Integrand size = 10, antiderivative size = 65 \[ \int \left (a \text {csch}^2(x)\right )^{5/2} \, dx=-\frac {3}{8} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right )+\frac {3}{8} a^2 \coth (x) \sqrt {a \text {csch}^2(x)}-\frac {1}{4} a \coth (x) \left (a \text {csch}^2(x)\right )^{3/2} \]
-3/8*a^(5/2)*arctanh(coth(x)*a^(1/2)/(a*csch(x)^2)^(1/2))-1/4*a*coth(x)*(a *csch(x)^2)^(3/2)+3/8*a^2*coth(x)*(a*csch(x)^2)^(1/2)
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.78 \[ \int \left (a \text {csch}^2(x)\right )^{5/2} \, dx=\frac {1}{64} \left (a \text {csch}^2(x)\right )^{5/2} \sinh (x) \left (-22 \cosh (x)+6 \left (\cosh (3 x)+4 \left (-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \sinh ^4(x)\right )\right ) \]
((a*Csch[x]^2)^(5/2)*Sinh[x]*(-22*Cosh[x] + 6*(Cosh[3*x] + 4*(-Log[Cosh[x/ 2]] + Log[Sinh[x/2]])*Sinh[x]^4)))/64
Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.26, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4610, 211, 211, 224, 219}
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.
\(\displaystyle \int \left (a \text {csch}^2(x)\right )^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (-a \sec \left (\frac {\pi }{2}+i x\right )^2\right )^{5/2}dx\) |
\(\Big \downarrow \) 4610 |
\(\displaystyle -a \int \left (a \coth ^2(x)-a\right )^{3/2}d\coth (x)\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -a \left (\frac {1}{4} \coth (x) \left (a \coth ^2(x)-a\right )^{3/2}-\frac {3}{4} a \int \sqrt {a \coth ^2(x)-a}d\coth (x)\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -a \left (\frac {1}{4} \coth (x) \left (a \coth ^2(x)-a\right )^{3/2}-\frac {3}{4} a \left (\frac {1}{2} \coth (x) \sqrt {a \coth ^2(x)-a}-\frac {1}{2} a \int \frac {1}{\sqrt {a \coth ^2(x)-a}}d\coth (x)\right )\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -a \left (\frac {1}{4} \coth (x) \left (a \coth ^2(x)-a\right )^{3/2}-\frac {3}{4} a \left (\frac {1}{2} \coth (x) \sqrt {a \coth ^2(x)-a}-\frac {1}{2} a \int \frac {1}{1-\frac {a \coth ^2(x)}{a \coth ^2(x)-a}}d\frac {\coth (x)}{\sqrt {a \coth ^2(x)-a}}\right )\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -a \left (\frac {1}{4} \coth (x) \left (a \coth ^2(x)-a\right )^{3/2}-\frac {3}{4} a \left (\frac {1}{2} \coth (x) \sqrt {a \coth ^2(x)-a}-\frac {1}{2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \coth ^2(x)-a}}\right )\right )\right )\) |
-(a*((Coth[x]*(-a + a*Coth[x]^2)^(3/2))/4 - (3*a*(-1/2*(Sqrt[a]*ArcTanh[(S qrt[a]*Coth[x])/Sqrt[-a + a*Coth[x]^2]]) + (Coth[x]*Sqrt[-a + a*Coth[x]^2] )/2))/4))
3.1.29.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFac tors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(122\) vs. \(2(49)=98\).
Time = 0.33 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.89
method | result | size |
risch | \(\frac {a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left (3 \,{\mathrm e}^{6 x}-11 \,{\mathrm e}^{4 x}-11 \,{\mathrm e}^{2 x}+3\right )}{4 \left ({\mathrm e}^{2 x}-1\right )^{3}}+\frac {3 a^{2} {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}-1\right )}{8}-\frac {3 a^{2} {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}+1\right )}{8}\) | \(123\) |
1/4*a^2/(exp(2*x)-1)^3*(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)*(3*exp(6*x)-11*ex p(4*x)-11*exp(2*x)+3)+3/8*a^2*exp(-x)*(exp(2*x)-1)*(a*exp(2*x)/(exp(2*x)-1 )^2)^(1/2)*ln(exp(x)-1)-3/8*a^2*exp(-x)*(exp(2*x)-1)*(a*exp(2*x)/(exp(2*x) -1)^2)^(1/2)*ln(exp(x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 1128 vs. \(2 (49) = 98\).
Time = 0.27 (sec) , antiderivative size = 1128, normalized size of antiderivative = 17.35 \[ \int \left (a \text {csch}^2(x)\right )^{5/2} \, dx=\text {Too large to display} \]
-1/8*(6*a^2*cosh(x)^7 - 6*(a^2*e^(2*x) - a^2)*sinh(x)^7 - 22*a^2*cosh(x)^5 - 42*(a^2*cosh(x)*e^(2*x) - a^2*cosh(x))*sinh(x)^6 + 2*(63*a^2*cosh(x)^2 - 11*a^2 - (63*a^2*cosh(x)^2 - 11*a^2)*e^(2*x))*sinh(x)^5 - 22*a^2*cosh(x) ^3 + 10*(21*a^2*cosh(x)^3 - 11*a^2*cosh(x) - (21*a^2*cosh(x)^3 - 11*a^2*co sh(x))*e^(2*x))*sinh(x)^4 + 2*(105*a^2*cosh(x)^4 - 110*a^2*cosh(x)^2 - 11* a^2 - (105*a^2*cosh(x)^4 - 110*a^2*cosh(x)^2 - 11*a^2)*e^(2*x))*sinh(x)^3 + 6*a^2*cosh(x) + 2*(63*a^2*cosh(x)^5 - 110*a^2*cosh(x)^3 - 33*a^2*cosh(x) - (63*a^2*cosh(x)^5 - 110*a^2*cosh(x)^3 - 33*a^2*cosh(x))*e^(2*x))*sinh(x )^2 - 2*(3*a^2*cosh(x)^7 - 11*a^2*cosh(x)^5 - 11*a^2*cosh(x)^3 + 3*a^2*cos h(x))*e^(2*x) + 3*(a^2*cosh(x)^8 - (a^2*e^(2*x) - a^2)*sinh(x)^8 - 4*a^2*c osh(x)^6 - 8*(a^2*cosh(x)*e^(2*x) - a^2*cosh(x))*sinh(x)^7 + 4*(7*a^2*cosh (x)^2 - a^2 - (7*a^2*cosh(x)^2 - a^2)*e^(2*x))*sinh(x)^6 + 6*a^2*cosh(x)^4 + 8*(7*a^2*cosh(x)^3 - 3*a^2*cosh(x) - (7*a^2*cosh(x)^3 - 3*a^2*cosh(x))* e^(2*x))*sinh(x)^5 + 2*(35*a^2*cosh(x)^4 - 30*a^2*cosh(x)^2 + 3*a^2 - (35* a^2*cosh(x)^4 - 30*a^2*cosh(x)^2 + 3*a^2)*e^(2*x))*sinh(x)^4 - 4*a^2*cosh( x)^2 + 8*(7*a^2*cosh(x)^5 - 10*a^2*cosh(x)^3 + 3*a^2*cosh(x) - (7*a^2*cosh (x)^5 - 10*a^2*cosh(x)^3 + 3*a^2*cosh(x))*e^(2*x))*sinh(x)^3 + 4*(7*a^2*co sh(x)^6 - 15*a^2*cosh(x)^4 + 9*a^2*cosh(x)^2 - a^2 - (7*a^2*cosh(x)^6 - 15 *a^2*cosh(x)^4 + 9*a^2*cosh(x)^2 - a^2)*e^(2*x))*sinh(x)^2 + a^2 - (a^2*co sh(x)^8 - 4*a^2*cosh(x)^6 + 6*a^2*cosh(x)^4 - 4*a^2*cosh(x)^2 + a^2)*e^...
\[ \int \left (a \text {csch}^2(x)\right )^{5/2} \, dx=\int \left (a \operatorname {csch}^{2}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \]
Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.42 \[ \int \left (a \text {csch}^2(x)\right )^{5/2} \, dx=\frac {3}{8} \, a^{\frac {5}{2}} \log \left (e^{\left (-x\right )} + 1\right ) - \frac {3}{8} \, a^{\frac {5}{2}} \log \left (e^{\left (-x\right )} - 1\right ) + \frac {3 \, a^{\frac {5}{2}} e^{\left (-x\right )} - 11 \, a^{\frac {5}{2}} e^{\left (-3 \, x\right )} - 11 \, a^{\frac {5}{2}} e^{\left (-5 \, x\right )} + 3 \, a^{\frac {5}{2}} e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} \]
3/8*a^(5/2)*log(e^(-x) + 1) - 3/8*a^(5/2)*log(e^(-x) - 1) + 1/4*(3*a^(5/2) *e^(-x) - 11*a^(5/2)*e^(-3*x) - 11*a^(5/2)*e^(-5*x) + 3*a^(5/2)*e^(-7*x))/ (4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1)
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.15 \[ \int \left (a \text {csch}^2(x)\right )^{5/2} \, dx=\frac {1}{16} \, a^{\frac {5}{2}} {\left (\frac {4 \, {\left (3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 20 \, e^{\left (-x\right )} - 20 \, e^{x}\right )}}{{\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{2}} - 3 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + 3 \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) \]
1/16*a^(5/2)*(4*(3*(e^(-x) + e^x)^3 - 20*e^(-x) - 20*e^x)/((e^(-x) + e^x)^ 2 - 4)^2 - 3*log(e^(-x) + e^x + 2) + 3*log(e^(-x) + e^x - 2))*sgn(e^(3*x) - e^x)
Timed out. \[ \int \left (a \text {csch}^2(x)\right )^{5/2} \, dx=\int {\left (\frac {a}{{\mathrm {sinh}\left (x\right )}^2}\right )}^{5/2} \,d x \]