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

Optimal. Leaf size=100 \[ \frac{i (e x)^{m+1}}{e (m+1)}-\frac{2 i (e x)^{m+1} \text{Hypergeometric2F1}\left (1,-\frac{i (m+1)}{2 b d n},1-\frac{i (m+1)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (m+1)} \]

[Out]

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

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Rubi [F]  time = 0.0477099, 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 = {} \[ \int (e x)^m \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 13.9691, size = 182, normalized size = 1.82 \[ -\frac{i x (e x)^m \left (\frac{(m+1) e^{2 i a d} \left (c x^n\right )^{2 i b d} \text{Hypergeometric2F1}\left (1,-\frac{i (2 i b d n+m+1)}{2 b d n},-\frac{i (4 i b d n+m+1)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{2 i b d n+m+1}+\text{Hypergeometric2F1}\left (1,-\frac{i (m+1)}{2 b d n},1-\frac{i (m+1)}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.946, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}\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((e*x)^m*cot(d*(a+b*ln(c*x^n))),x)

[Out]

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

[Out]

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

[Out]

integral((e*x)^m*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 \left (e x\right )^{m} \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((e*x)**m*cot(d*(a+b*ln(c*x**n))),x)

[Out]

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

[Out]

Exception raised: TypeError