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

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

[Out]

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

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Rubi [F]  time = 0.0225825, 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 = {} \[ \int x \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*Cot[d*(a + b*Log[c*x^n])],x]

[Out]

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

Rubi steps

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

Mathematica [B]  time = 5.71875, size = 219, normalized size = 3.22 \[ -\frac{x^2 \left (e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,1-\frac{i}{b d n},2-\frac{i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(b d n-i) \left (i \text{Hypergeometric2F1}\left (1,-\frac{i}{b d n},1-\frac{i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\cot \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+\sin (b d n \log (x)) \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )\right )}{2 b d n-2 i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((x^2*(E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - I/(b*d*n), 2 - I/(b*d*n), E^((2*I)*d*(a + b*Lo
g[c*x^n]))] + (-I + b*d*n)*(Cot[d*(a + b*Log[c*x^n])] - Cot[d*(a - b*n*Log[x] + b*Log[c*x^n])] + I*Hypergeomet
ric2F1[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*I + 2*b*d*n))

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Maple [F]  time = 1.45, size = 0, normalized size = 0. \begin{align*} \int x\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*cot(d*(a+b*ln(c*x^n))),x)

[Out]

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

[Out]

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

[Out]

integral(x*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 \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*cot(d*(a+b*ln(c*x**n))),x)

[Out]

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

[Out]

Exception raised: TypeError