\(\int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx\) [146]

Optimal result

Integrand size = 15, antiderivative size = 162 \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {4}{15 c^4 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \] Output:

4/15/c^4/(c^4-1/x^4)/x^2/csch(2*ln(c*x))^(3/2)-2/15*x^2/(c^4-1/x^4)/csch(2 
*ln(c*x))^(3/2)+1/9*x^6/csch(2*ln(c*x))^(3/2)+4/15*EllipticE(1/c/x,I)/c^9/ 
(1-1/c^4/x^4)^(3/2)/x^3/csch(2*ln(c*x))^(3/2)-4/15*InverseJacobiAM(arccsc( 
c*x),I)/c^9/(1-1/c^4/x^4)^(3/2)/x^3/csch(2*ln(c*x))^(3/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.39 \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=-\frac {x^4 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},c^4 x^4\right )}{6 c^2 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \] Input:

Integrate[x^5/Csch[2*Log[c*x]]^(3/2),x]
 

Output:

-1/6*(x^4*Hypergeometric2F1[-3/2, 3/4, 7/4, c^4*x^4])/(c^2*Sqrt[2 - 2*c^4* 
x^4]*Sqrt[(c^2*x^2)/(-1 + c^4*x^4)])
 

Rubi [A] (warning: unable to verify)

Time = 0.41 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.84, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6086, 6084, 858, 809, 809, 847, 836, 762, 1388, 327}

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.

\(\displaystyle \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx\)

\(\Big \downarrow \) 6086

\(\displaystyle \frac {\int \frac {c^5 x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))}d(c x)}{c^6}\)

\(\Big \downarrow \) 6084

\(\displaystyle \frac {\int c^8 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^8d(c x)}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\)

\(\Big \downarrow \) 858

\(\displaystyle -\frac {\int \frac {\left (1-c^4 x^4\right )^{3/2}}{c^{10} x^{10}}d\frac {1}{c x}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\)

\(\Big \downarrow \) 809

\(\displaystyle -\frac {-\frac {2}{3} \int \frac {\sqrt {1-c^4 x^4}}{c^6 x^6}d\frac {1}{c x}-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\)

\(\Big \downarrow \) 809

\(\displaystyle -\frac {-\frac {2}{3} \left (-\frac {2}{5} \int \frac {1}{c^2 x^2 \sqrt {1-c^4 x^4}}d\frac {1}{c x}-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\)

\(\Big \downarrow \) 847

\(\displaystyle -\frac {-\frac {2}{3} \left (-\frac {2}{5} \left (-\int \frac {c^2 x^2}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\)

\(\Big \downarrow \) 836

\(\displaystyle -\frac {-\frac {2}{3} \left (-\frac {2}{5} \left (\int \frac {1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}-\int \frac {c^2 x^2+1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\)

\(\Big \downarrow \) 762

\(\displaystyle -\frac {-\frac {2}{3} \left (-\frac {2}{5} \left (-\int \frac {c^2 x^2+1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}+\operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\)

\(\Big \downarrow \) 1388

\(\displaystyle -\frac {-\frac {2}{3} \left (-\frac {2}{5} \left (-\int \frac {\sqrt {c^2 x^2+1}}{\sqrt {1-c^2 x^2}}d\frac {1}{c x}+\operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {-\frac {2}{3} \left (-\frac {2}{5} \left (\operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )-E\left (\left .\arcsin \left (\frac {1}{c x}\right )\right |-1\right )-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\)

Input:

Int[x^5/Csch[2*Log[c*x]]^(3/2),x]
 

Output:

-((-1/9*(1 - c^4*x^4)^(3/2)/(c^9*x^9) - (2*(-1/5*Sqrt[1 - c^4*x^4]/(c^5*x^ 
5) - (2*(-(Sqrt[1 - c^4*x^4]/(c*x)) - EllipticE[ArcSin[1/(c*x)], -1] + Ell 
ipticF[ArcSin[1/(c*x)], -1]))/5))/3)/(c^9*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch 
[2*Log[c*x]]^(3/2)))
 

Defintions of rubi rules used

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 809

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* 
x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1)))   I 
nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ 
[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntB 
inomialQ[a, b, c, n, m, p, x]
 

rule 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

rule 847

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x 
)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) 
+ 1)/(a*c^n*(m + 1)))   Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a 
, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 858

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + 
b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int 
egerQ[m]
 

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

rule 6084

Int[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[Csch[d*(a + b*Log[x])]^p*((1 - 1/(E^(2*a*d)*x^(2*b*d)))^p/x^((-b)* 
d*p))   Int[(e*x)^m*(1/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p)), x], x] 
 /; FreeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]
 

rule 6086

Int[Csch[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m 
_.), x_Symbol] :> Simp[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))   Subst[Int[ 
x^((m + 1)/n - 1)*Csch[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, 
b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.86

method	result	size
risch	\(\frac {x^{4} \left (5 c^{4} x^{4}-11\right ) \sqrt {2}}{180 c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}+\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-c^{2}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-c^{2}}, i\right )\right ) \sqrt {2}\, x}{15 \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) c^{4} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\)	\(140\)

Input:

int(x^5/csch(2*ln(x*c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/180*x^4*(5*c^4*x^4-11)*2^(1/2)/c^2/(c^2*x^2/(c^4*x^4-1))^(1/2)+1/15/(-c^ 
2)^(1/2)*(c^2*x^2+1)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^4*x^4-1)/c^4*(EllipticF(x 
*(-c^2)^(1/2),I)-EllipticE(x*(-c^2)^(1/2),I))*2^(1/2)*x/(c^2*x^2/(c^4*x^4- 
1))^(1/2)
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.64 \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\sqrt {2} {\left (5 \, c^{14} x^{12} - 16 \, c^{10} x^{8} + 23 \, c^{6} x^{4} - 12 \, c^{2}\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} + 24 \, \sqrt {\frac {1}{2}} \sqrt {c^{4}} {\left (x^{2} E(\arcsin \left (\frac {1}{c x}\right )\,|\,-1) - x^{2} F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)\right )}}{180 \, c^{10} x^{2}} \] Input:

integrate(x^5/csch(2*log(c*x))^(3/2),x, algorithm="fricas")
 

Output:

1/180*(sqrt(2)*(5*c^14*x^12 - 16*c^10*x^8 + 23*c^6*x^4 - 12*c^2)*sqrt(c^2* 
x^2/(c^4*x^4 - 1)) + 24*sqrt(1/2)*sqrt(c^4)*(x^2*elliptic_e(arcsin(1/(c*x) 
), -1) - x^2*elliptic_f(arcsin(1/(c*x)), -1)))/(c^10*x^2)
 

Sympy [F]

\[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^{5}}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \] Input:

integrate(x**5/csch(2*ln(c*x))**(3/2),x)
 

Output:

Integral(x**5/csch(2*log(c*x))**(3/2), x)
 

Maxima [F]

\[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int { \frac {x^{5}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^5/csch(2*log(c*x))^(3/2),x, algorithm="maxima")
 

Output:

integrate(x^5/csch(2*log(c*x))^(3/2), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Timed out} \] Input:

integrate(x^5/csch(2*log(c*x))^(3/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^5}{{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \] Input:

int(x^5/(1/sinh(2*log(c*x)))^(3/2),x)
 

Output:

int(x^5/(1/sinh(2*log(c*x)))^(3/2), x)
 

Reduce [F]

\[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {\sqrt {\mathrm {csch}\left (2 \,\mathrm {log}\left (c x \right )\right )}\, x^{5}}{\mathrm {csch}\left (2 \,\mathrm {log}\left (c x \right )\right )^{2}}d x \] Input:

int(x^5/csch(2*log(c*x))^(3/2),x)
 

Output:

int((sqrt(csch(2*log(c*x)))*x**5)/csch(2*log(c*x))**2,x)