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

Optimal. Leaf size=66 \[ i x-2 i x \text{Hypergeometric2F1}\left (1,-\frac{i}{2 b d n},1-\frac{i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]

[Out]

I*x - (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)]

________________________________________________________________________________________

Rubi [F]  time = 0.01151, 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 = {} \[ \int \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[d*(a + b*Log[c*x^n])],x]

[Out]

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

Rubi steps

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

Mathematica [B]  time = 10.6454, size = 141, normalized size = 2.14 \[ x \left (-\frac{e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,1-\frac{i}{2 b d n},2-\frac{i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b d n-i}-i \text{Hypergeometric2F1}\left (1,-\frac{i}{2 b d n},1-\frac{i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

________________________________________________________________________________________

Maple [F]  time = 1.309, size = 0, normalized size = 0. \begin{align*} \int \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(cot(d*(a+b*ln(c*x^n))),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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(cot(d*(a+b*log(c*x^n))),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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 ), 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, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cot{\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))),x)

[Out]

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

________________________________________________________________________________________

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))),x, algorithm="giac")

[Out]

Exception raised: TypeError