3.196 \(\int (e x)^m \coth ^3(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=306 \[ -\frac{(e x)^{m+1} \left (2 b^2 d^2 n^2+m^2+2 m+1\right ) \, _2F_1\left (1,\frac{m+1}{2 b d n};\frac{m+1}{2 b d n}+1;e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b^2 d^2 e (m+1) n^2}+\frac{e^{-2 a d} (e x)^{m+1} \left (\frac{e^{4 a d} (2 b d n+m+1) \left (c x^n\right )^{2 b d}}{n}+\frac{e^{2 a d} (-2 b d n+m+1)}{n}\right )}{2 b^2 d^2 e n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac{(e x)^{m+1} \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )^2}{2 b d e n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^2}+\frac{(e x)^{m+1} (b d n+m+1) (2 b d n+m+1)}{2 b^2 d^2 e (m+1) n^2} \]

[Out]

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

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Rubi [F]  time = 0.0694341, 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 = {} \[ \int (e x)^m \coth ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*x)^m*Coth[d*(a + b*Log[c*x^n])]^3,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 16.5856, size = 600, normalized size = 1.96 \[ -\frac{x^{-m} (e x)^m \left (2 b^2 d^2 n^2+m^2+2 m+1\right ) \text{csch}\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \left (\frac{\sinh \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \exp \left (-\frac{(2 m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \left ((2 b d n+m+1) \exp \left (\frac{2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right ) \, _2F_1\left (1,\frac{m+1}{2 b d n};\frac{m+1}{2 b d n}+1;e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(m+1) \exp \left (\frac{a (2 b d n+2 m+1)}{b n}+\frac{(2 b d n+2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )}{n}+\log (x) (2 b d n+m+1)\right ) \, _2F_1\left (1,\frac{m+2 b d n+1}{2 b d n};\frac{m+4 b d n+1}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n+m+1) \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \exp \left (\frac{2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right )}{(m+1) (2 b d n+m+1)}+\frac{x^{m+1} \sinh (b d n \log (x)) \text{csch}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{m+1}\right )}{2 b^2 d^2 n^2}+\frac{(m+1) x (e x)^m \sinh (b d n \log (x)) \text{csch}\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \text{csch}\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b d n \log (x)\right )}{2 b^2 d^2 n^2}+\frac{x (e x)^m \coth \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )}{m+1}-\frac{x (e x)^m \text{csch}^2\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b d n \log (x)\right )}{2 b d n} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.24, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ({\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*coth(d*(a+b*ln(c*x^n)))^3,x)

[Out]

int((e*x)^m*coth(d*(a+b*ln(c*x^n)))^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -{\left (2 \, b^{2} d^{2} e^{m} n^{2} +{\left (m^{2} + 2 \, m + 1\right )} e^{m}\right )} \int \frac{x^{m}}{2 \,{\left (b^{2} c^{b d} d^{2} n^{2} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} + b^{2} d^{2} n^{2}\right )}}\,{d x} +{\left (2 \, b^{2} d^{2} e^{m} n^{2} +{\left (m^{2} + 2 \, m + 1\right )} e^{m}\right )} \int \frac{x^{m}}{2 \,{\left (b^{2} c^{b d} d^{2} n^{2} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} - b^{2} d^{2} n^{2}\right )}}\,{d x} + \frac{b^{2} c^{4 \, b d} d^{2} e^{m} n^{2} x e^{\left (4 \, b d \log \left (x^{n}\right ) + 4 \, a d + m \log \left (x\right )\right )} +{\left (b^{2} d^{2} e^{m} n^{2} +{\left (m^{2} + 2 \, m + 1\right )} e^{m}\right )} x x^{m} -{\left (2 \, b^{2} c^{2 \, b d} d^{2} e^{m} n^{2} e^{\left (2 \, a d\right )} + 2 \,{\left (m n + n\right )} b c^{2 \, b d} d e^{m} e^{\left (2 \, a d\right )} +{\left (m^{2} + 2 \, m + 1\right )} c^{2 \, b d} e^{m} e^{\left (2 \, a d\right )}\right )} x e^{\left (2 \, b d \log \left (x^{n}\right ) + m \log \left (x\right )\right )}}{{\left (m n^{2} + n^{2}\right )} b^{2} c^{4 \, b d} d^{2} e^{\left (4 \, b d \log \left (x^{n}\right ) + 4 \, a d\right )} - 2 \,{\left (m n^{2} + n^{2}\right )} b^{2} c^{2 \, b d} d^{2} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} +{\left (m n^{2} + n^{2}\right )} b^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*coth(d*(a+b*log(c*x^n)))^3,x, algorithm="maxima")

[Out]

-(2*b^2*d^2*e^m*n^2 + (m^2 + 2*m + 1)*e^m)*integrate(1/2*x^m/(b^2*c^(b*d)*d^2*n^2*e^(b*d*log(x^n) + a*d) + b^2
*d^2*n^2), x) + (2*b^2*d^2*e^m*n^2 + (m^2 + 2*m + 1)*e^m)*integrate(1/2*x^m/(b^2*c^(b*d)*d^2*n^2*e^(b*d*log(x^
n) + a*d) - b^2*d^2*n^2), x) + (b^2*c^(4*b*d)*d^2*e^m*n^2*x*e^(4*b*d*log(x^n) + 4*a*d + m*log(x)) + (b^2*d^2*e
^m*n^2 + (m^2 + 2*m + 1)*e^m)*x*x^m - (2*b^2*c^(2*b*d)*d^2*e^m*n^2*e^(2*a*d) + 2*(m*n + n)*b*c^(2*b*d)*d*e^m*e
^(2*a*d) + (m^2 + 2*m + 1)*c^(2*b*d)*e^m*e^(2*a*d))*x*e^(2*b*d*log(x^n) + m*log(x)))/((m*n^2 + n^2)*b^2*c^(4*b
*d)*d^2*e^(4*b*d*log(x^n) + 4*a*d) - 2*(m*n^2 + n^2)*b^2*c^(2*b*d)*d^2*e^(2*b*d*log(x^n) + 2*a*d) + (m*n^2 + n
^2)*b^2*d^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{m} \coth \left (b d \log \left (c x^{n}\right ) + a d\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*coth(d*(a+b*log(c*x^n)))^3,x, algorithm="fricas")

[Out]

integral((e*x)^m*coth(b*d*log(c*x^n) + a*d)^3, 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*coth(d*(a+b*ln(c*x**n)))**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*coth(d*(a+b*log(c*x^n)))^3,x, algorithm="giac")

[Out]

integrate((e*x)^m*coth((b*log(c*x^n) + a)*d)^3, x)