3.209 \(\int x^3 \cot (d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=70 \[ \frac{i x^4}{4}-\frac{1}{2} i x^4 \text{Hypergeometric2F1}\left (1,-\frac{2 i}{b d n},1-\frac{2 i}{b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]

[Out]

(I/4)*x^4 - (I/2)*x^4*Hypergeometric2F1[1, (-2*I)/(b*d*n), 1 - (2*I)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d
)]

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Rubi [F]  time = 0.0352325, 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 = {} \[ \int x^3 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^3*Cot[d*(a + b*Log[c*x^n])],x]

[Out]

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

Rubi steps

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

Mathematica [B]  time = 5.39956, size = 220, normalized size = 3.14 \[ -\frac{x^4 \left (2 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,1-\frac{2 i}{b d n},2-\frac{2 i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(b d n-2 i) \left (i \text{Hypergeometric2F1}\left (1,-\frac{2 i}{b d n},1-\frac{2 i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\cot \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+\sin (b d n \log (x)) \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )\right )}{4 b d n-8 i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((x^4*(2*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - (2*I)/(b*d*n), 2 - (2*I)/(b*d*n), E^((2*I)*d
*(a + b*Log[c*x^n]))] + (-2*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
*Hypergeometric2F1[1, (-2*I)/(b*d*n), 1 - (2*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]])))/(-8*I + 4*b*d*n))

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Maple [F]  time = 1.831, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}\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^3*cot(d*(a+b*ln(c*x^n))),x)

[Out]

int(x^3*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^{3} \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^3*cot(d*(a+b*log(c*x^n))),x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [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(x**3*cot(d*(a+b*ln(c*x**n))),x)

[Out]

Timed out

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

[Out]

Exception raised: TypeError