Integrand size = 9, antiderivative size = 31 \[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\frac {2}{3} \cosh (x) \sqrt {\cosh (x) \coth (x)}-\frac {8}{3} \sqrt {\cosh (x) \coth (x)} \text {sech}(x) \] Output:
2/3*cosh(x)*(cosh(x)*coth(x))^(1/2)-8/3*(cosh(x)*coth(x))^(1/2)*sech(x)
Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\frac {2}{3} \left (-4+\cosh ^2(x)\right ) \sqrt {\cosh (x) \coth (x)} \text {sech}(x) \] Input:
Integrate[(Cosh[x]*Coth[x])^(3/2),x]
Output:
(2*(-4 + Cosh[x]^2)*Sqrt[Cosh[x]*Coth[x]]*Sech[x])/3
Result contains complex when optimal does not.
Time = 0.67 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.39, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {3042, 4898, 3042, 4900, 3042, 3078, 3042, 3069}
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.
\(\displaystyle \int (\cosh (x) \coth (x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (i \cos (i x) \cot (i x))^{3/2}dx\) |
\(\Big \downarrow \) 4898 |
\(\displaystyle \frac {i \sqrt {\cosh (x) \coth (x)} \int (-i \cosh (x) \coth (x))^{3/2}dx}{\sqrt {-i \cosh (x) \coth (x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {i \sqrt {\cosh (x) \coth (x)} \int (\cos (i x) \cot (i x))^{3/2}dx}{\sqrt {-i \cosh (x) \coth (x)}}\) |
\(\Big \downarrow \) 4900 |
\(\displaystyle \frac {i \sqrt {\cosh (x) \coth (x)} \int \cosh ^{\frac {3}{2}}(x) (-i \coth (x))^{3/2}dx}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {i \sqrt {\cosh (x) \coth (x)} \int \sin \left (i x+\frac {\pi }{2}\right )^{3/2} \left (-\tan \left (i x+\frac {\pi }{2}\right )\right )^{3/2}dx}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\) |
\(\Big \downarrow \) 3078 |
\(\displaystyle \frac {i \sqrt {\cosh (x) \coth (x)} \left (\frac {4}{3} \int \frac {(-i \coth (x))^{3/2}}{\sqrt {\cosh (x)}}dx-\frac {2}{3} i \cosh ^{\frac {3}{2}}(x) \sqrt {-i \coth (x)}\right )}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {i \sqrt {\cosh (x) \coth (x)} \left (\frac {4}{3} \int \frac {\left (-\tan \left (i x+\frac {\pi }{2}\right )\right )^{3/2}}{\sqrt {\sin \left (i x+\frac {\pi }{2}\right )}}dx-\frac {2}{3} i \cosh ^{\frac {3}{2}}(x) \sqrt {-i \coth (x)}\right )}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\) |
\(\Big \downarrow \) 3069 |
\(\displaystyle \frac {i \left (\frac {8 i \sqrt {-i \coth (x)}}{3 \sqrt {\cosh (x)}}-\frac {2}{3} i \cosh ^{\frac {3}{2}}(x) \sqrt {-i \coth (x)}\right ) \sqrt {\cosh (x) \coth (x)}}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\) |
Input:
Int[(Cosh[x]*Coth[x])^(3/2),x]
Output:
(I*((((8*I)/3)*Sqrt[(-I)*Coth[x]])/Sqrt[Cosh[x]] - ((2*I)/3)*Cosh[x]^(3/2) *Sqrt[(-I)*Coth[x]])*Sqrt[Cosh[x]*Coth[x]])/(Sqrt[Cosh[x]]*Sqrt[(-I)*Coth[ x]])
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-b)*(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f* m)), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 1, 0]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[(-b)*(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/( f*m)), x] + Simp[a^2*((m + n - 1)/m) Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[ e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1 ] && EqQ[n, 1/2])) && IntegersQ[2*m, 2*n]
Int[(u_.)*((a_)*(v_))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = A ctivateTrig[v]}, Simp[a^IntPart[p]*((a*vv)^FracPart[p]/vv^FracPart[p]) In t[uu*vv^p, x], x]] /; FreeQ[{a, p}, x] && !IntegerQ[p] && !InertTrigFreeQ [v]
Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTri g[u], vv = ActivateTrig[v], ww = ActivateTrig[w]}, Simp[(vv^m*ww^n)^FracPar t[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])) Int[uu*vv^(m*p)*ww^(n*p), x] , x]] /; FreeQ[{m, n, p}, x] && !IntegerQ[p] && ( !InertTrigFreeQ[v] || ! InertTrigFreeQ[w])
\[\int \left (\cosh \left (x \right ) \coth \left (x \right )\right )^{\frac {3}{2}}d x\]
Input:
int((cosh(x)*coth(x))^(3/2),x)
Output:
int((cosh(x)*coth(x))^(3/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (23) = 46\).
Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.13 \[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 7\right )} \sinh \left (x\right )^{2} - 14 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}}{3 \, \sqrt {\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \] Input:
integrate((cosh(x)*coth(x))^(3/2),x, algorithm="fricas")
Output:
1/3*sqrt(1/2)*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^ 2 - 7)*sinh(x)^2 - 14*cosh(x)^2 + 4*(cosh(x)^3 - 7*cosh(x))*sinh(x) + 1)/( sqrt(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)^3 + (3*cosh(x)^2 - 1)*sinh( x) - cosh(x))*(cosh(x) + sinh(x)))
Timed out. \[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate((cosh(x)*coth(x))**(3/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (23) = 46\).
Time = 0.16 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.52 \[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\frac {\sqrt {2} e^{\left (\frac {3}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} e^{\left (-\frac {1}{2} \, x\right )}}{2 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {5 \, \sqrt {2} e^{\left (-\frac {5}{2} \, x\right )}}{2 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {2} e^{\left (-\frac {9}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} \] Input:
integrate((cosh(x)*coth(x))^(3/2),x, algorithm="maxima")
Output:
1/6*sqrt(2)*e^(3/2*x)/((e^(-x) + 1)^(3/2)*(-e^(-x) + 1)^(3/2)) - 5/2*sqrt( 2)*e^(-1/2*x)/((e^(-x) + 1)^(3/2)*(-e^(-x) + 1)^(3/2)) + 5/2*sqrt(2)*e^(-5 /2*x)/((e^(-x) + 1)^(3/2)*(-e^(-x) + 1)^(3/2)) - 1/6*sqrt(2)*e^(-9/2*x)/(( e^(-x) + 1)^(3/2)*(-e^(-x) + 1)^(3/2))
\[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\int { \left (\cosh \left (x\right ) \coth \left (x\right )\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((cosh(x)*coth(x))^(3/2),x, algorithm="giac")
Output:
integrate((cosh(x)*coth(x))^(3/2), x)
Timed out. \[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\int {\left (\mathrm {cosh}\left (x\right )\,\mathrm {coth}\left (x\right )\right )}^{3/2} \,d x \] Input:
int((cosh(x)*coth(x))^(3/2),x)
Output:
int((cosh(x)*coth(x))^(3/2), x)
\[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\int \sqrt {\coth \left (x \right )}\, \sqrt {\cosh \left (x \right )}\, \cosh \left (x \right ) \coth \left (x \right )d x \] Input:
int((cosh(x)*coth(x))^(3/2),x)
Output:
int(sqrt(coth(x))*sqrt(cosh(x))*cosh(x)*coth(x),x)