\(\int (a \text {sech}^4(x))^{3/2} \, dx\) [47]

Optimal result

Integrand size = 10, antiderivative size = 61 \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=a \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {2}{3} a \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh (x)+\frac {1}{5} a \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^3(x) \]

[Out]

a*cosh(x)*sinh(x)*(a*sech(x)^4)^(1/2)-2/3*a*sinh(x)^2*(a*sech(x)^4)^(1/2)*tanh(x)+1/5*a*sinh(x)^2*(a*sech(x)^4
)^(1/2)*tanh(x)^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4208, 3852} \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=a \sinh (x) \cosh (x) \sqrt {a \text {sech}^4(x)}+\frac {1}{5} a \sinh ^2(x) \tanh ^3(x) \sqrt {a \text {sech}^4(x)}-\frac {2}{3} a \sinh ^2(x) \tanh (x) \sqrt {a \text {sech}^4(x)} \]

[In]

Int[(a*Sech[x]^4)^(3/2),x]

[Out]

a*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x] - (2*a*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x])/3 + (a*Sqrt[a*Sech[x]^4]*Sinh[
x]^2*Tanh[x]^3)/5

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4208

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Sec[e + f*x])^n)^
FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \left (a \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int \text {sech}^6(x) \, dx \\ & = \left (i a \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \tanh (x)\right ) \\ & = a \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {2}{3} a \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh (x)+\frac {1}{5} a \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^3(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=\frac {1}{15} \cosh (x) (8+6 \cosh (2 x)+\cosh (4 x)) \left (a \text {sech}^4(x)\right )^{3/2} \sinh (x) \]

[In]

Integrate[(a*Sech[x]^4)^(3/2),x]

[Out]

(Cosh[x]*(8 + 6*Cosh[2*x] + Cosh[4*x])*(a*Sech[x]^4)^(3/2)*Sinh[x])/15

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75

method	result	size
risch	\(-\frac {16 a \,{\mathrm e}^{-2 x} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (10 \,{\mathrm e}^{4 x}+5 \,{\mathrm e}^{2 x}+1\right )}{15 \left (1+{\mathrm e}^{2 x}\right )^{3}}\)	\(46\)

[In]

int((sech(x)^4*a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-16/15*a*exp(-2*x)/(1+exp(2*x))^3*(exp(4*x)*a/(1+exp(2*x))^4)^(1/2)*(10*exp(4*x)+5*exp(2*x)+1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (51) = 102\).

Time = 0.26 (sec) , antiderivative size = 516, normalized size of antiderivative = 8.46 \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=-\frac {16 \, {\left (10 \, a \cosh \left (x\right )^{4} + 10 \, {\left (a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{4} + 40 \, {\left (a \cosh \left (x\right ) e^{\left (4 \, x\right )} + 2 \, a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 5 \, a \cosh \left (x\right )^{2} + 5 \, {\left (12 \, a \cosh \left (x\right )^{2} + {\left (12 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (4 \, x\right )} + 2 \, {\left (12 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} + {\left (10 \, a \cosh \left (x\right )^{4} + 5 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (4 \, x\right )} + 2 \, {\left (10 \, a \cosh \left (x\right )^{4} + 5 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + 10 \, {\left (4 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right ) + {\left (4 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} e^{\left (4 \, x\right )} + 2 \, {\left (4 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right ) + a\right )} \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} e^{\left (2 \, x\right )}}{15 \, {\left (10 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} \sinh \left (x\right )^{9} + e^{\left (2 \, x\right )} \sinh \left (x\right )^{10} + 5 \, {\left (9 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{8} + 40 \, {\left (3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{7} + 10 \, {\left (21 \, \cosh \left (x\right )^{4} + 14 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{6} + 4 \, {\left (63 \, \cosh \left (x\right )^{5} + 70 \, \cosh \left (x\right )^{3} + 15 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{5} + 10 \, {\left (21 \, \cosh \left (x\right )^{6} + 35 \, \cosh \left (x\right )^{4} + 15 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{4} + 40 \, {\left (3 \, \cosh \left (x\right )^{7} + 7 \, \cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{3} + 5 \, {\left (9 \, \cosh \left (x\right )^{8} + 28 \, \cosh \left (x\right )^{6} + 30 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{2} + 10 \, {\left (\cosh \left (x\right )^{9} + 4 \, \cosh \left (x\right )^{7} + 6 \, \cosh \left (x\right )^{5} + 4 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} \sinh \left (x\right ) + {\left (\cosh \left (x\right )^{10} + 5 \, \cosh \left (x\right )^{8} + 10 \, \cosh \left (x\right )^{6} + 10 \, \cosh \left (x\right )^{4} + 5 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )}\right )}} \]

[In]

integrate((a*sech(x)^4)^(3/2),x, algorithm="fricas")

[Out]

-16/15*(10*a*cosh(x)^4 + 10*(a*e^(4*x) + 2*a*e^(2*x) + a)*sinh(x)^4 + 40*(a*cosh(x)*e^(4*x) + 2*a*cosh(x)*e^(2
*x) + a*cosh(x))*sinh(x)^3 + 5*a*cosh(x)^2 + 5*(12*a*cosh(x)^2 + (12*a*cosh(x)^2 + a)*e^(4*x) + 2*(12*a*cosh(x
)^2 + a)*e^(2*x) + a)*sinh(x)^2 + (10*a*cosh(x)^4 + 5*a*cosh(x)^2 + a)*e^(4*x) + 2*(10*a*cosh(x)^4 + 5*a*cosh(
x)^2 + a)*e^(2*x) + 10*(4*a*cosh(x)^3 + a*cosh(x) + (4*a*cosh(x)^3 + a*cosh(x))*e^(4*x) + 2*(4*a*cosh(x)^3 + a
*cosh(x))*e^(2*x))*sinh(x) + a)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)/(10*cosh(x)*
e^(2*x)*sinh(x)^9 + e^(2*x)*sinh(x)^10 + 5*(9*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^8 + 40*(3*cosh(x)^3 + cosh(x))*e^
(2*x)*sinh(x)^7 + 10*(21*cosh(x)^4 + 14*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^6 + 4*(63*cosh(x)^5 + 70*cosh(x)^3 + 15
*cosh(x))*e^(2*x)*sinh(x)^5 + 10*(21*cosh(x)^6 + 35*cosh(x)^4 + 15*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^4 + 40*(3*co
sh(x)^7 + 7*cosh(x)^5 + 5*cosh(x)^3 + cosh(x))*e^(2*x)*sinh(x)^3 + 5*(9*cosh(x)^8 + 28*cosh(x)^6 + 30*cosh(x)^
4 + 12*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^2 + 10*(cosh(x)^9 + 4*cosh(x)^7 + 6*cosh(x)^5 + 4*cosh(x)^3 + cosh(x))*e
^(2*x)*sinh(x) + (cosh(x)^10 + 5*cosh(x)^8 + 10*cosh(x)^6 + 10*cosh(x)^4 + 5*cosh(x)^2 + 1)*e^(2*x))

Sympy [F]

\[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=\int \left (a \operatorname {sech}^{4}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((a*sech(x)**4)**(3/2),x)

[Out]

Integral((a*sech(x)**4)**(3/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (51) = 102\).

Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.97 \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=\frac {16 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} + \frac {32 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} + \frac {16 \, a^{\frac {3}{2}}}{15 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} \]

[In]

integrate((a*sech(x)^4)^(3/2),x, algorithm="maxima")

[Out]

16/3*a^(3/2)*e^(-2*x)/(5*e^(-2*x) + 10*e^(-4*x) + 10*e^(-6*x) + 5*e^(-8*x) + e^(-10*x) + 1) + 32/3*a^(3/2)*e^(
-4*x)/(5*e^(-2*x) + 10*e^(-4*x) + 10*e^(-6*x) + 5*e^(-8*x) + e^(-10*x) + 1) + 16/15*a^(3/2)/(5*e^(-2*x) + 10*e
^(-4*x) + 10*e^(-6*x) + 5*e^(-8*x) + e^(-10*x) + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.44 \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=-\frac {16 \, a^{\frac {3}{2}} {\left (10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} + 1\right )}}{15 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{5}} \]

[In]

integrate((a*sech(x)^4)^(3/2),x, algorithm="giac")

[Out]

-16/15*a^(3/2)*(10*e^(4*x) + 5*e^(2*x) + 1)/(e^(2*x) + 1)^5

Mupad [B] (verification not implemented)

Time = 1.96 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=-\frac {4\,a\,{\mathrm {e}}^{-2\,x}\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+1\right )}{15\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3} \]

[In]

int((a/cosh(x)^4)^(3/2),x)

[Out]

-(4*a*exp(-2*x)*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(5*exp(2*x) + 10*exp(4*x) + 1))/(15*(exp(2*x) + 1)^3)