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

Optimal result

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

[Out]

-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/x/(1-1/c^4/x^4)^(1/2)/
csch(2*ln(c*x))^(1/2)+2/5*EllipticF(1/c/x,I)/c^5/x/(1-1/c^4/x^4)^(1/2)/csch(2*ln(c*x))^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5671, 5669, 342, 283, 331, 313, 227, 1195, 435} \[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=-\frac {2}{5 c^4 \sqrt {\text {csch}(2 \log (c x))}}+\frac {2 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{5 c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}-\frac {2 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {csch}(2 \log (c x))}} \]

[In]

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

[Out]

-2/(5*c^4*Sqrt[Csch[2*Log[c*x]]]) + x^4/(5*Sqrt[Csch[2*Log[c*x]]]) - (2*EllipticE[ArcCsc[c*x], -1])/(5*c^5*Sqr
t[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]]) + (2*EllipticF[ArcCsc[c*x], -1])/(5*c^5*Sqrt[1 - 1/(c^4*x^4)]*x*S
qrt[Csch[2*Log[c*x]]])

Rule 227

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

Rule 283

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] - Dist[b*n*(p/(c^n*(m + 1))), Int[(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] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 313

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

Rule 331

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] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 342

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] && IntegerQ[m]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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 1195

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c], Int
[(d + e*x^2)/(Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c, 0]

Rule 5669

Int[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[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 5671

Int[Csch[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(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])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3}{\sqrt {\text {csch}(2 \log (x))}} \, dx,x,c x\right )}{c^4} \\ & = \frac {\text {Subst}\left (\int \sqrt {1-\frac {1}{x^4}} x^4 \, dx,x,c x\right )}{c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \\ & = -\frac {\text {Subst}\left (\int \frac {\sqrt {1-x^4}}{x^6} \, dx,x,\frac {1}{c x}\right )}{c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \\ & = \frac {x^4}{5 \sqrt {\text {csch}(2 \log (c x))}}+\frac {2 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \\ & = -\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 \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \\ & = -\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 \text {Subst}\left (\int \frac {1}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}-\frac {2 \text {Subst}\left (\int \frac {1+x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \\ & = -\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 \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))}}-\frac {2 \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \\ & = -\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))}} \\ \end{align*}

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 \[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {x^4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},c^4 x^4\right )}{3 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.07

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

[In]

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

[Out]

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)-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)

none

Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {\sqrt {2} {\left (c^{10} x^{8} - 3 \, c^{6} x^{4} + 2 \, c^{2}\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} - 2 \, \sqrt {c^{4}} {\left (\sqrt {2} x^{2} E(\arcsin \left (\frac {1}{c x}\right )\,|\,-1) - \sqrt {2} x^{2} F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)\right )}}{10 \, c^{8} x^{2}} \]

[In]

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

[Out]

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)) - 2*sqrt(c^4)*(sqrt(2)*x^2*elliptic_e
(arcsin(1/(c*x)), -1) - sqrt(2)*x^2*elliptic_f(arcsin(1/(c*x)), -1)))/(c^8*x^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly exception caught Unable to convert to real %%{poly1[1.0000000000000000000000000000000,0.0000000000
00000000000

Mupad [F(-1)]

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

[In]

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

[Out]

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