Integrand size = 11, antiderivative size = 33 \[ \int (-\cosh (x)+\text {sech}(x))^{3/2} \, dx=-\frac {8}{3} \text {csch}(x) \sqrt {-\sinh (x) \tanh (x)}-\frac {2}{3} \sinh (x) \sqrt {-\sinh (x) \tanh (x)} \] Output:
-8/3*csch(x)*(-sinh(x)*tanh(x))^(1/2)-2/3*sinh(x)*(-sinh(x)*tanh(x))^(1/2)
Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int (-\cosh (x)+\text {sech}(x))^{3/2} \, dx=\frac {2}{3} \coth (x) \left (1+4 \text {csch}^2(x)\right ) (-\sinh (x) \tanh (x))^{3/2} \] Input:
Integrate[(-Cosh[x] + Sech[x])^(3/2),x]
Output:
(2*Coth[x]*(1 + 4*Csch[x]^2)*(-(Sinh[x]*Tanh[x]))^(3/2))/3
Result contains complex when optimal does not.
Time = 0.74 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.55, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3042, 4897, 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.
\(\displaystyle \int (\text {sech}(x)-\cosh (x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (\sec (i x)-\cos (i x))^{3/2}dx\) |
\(\Big \downarrow \) 4897 |
\(\displaystyle \int (-\sinh (x) \tanh (x))^{3/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (\sin (i x) \tan (i x))^{3/2}dx\) |
\(\Big \downarrow \) 4900 |
\(\displaystyle \frac {\sqrt {-\sinh (x) \tanh (x)} \int (i \sinh (x))^{3/2} (i \tanh (x))^{3/2}dx}{\sqrt {i \sinh (x)} \sqrt {i \tanh (x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {-\sinh (x) \tanh (x)} \int \sin (i x)^{3/2} \tan (i x)^{3/2}dx}{\sqrt {i \sinh (x)} \sqrt {i \tanh (x)}}\) |
\(\Big \downarrow \) 3078 |
\(\displaystyle \frac {\sqrt {-\sinh (x) \tanh (x)} \left (\frac {4}{3} \int \frac {(i \tanh (x))^{3/2}}{\sqrt {i \sinh (x)}}dx+\frac {2}{3} i (i \sinh (x))^{3/2} \sqrt {i \tanh (x)}\right )}{\sqrt {i \sinh (x)} \sqrt {i \tanh (x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {-\sinh (x) \tanh (x)} \left (\frac {4}{3} \int \frac {\tan (i x)^{3/2}}{\sqrt {\sin (i x)}}dx+\frac {2}{3} i (i \sinh (x))^{3/2} \sqrt {i \tanh (x)}\right )}{\sqrt {i \sinh (x)} \sqrt {i \tanh (x)}}\) |
\(\Big \downarrow \) 3069 |
\(\displaystyle \frac {\left (\frac {2}{3} i (i \sinh (x))^{3/2} \sqrt {i \tanh (x)}-\frac {8 i \sqrt {i \tanh (x)}}{3 \sqrt {i \sinh (x)}}\right ) \sqrt {-\sinh (x) \tanh (x)}}{\sqrt {i \sinh (x)} \sqrt {i \tanh (x)}}\) |
Input:
Int[(-Cosh[x] + Sech[x])^(3/2),x]
Output:
(((((-8*I)/3)*Sqrt[I*Tanh[x]])/Sqrt[I*Sinh[x]] + ((2*I)/3)*(I*Sinh[x])^(3/ 2)*Sqrt[I*Tanh[x]])*Sqrt[-(Sinh[x]*Tanh[x])])/(Sqrt[I*Sinh[x]]*Sqrt[I*Tanh [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_.)*((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])
\[\int \left (-\cosh \left (x \right )+\operatorname {sech}\left (x \right )\right )^{\frac {3}{2}}d x\]
Input:
int((-cosh(x)+sech(x))^(3/2),x)
Output:
int((-cosh(x)+sech(x))^(3/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (25) = 50\).
Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.00 \[ \int (-\cosh (x)+\text {sech}(x))^{3/2} \, dx=-\frac {\sqrt {\frac {1}{2}} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 7\right )} \sinh \left (x\right )^{2} + 14 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \sqrt {-\frac {1}{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )}}}{3 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \] Input:
integrate((-cosh(x)+sech(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(-1/(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)^3 + (3*cosh(x)^2 + 1)*s inh(x) + cosh(x)))/(cosh(x) + sinh(x))
\[ \int (-\cosh (x)+\text {sech}(x))^{3/2} \, dx=\int \left (- \cosh {\left (x \right )} + \operatorname {sech}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((-cosh(x)+sech(x))**(3/2),x)
Output:
Integral((-cosh(x) + sech(x))**(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (25) = 50\).
Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.33 \[ \int (-\cosh (x)+\text {sech}(x))^{3/2} \, dx=-\frac {\sqrt {2} e^{\left (\frac {3}{2} \, x\right )}}{6 \, {\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} e^{\left (-\frac {1}{2} \, x\right )}}{2 \, {\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} e^{\left (-\frac {5}{2} \, x\right )}}{2 \, {\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {2} e^{\left (-\frac {9}{2} \, x\right )}}{6 \, {\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}}} \] Input:
integrate((-cosh(x)+sech(x))^(3/2),x, algorithm="maxima")
Output:
-1/6*sqrt(2)*e^(3/2*x)/(-e^(-2*x) - 1)^(3/2) - 5/2*sqrt(2)*e^(-1/2*x)/(-e^ (-2*x) - 1)^(3/2) - 5/2*sqrt(2)*e^(-5/2*x)/(-e^(-2*x) - 1)^(3/2) - 1/6*sqr t(2)*e^(-9/2*x)/(-e^(-2*x) - 1)^(3/2)
\[ \int (-\cosh (x)+\text {sech}(x))^{3/2} \, dx=\int { {\left (-\cosh \left (x\right ) + \operatorname {sech}\left (x\right )\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((-cosh(x)+sech(x))^(3/2),x, algorithm="giac")
Output:
integrate((-cosh(x) + sech(x))^(3/2), x)
Timed out. \[ \int (-\cosh (x)+\text {sech}(x))^{3/2} \, dx=\int {\left (\frac {1}{\mathrm {cosh}\left (x\right )}-\mathrm {cosh}\left (x\right )\right )}^{3/2} \,d x \] Input:
int((1/cosh(x) - cosh(x))^(3/2),x)
Output:
int((1/cosh(x) - cosh(x))^(3/2), x)
\[ \int (-\cosh (x)+\text {sech}(x))^{3/2} \, dx=-\left (\int \sqrt {-\cosh \left (x \right )+\mathrm {sech}\left (x \right )}\, \cosh \left (x \right )d x \right )+\int \sqrt {-\cosh \left (x \right )+\mathrm {sech}\left (x \right )}\, \mathrm {sech}\left (x \right )d x \] Input:
int((-cosh(x)+sech(x))^(3/2),x)
Output:
- int(sqrt( - cosh(x) + sech(x))*cosh(x),x) + int(sqrt( - cosh(x) + sech( x))*sech(x),x)