3.1.83 \(\int \frac {(e x)^{-1+3 n}}{(a+b \text {csch}(c+d x^n))^2} \, dx\) [83]

3.1.83.1 Optimal result

Integrand size = 24, antiderivative size = 1218 \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2+b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac {2 b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac {2 b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}+\frac {2 b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}+\frac {2 b^3 x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}-\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}+\frac {2 b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}-\frac {2 b^3 x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {2 b^3 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}+\frac {4 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3 e n}+\frac {2 b^3 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}-\frac {4 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3 e n}-\frac {b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )} \]

output

1/3*(e*x)^(3*n)/a^2/e/n-b^2*(e*x)^(3*n)/a^2/(a^2+b^2)/d/e/n/(x^n)+2*b^2*(e 
*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^2/e/n/( 
x^(2*n))+b^3*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b-(a^2+b^2)^(1/2)))/a^2/(a^2 
+b^2)^(3/2)/d/e/n/(x^n)+2*b^2*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b+(a^2+b^2) 
^(1/2)))/a^2/(a^2+b^2)/d^2/e/n/(x^(2*n))-b^3*(e*x)^(3*n)*ln(1+a*exp(c+d*x^ 
n)/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d/e/n/(x^n)+2*b^2*(e*x)^(3*n)* 
polylog(2,-a*exp(c+d*x^n)/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^3/e/n/(x^(3 
*n))+2*b^3*(e*x)^(3*n)*polylog(2,-a*exp(c+d*x^n)/(b-(a^2+b^2)^(1/2)))/a^2/ 
(a^2+b^2)^(3/2)/d^2/e/n/(x^(2*n))+2*b^2*(e*x)^(3*n)*polylog(2,-a*exp(c+d*x 
^n)/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^3/e/n/(x^(3*n))-2*b^3*(e*x)^(3*n) 
*polylog(2,-a*exp(c+d*x^n)/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^2/e/ 
n/(x^(2*n))-2*b^3*(e*x)^(3*n)*polylog(3,-a*exp(c+d*x^n)/(b-(a^2+b^2)^(1/2) 
))/a^2/(a^2+b^2)^(3/2)/d^3/e/n/(x^(3*n))+2*b^3*(e*x)^(3*n)*polylog(3,-a*ex 
p(c+d*x^n)/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^3/e/n/(x^(3*n))-b^2* 
(e*x)^(3*n)*cosh(c+d*x^n)/a/(a^2+b^2)/d/e/n/(x^n)/(b+a*sinh(c+d*x^n))-2*b* 
(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b-(a^2+b^2)^(1/2)))/a^2/d/e/n/(x^n)/(a^2+ 
b^2)^(1/2)+2*b*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b+(a^2+b^2)^(1/2)))/a^2/d/ 
e/n/(x^n)/(a^2+b^2)^(1/2)-4*b*(e*x)^(3*n)*polylog(2,-a*exp(c+d*x^n)/(b-(a^ 
2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(a^2+b^2)^(1/2)+4*b*(e*x)^(3*n)*polyl 
og(2,-a*exp(c+d*x^n)/(b+(a^2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(a^2+b^...
 

3.1.83.2 Mathematica [F]

\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\int \frac {(e x)^{-1+3 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx \]

input

Integrate[(e*x)^(-1 + 3*n)/(a + b*Csch[c + d*x^n])^2,x]
 

output

Integrate[(e*x)^(-1 + 3*n)/(a + b*Csch[c + d*x^n])^2, x]
 

3.1.83.3 Rubi [A] (verified)

Time = 2.26 (sec) , antiderivative size = 953, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5964, 5960, 3042, 4679, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(e x)^{3 n-1}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx\)

\(\Big \downarrow \) 5964

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \frac {x^{3 n-1}}{\left (a+b \text {csch}\left (d x^n+c\right )\right )^2}dx}{e}\)

\(\Big \downarrow \) 5960

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \frac {x^{2 n}}{\left (a+b \text {csch}\left (d x^n+c\right )\right )^2}dx^n}{e n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \frac {x^{2 n}}{\left (a+i b \csc \left (i d x^n+i c\right )\right )^2}dx^n}{e n}\)

\(\Big \downarrow \) 4679

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \left (-\frac {2 b x^{2 n}}{a^2 \left (b+a \sinh \left (d x^n+c\right )\right )}+\frac {x^{2 n}}{a^2}+\frac {b^2 x^{2 n}}{a^2 \left (b+a \sinh \left (d x^n+c\right )\right )^2}\right )dx^n}{e n}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \left (\frac {2 b^2 \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {a^2+b^2}}+1\right ) x^n}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^2 \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {a^2+b^2}}+1\right ) x^n}{a^2 \left (a^2+b^2\right ) d^2}-\frac {4 b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right ) x^n}{a^2 \sqrt {a^2+b^2} d^2}+\frac {2 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right ) x^n}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {4 b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right ) x^n}{a^2 \sqrt {a^2+b^2} d^2}-\frac {2 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right ) x^n}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {a^2+b^2}}+1\right ) x^{2 n}}{a^2 \sqrt {a^2+b^2} d}+\frac {b^3 \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {a^2+b^2}}+1\right ) x^{2 n}}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {2 b \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {a^2+b^2}}+1\right ) x^{2 n}}{a^2 \sqrt {a^2+b^2} d}-\frac {b^3 \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {a^2+b^2}}+1\right ) x^{2 n}}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^2 x^{2 n}}{a^2 \left (a^2+b^2\right ) d}-\frac {b^2 \cosh \left (d x^n+c\right ) x^{2 n}}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (d x^n+c\right )\right )}+\frac {x^{3 n}}{3 a^2}+\frac {2 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {4 b \operatorname {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {2 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {4 b \operatorname {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {2 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}\right )}{e n}\)

input

Int[(e*x)^(-1 + 3*n)/(a + b*Csch[c + d*x^n])^2,x]
 

output

((e*x)^(3*n)*(-((b^2*x^(2*n))/(a^2*(a^2 + b^2)*d)) + x^(3*n)/(3*a^2) + (2* 
b^2*x^n*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2) 
*d^2) + (b^3*x^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2])])/(a^ 
2*(a^2 + b^2)^(3/2)*d) - (2*b*x^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[ 
a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (2*b^2*x^n*Log[1 + (a*E^(c + d*x^n 
))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) - (b^3*x^(2*n)*Log[1 + (a 
*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) + (2*b*x 
^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b 
^2]*d) + (2*b^2*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a 
^2*(a^2 + b^2)*d^3) + (2*b^3*x^n*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[ 
a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (4*b*x^n*PolyLog[2, -((a*E^(c 
 + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) + (2*b^2*Pol 
yLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) 
 - (2*b^3*x^n*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a^2 
*(a^2 + b^2)^(3/2)*d^2) + (4*b*x^n*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqr 
t[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (2*b^3*PolyLog[3, -((a*E^(c + 
 d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) + (4*b*PolyL 
og[3, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^ 
3) + (2*b^3*PolyLog[3, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a^2*( 
a^2 + b^2)^(3/2)*d^3) - (4*b*PolyLog[3, -((a*E^(c + d*x^n))/(b + Sqrt[a...
 

3.1.83.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4679

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Si 
n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt 
Q[m, 0]
 

rule 5960

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo 
l] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csch[c + d*x] 
)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m 
 + 1)/n], 0] && IntegerQ[p]
 

rule 5964

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_.), 
x_Symbol] :> Simp[e^IntPart[m]*((e*x)^FracPart[m]/x^FracPart[m])   Int[x^m* 
(a + b*Csch[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
 

3.1.83.4 Maple [F]

\[\int \frac {\left (e x \right )^{-1+3 n}}{{\left (a +b \,\operatorname {csch}\left (c +d \,x^{n}\right )\right )}^{2}}d x\]

input

int((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n))^2,x)
 

output

int((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n))^2,x)
 

3.1.83.5 Fricas [F(-1)]

Timed out. \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\text {Timed out} \]

input

integrate((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n))^2,x, algorithm="fricas")
 

output

Timed out
 

3.1.83.6 Sympy [F]

\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\int \frac {\left (e x\right )^{3 n - 1}}{\left (a + b \operatorname {csch}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \]

input

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

output

Integral((e*x)**(3*n - 1)/(a + b*csch(c + d*x**n))**2, x)
 

3.1.83.7 Maxima [F]

\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{3 \, n - 1}}{{\left (b \operatorname {csch}\left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]

input

integrate((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n))^2,x, algorithm="maxima")
 

output

1/3*(6*a*b^2*e^(3*n)*x^(2*n) + (a^3*d*e^(3*n) + a*b^2*d*e^(3*n))*x^(3*n) - 
 (a^3*d*e^(3*n)*e^(2*c) + a*b^2*d*e^(3*n)*e^(2*c))*e^(2*d*x^n + 3*n*log(x) 
) - 2*(3*b^3*e^(3*n)*e^(2*n*log(x) + c) + (a^2*b*d*e^(3*n)*e^c + b^3*d*e^( 
3*n)*e^c)*x^(3*n))*e^(d*x^n))/(a^5*d*e*n + a^3*b^2*d*e*n - (a^5*d*e*n*e^(2 
*c) + a^3*b^2*d*e*n*e^(2*c))*e^(2*d*x^n) - 2*(a^4*b*d*e*n*e^c + a^2*b^3*d* 
e*n*e^c)*e^(d*x^n)) - integrate(-2*(2*a*b^2*e^(3*n)*x^(2*n) - (2*b^3*e^(3* 
n)*e^(2*n*log(x) + c) + (2*a^2*b*d*e^(3*n)*e^c + b^3*d*e^(3*n)*e^c)*x^(3*n 
))*e^(d*x^n))/((a^5*d*e*e^(2*c) + a^3*b^2*d*e*e^(2*c))*x*e^(2*d*x^n) + 2*( 
a^4*b*d*e*e^c + a^2*b^3*d*e*e^c)*x*e^(d*x^n) - (a^5*d*e + a^3*b^2*d*e)*x), 
 x)
 

3.1.83.8 Giac [F]

\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{3 \, n - 1}}{{\left (b \operatorname {csch}\left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]

input

integrate((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n))^2,x, algorithm="giac")
 

output

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

3.1.83.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{3\,n-1}}{{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,x^n\right )}\right )}^2} \,d x \]

input

int((e*x)^(3*n - 1)/(a + b/sinh(c + d*x^n))^2,x)
 

output

int((e*x)^(3*n - 1)/(a + b/sinh(c + d*x^n))^2, x)