3.227 \(\int (e x)^m \cot ^2(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=195 \[ -\frac{2 i (e x)^{m+1} \text{Hypergeometric2F1}\left (1,-\frac{i (m+1)}{2 b d n},1-\frac{i (m+1)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d e n}+\frac{i (e x)^{m+1} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d e n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac{(e x)^{m+1} (-b d n+i (m+1))}{b d e (m+1) n} \]

[Out]

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

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Rubi [F]  time = 0.0784244, 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 = {} \[ \int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^2,x]

[Out]

Defer[Int][(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^2, x]

Rubi steps

\begin{align*} \int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end{align*}

Mathematica [B]  time = 16.5994, size = 547, normalized size = 2.81 \[ -\frac{(m+1) x^{-m} (e x)^m \csc \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \left (\frac{x^{m+1} \sin (b d n \log (x)) \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{m+1}-\frac{i \sin \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \exp \left (-\frac{(2 m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \left (-(2 i b d n+m+1) \exp \left (\frac{2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right ) \text{Hypergeometric2F1}\left (1,-\frac{i (m+1)}{2 b d n},1-\frac{i (m+1)}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-(m+1) \exp \left (\frac{a (2 i b d n+2 m+1)}{b n}+\frac{(2 i b d n+2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )}{n}+\log (x) (2 i b d n+m+1)\right ) \text{Hypergeometric2F1}\left (1,-\frac{i (2 i b d n+m+1)}{2 b d n},-\frac{i (4 i b d n+m+1)}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+i (2 i b d n+m+1) \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \exp \left (\frac{2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right )}{(m+1) (2 i b d n+m+1)}\right )}{b d n}+\frac{x (e x)^m \sin (b d n \log (x)) \csc \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \csc \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b d n \log (x)\right )}{b d n}-\frac{x (e x)^m}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^2,x]

[Out]

-((x*(e*x)^m)/(1 + m)) + (x*(e*x)^m*Csc[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Csc[b*d*n*Log[x] + d*(a + b*(-(n
*Log[x]) + Log[c*x^n]))]*Sin[b*d*n*Log[x]])/(b*d*n) - ((1 + m)*(e*x)^m*Csc[d*(a + b*(-(n*Log[x]) + Log[c*x^n])
)]*((x^(1 + m)*Csc[d*(a + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])/(1 + m) - (I*(I*E^((a + 2*a*m + b*(1 + m)*n*Log[x]
+ b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Cot[d*(a + b*Log[c*x^n])] - E^((a + 2*
a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Hypergeometric
2F1[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^((a*(1 + 2*m
+ (2*I)*b*d*n))/(b*n) + (1 + m + (2*I)*b*d*n)*Log[x] + ((1 + 2*m + (2*I)*b*d*n)*(-(n*Log[x]) + Log[c*x^n]))/n
)*(1 + m)*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)*d*(a + b*Log[c*x^n]))])*Sin[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))])/(E^(((1 + 2*m)*(a + b*(-(n*Log[x]
) + Log[c*x^n])))/(b*n))*(1 + m)*(1 + m + (2*I)*b*d*n))))/(b*d*n*x^m)

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Maple [F]  time = 1.967, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \cot \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{2}\, 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)))^2,x)

[Out]

int((e*x)^m*cot(d*(a+b*ln(c*x^n)))^2,x)

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Maxima [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((e*x)^m*cot(d*(a+b*log(c*x^n)))^2,x, algorithm="maxima")

[Out]

Timed out

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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 )^{2}, 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)))^2,x, algorithm="fricas")

[Out]

integral((e*x)^m*cot(b*d*log(c*x^n) + a*d)^2, 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((e*x)**m*cot(d*(a+b*ln(c*x**n)))**2,x)

[Out]

Timed out

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Giac [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((e*x)^m*cot(d*(a+b*log(c*x^n)))^2,x, algorithm="giac")

[Out]

Timed out