Integrand size = 10, antiderivative size = 46 \[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right )-\frac {1}{2} a \coth (x) \sqrt {a \text {csch}^2(x)} \] Output:
1/2*a^(3/2)*arctanh(a^(1/2)*coth(x)/(a*csch(x)^2)^(1/2))-1/2*a*coth(x)*(a* csch(x)^2)^(1/2)
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=-\frac {1}{2} a \sqrt {a \text {csch}^2(x)} \left (\coth (x) \text {csch}(x)-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \sinh (x) \] Input:
Integrate[(a*Csch[x]^2)^(3/2),x]
Output:
-1/2*(a*Sqrt[a*Csch[x]^2]*(Coth[x]*Csch[x] - Log[Cosh[x/2]] + Log[Sinh[x/2 ]])*Sinh[x])
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4610, 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.
\(\displaystyle \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (-a \sec \left (\frac {\pi }{2}+i x\right )^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4610 |
\(\displaystyle -a \int \sqrt {a \coth ^2(x)-a}d\coth (x)\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -a \left (\frac {1}{2} \coth (x) \sqrt {a \coth ^2(x)-a}-\frac {1}{2} a \int \frac {1}{\sqrt {a \coth ^2(x)-a}}d\coth (x)\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -a \left (\frac {1}{2} \coth (x) \sqrt {a \coth ^2(x)-a}-\frac {1}{2} a \int \frac {1}{1-\frac {a \coth ^2(x)}{a \coth ^2(x)-a}}d\frac {\coth (x)}{\sqrt {a \coth ^2(x)-a}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -a \left (\frac {1}{2} \coth (x) \sqrt {a \coth ^2(x)-a}-\frac {1}{2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \coth ^2(x)-a}}\right )\right )\) |
Input:
Int[(a*Csch[x]^2)^(3/2),x]
Output:
-(a*(-1/2*(Sqrt[a]*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[-a + a*Coth[x]^2]]) + (C oth[x]*Sqrt[-a + a*Coth[x]^2])/2))
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. \(102\) vs. \(2(34)=68\).
Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.24
method | result | size |
risch | \(-\frac {a \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}+1\right )}{{\mathrm e}^{2 x}-1}+\frac {a \,{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left (1+{\mathrm e}^{x}\right )}{2}-\frac {a \,{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}-1\right )}{2}\) | \(103\) |
Input:
int((a*csch(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-a/(exp(2*x)-1)*(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)*(exp(2*x)+1)+1/2*a*exp(- x)*(exp(2*x)-1)*(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)*ln(1+exp(x))-1/2*a*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. 340 vs. \(2 (34) = 68\).
Time = 0.10 (sec) , antiderivative size = 340, normalized size of antiderivative = 7.39 \[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=\frac {{\left (2 \, a \cosh \left (x\right )^{3} - 2 \, {\left (a e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )^{3} - 6 \, {\left (a \cosh \left (x\right ) e^{\left (2 \, x\right )} - a \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) - 2 \, {\left (a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} - {\left (a \cosh \left (x\right )^{4} - {\left (a e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )^{4} - 4 \, {\left (a \cosh \left (x\right ) e^{\left (2 \, x\right )} - a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 2 \, a \cosh \left (x\right )^{2} + 2 \, {\left (3 \, a \cosh \left (x\right )^{2} - {\left (3 \, a \cosh \left (x\right )^{2} - a\right )} e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )^{2} - {\left (a \cosh \left (x\right )^{4} - 2 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + 4 \, {\left (a \cosh \left (x\right )^{3} - a \cosh \left (x\right ) - {\left (a \cosh \left (x\right )^{3} - a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right ) + a\right )} \log \left (\frac {\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) + 2 \, {\left (3 \, a \cosh \left (x\right )^{2} - {\left (3 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \, {\left (4 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{3} + e^{x} \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} e^{x} \sinh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right ) + {\left (\cosh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{2} + 1\right )} e^{x}\right )}} \] Input:
integrate((a*csch(x)^2)^(3/2),x, algorithm="fricas")
Output:
1/2*(2*a*cosh(x)^3 - 2*(a*e^(2*x) - a)*sinh(x)^3 - 6*(a*cosh(x)*e^(2*x) - a*cosh(x))*sinh(x)^2 + 2*a*cosh(x) - 2*(a*cosh(x)^3 + a*cosh(x))*e^(2*x) - (a*cosh(x)^4 - (a*e^(2*x) - a)*sinh(x)^4 - 4*(a*cosh(x)*e^(2*x) - a*cosh( x))*sinh(x)^3 - 2*a*cosh(x)^2 + 2*(3*a*cosh(x)^2 - (3*a*cosh(x)^2 - a)*e^( 2*x) - a)*sinh(x)^2 - (a*cosh(x)^4 - 2*a*cosh(x)^2 + a)*e^(2*x) + 4*(a*cos h(x)^3 - a*cosh(x) - (a*cosh(x)^3 - a*cosh(x))*e^(2*x))*sinh(x) + a)*log(( cosh(x) + sinh(x) + 1)/(cosh(x) + sinh(x) - 1)) + 2*(3*a*cosh(x)^2 - (3*a* cosh(x)^2 + a)*e^(2*x) + a)*sinh(x))*sqrt(a/(e^(4*x) - 2*e^(2*x) + 1))*e^x /(4*cosh(x)*e^x*sinh(x)^3 + e^x*sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*e^x*sinh(x )^2 + 4*(cosh(x)^3 - cosh(x))*e^x*sinh(x) + (cosh(x)^4 - 2*cosh(x)^2 + 1)* e^x)
\[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=\int \left (a \operatorname {csch}^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((a*csch(x)**2)**(3/2),x)
Output:
Integral((a*csch(x)**2)**(3/2), x)
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \, a^{\frac {3}{2}} \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{2} \, a^{\frac {3}{2}} \log \left (e^{\left (-x\right )} - 1\right ) - \frac {a^{\frac {3}{2}} e^{\left (-x\right )} + a^{\frac {3}{2}} e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \] Input:
integrate((a*csch(x)^2)^(3/2),x, algorithm="maxima")
Output:
-1/2*a^(3/2)*log(e^(-x) + 1) + 1/2*a^(3/2)*log(e^(-x) - 1) - (a^(3/2)*e^(- x) + a^(3/2)*e^(-3*x))/(2*e^(-2*x) - e^(-4*x) - 1)
Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26 \[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=-\frac {1}{4} \, a^{\frac {3}{2}} {\left (\frac {4 \, {\left (e^{\left (-x\right )} + e^{x}\right )}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} - \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \log \left (e^{\left (-x\right )} + e^{x} - 2\right )\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) \] Input:
integrate((a*csch(x)^2)^(3/2),x, algorithm="giac")
Output:
-1/4*a^(3/2)*(4*(e^(-x) + e^x)/((e^(-x) + e^x)^2 - 4) - log(e^(-x) + e^x + 2) + log(e^(-x) + e^x - 2))*sgn(e^(3*x) - e^x)
Timed out. \[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=\int {\left (\frac {a}{{\mathrm {sinh}\left (x\right )}^2}\right )}^{3/2} \,d x \] Input:
int((a/sinh(x)^2)^(3/2),x)
Output:
int((a/sinh(x)^2)^(3/2), x)
Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.17 \[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=\frac {\sqrt {a}\, a \left (-e^{4 x} \mathrm {log}\left (e^{x}-1\right )+e^{4 x} \mathrm {log}\left (e^{x}+1\right )-2 e^{3 x}+2 e^{2 x} \mathrm {log}\left (e^{x}-1\right )-2 e^{2 x} \mathrm {log}\left (e^{x}+1\right )-2 e^{x}-\mathrm {log}\left (e^{x}-1\right )+\mathrm {log}\left (e^{x}+1\right )\right )}{2 e^{4 x}-4 e^{2 x}+2} \] Input:
int((a*csch(x)^2)^(3/2),x)
Output:
(sqrt(a)*a*( - e**(4*x)*log(e**x - 1) + e**(4*x)*log(e**x + 1) - 2*e**(3*x ) + 2*e**(2*x)*log(e**x - 1) - 2*e**(2*x)*log(e**x + 1) - 2*e**x - log(e** x - 1) + log(e**x + 1)))/(2*(e**(4*x) - 2*e**(2*x) + 1))