Log[c*x^n])] - (E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeo
metric2F1[1, 1 + I/(b*d*n), 2 + I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])/(I + b*d*n) + I*Hypergeometric2F1[
1, I/(b*d*n), 1 + I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + Csc[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]])/(2*x^2)

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Maple [F]  time = 1.779, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cot \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*(a+b*ln(c*x^n)))/x^3,x)

[Out]

int(cot(d*(a+b*ln(c*x^n)))/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*(a+b*log(c*x^n)))/x^3,x, algorithm="maxima")

[Out]

integrate(cot((b*log(c*x^n) + a)*d)/x^3, x)

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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 )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*(a+b*log(c*x^n)))/x^3,x, algorithm="fricas")

[Out]

integral(cot(b*d*log(c*x^n) + a*d)/x^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (a d + b d \log{\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*(a+b*ln(c*x**n)))/x**3,x)

[Out]

Integral(cot(a*d + b*d*log(c*x**n))/x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*(a+b*log(c*x^n)))/x^3,x, algorithm="giac")

[Out]

integrate(cot((b*log(c*x^n) + a)*d)/x^3, x)