\(\int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx\) [134]

Optimal result

Integrand size = 15, antiderivative size = 119 \[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=-\frac {2}{5 c^4 \sqrt {\text {csch}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {csch}(2 \log (c x))}}-\frac {2 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}+\frac {2 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \] Output:

-2/5/c^4/csch(2*ln(c*x))^(1/2)+1/5*x^4/csch(2*ln(c*x))^(1/2)-2/5*EllipticE 
(1/c/x,I)/c^5/(1-1/c^4/x^4)^(1/2)/x/csch(2*ln(c*x))^(1/2)+2/5*InverseJacob 
iAM(arccsc(c*x),I)/c^5/(1-1/c^4/x^4)^(1/2)/x/csch(2*ln(c*x))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

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

Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.50 \[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {x^4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},c^4 x^4\right )}{3 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \] Input:

Integrate[x^3/Sqrt[Csch[2*Log[c*x]]],x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.42 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6086, 6084, 858, 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.

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

\(\Big \downarrow \) 6086

\(\displaystyle \frac {\int \frac {c^3 x^3}{\sqrt {\text {csch}(2 \log (c x))}}d(c x)}{c^4}\)

\(\Big \downarrow \) 6084

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

\(\Big \downarrow \) 858

\(\displaystyle -\frac {\int \frac {\sqrt {1-c^4 x^4}}{c^6 x^6}d\frac {1}{c x}}{c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\)

\(\Big \downarrow \) 809

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

\(\Big \downarrow \) 847

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

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 327

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

Input:

Int[x^3/Sqrt[Csch[2*Log[c*x]]],x]
 

Output:

-((-1/5*Sqrt[1 - c^4*x^4]/(c^5*x^5) - (2*(-(Sqrt[1 - c^4*x^4]/(c*x)) - Ell 
ipticE[ArcSin[1/(c*x)], -1] + EllipticF[ArcSin[1/(c*x)], -1]))/5)/(c^5*Sqr 
t[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]]))
 

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 = 127, normalized size of antiderivative = 1.07

method	result	size
risch	\(\frac {\sqrt {2}\, x^{4}}{10 \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}-\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-c^{2}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-c^{2}}, i\right )\right ) \sqrt {2}\, x}{5 \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\)	\(127\)

Input:

int(x^3/csch(2*ln(x*c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/10*2^(1/2)*x^4/(c^2*x^2/(c^4*x^4-1))^(1/2)-1/5/(-c^2)^(1/2)*(c^2*x^2+1)^ 
(1/2)*(-c^2*x^2+1)^(1/2)/(c^4*x^4-1)/c^2*(EllipticF(x*(-c^2)^(1/2),I)-Elli 
pticE(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 = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.79 \[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {\sqrt {2} {\left (c^{10} x^{8} - 3 \, c^{6} x^{4} + 2 \, c^{2}\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} - 4 \, \sqrt {\frac {1}{2}} \sqrt {c^{4}} {\left (x^{2} E(\arcsin \left (\frac {1}{c x}\right )\,|\,-1) - x^{2} F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)\right )}}{10 \, c^{8} x^{2}} \] Input:

integrate(x^3/csch(2*log(c*x))^(1/2),x, algorithm="fricas")
 

Output:

1/10*(sqrt(2)*(c^10*x^8 - 3*c^6*x^4 + 2*c^2)*sqrt(c^2*x^2/(c^4*x^4 - 1)) - 
 4*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^8*x^2)
 

Sympy [F]

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

integrate(x**3/csch(2*ln(c*x))**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(x^3/csch(2*log(c*x))^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3/csch(2*log(c*x))^(1/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly exception caught Unable to 
 convert to real %%{poly1[1.0000000000000000000000000000000,0.000000000000 
000000000
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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