3.190 \(\int \frac{\coth ^2(d (a+b \log (c x^n)))}{x^3} \, dx\)

Optimal. Leaf size=135 \[ -\frac{2 \, _2F_1\left (1,-\frac{1}{b d n};1-\frac{1}{b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n x^2}+\frac{e^{2 a d} \left (c x^n\right )^{2 b d}+1}{b d n x^2 \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}+\frac{2-b d n}{2 b d n x^2} \]

[Out]

(2 - b*d*n)/(2*b*d*n*x^2) + (1 + E^(2*a*d)*(c*x^n)^(2*b*d))/(b*d*n*x^2*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*H
ypergeometric2F1[1, -(1/(b*d*n)), 1 - 1/(b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/(b*d*n*x^2)

________________________________________________________________________________________

Rubi [F]  time = 0.0530002, 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 = {} \[ \int \frac{\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[d*(a + b*Log[c*x^n])]^2/x^3,x]

[Out]

Defer[Int][Coth[d*(a + b*Log[c*x^n])]^2/x^3, x]

Rubi steps

\begin{align*} \int \frac{\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=\int \frac{\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\\ \end{align*}

Mathematica [A]  time = 3.55429, size = 156, normalized size = 1.16 \[ \frac{2 e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac{1}{b d n};2-\frac{1}{b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-(b d n-1) \left (2 \, _2F_1\left (1,-\frac{1}{b d n};1-\frac{1}{b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+2 \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )}{2 b d n x^2 (b d n-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[d*(a + b*Log[c*x^n])]^2/x^3,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.127, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{2}}{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*(a+b*ln(c*x^n)))^2/x^3,x)

[Out]

int(coth(d*(a+b*ln(c*x^n)))^2/x^3,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - b d n + 4}{2 \,{\left (b c^{2 \, b d} d n x^{2} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - b d n x^{2}\right )}} + 2 \, \int \frac{1}{b c^{b d} d n x^{3} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} + b d n x^{3}}\,{d x} - 2 \, \int \frac{1}{b c^{b d} d n x^{3} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} - b d n x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*(a+b*log(c*x^n)))^2/x^3,x, algorithm="maxima")

[Out]

-1/2*(b*c^(2*b*d)*d*n*e^(2*b*d*log(x^n) + 2*a*d) - b*d*n + 4)/(b*c^(2*b*d)*d*n*x^2*e^(2*b*d*log(x^n) + 2*a*d)
- b*d*n*x^2) + 2*integrate(1/(b*c^(b*d)*d*n*x^3*e^(b*d*log(x^n) + a*d) + b*d*n*x^3), x) - 2*integrate(1/(b*c^(
b*d)*d*n*x^3*e^(b*d*log(x^n) + a*d) - b*d*n*x^3), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\coth \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*(a+b*log(c*x^n)))^2/x^3,x, algorithm="fricas")

[Out]

integral(coth(b*d*log(c*x^n) + a*d)^2/x^3, x)

________________________________________________________________________________________

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(coth(d*(a+b*ln(c*x**n)))**2/x**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*(a+b*log(c*x^n)))^2/x^3,x, algorithm="giac")

[Out]

integrate(coth((b*log(c*x^n) + a)*d)^2/x^3, x)