3.1.37 \(\int \frac {1}{(a \text {sech}^2(x))^{5/2}} \, dx\) [37]

Optimal. Leaf size=55 \[ \frac {\tanh (x)}{5 \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {4 \tanh (x)}{15 a \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {8 \tanh (x)}{15 a^2 \sqrt {a \text {sech}^2(x)}} \]

[Out]

1/5*tanh(x)/(a*sech(x)^2)^(5/2)+4/15*tanh(x)/a/(a*sech(x)^2)^(3/2)+8/15*tanh(x)/a^2/(a*sech(x)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4207, 198, 197} \begin {gather*} \frac {8 \tanh (x)}{15 a^2 \sqrt {a \text {sech}^2(x)}}+\frac {4 \tanh (x)}{15 a \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {\tanh (x)}{5 \left (a \text {sech}^2(x)\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*Sech[x]^2)^(-5/2),x]

[Out]

Tanh[x]/(5*(a*Sech[x]^2)^(5/2)) + (4*Tanh[x])/(15*a*(a*Sech[x]^2)^(3/2)) + (8*Tanh[x])/(15*a^2*Sqrt[a*Sech[x]^
2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 4207

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[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
]

Rubi steps

\begin {align*} \int \frac {1}{\left (a \text {sech}^2(x)\right )^{5/2}} \, dx &=a \text {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{7/2}} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh (x)}{5 \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {4}{5} \text {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh (x)}{5 \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {4 \tanh (x)}{15 a \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {8 \text {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )}{15 a}\\ &=\frac {\tanh (x)}{5 \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {4 \tanh (x)}{15 a \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {8 \tanh (x)}{15 a^2 \sqrt {a \text {sech}^2(x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 36, normalized size = 0.65 \begin {gather*} \frac {\cosh (x) \sqrt {a \text {sech}^2(x)} (150 \sinh (x)+25 \sinh (3 x)+3 \sinh (5 x))}{240 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*Sech[x]^2)^(-5/2),x]

[Out]

(Cosh[x]*Sqrt[a*Sech[x]^2]*(150*Sinh[x] + 25*Sinh[3*x] + 3*Sinh[5*x]))/(240*a^3)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(195\) vs. \(2(43)=86\).
time = 0.85, size = 196, normalized size = 3.56

method	result	size
risch	\(\frac {{\mathrm e}^{6 x}}{160 a^{2} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}+\frac {5 \,{\mathrm e}^{4 x}}{96 a^{2} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}+\frac {5 \,{\mathrm e}^{2 x}}{16 a^{2} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}-\frac {5}{16 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right ) a^{2}}-\frac {5 \,{\mathrm e}^{-2 x}}{96 a^{2} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}-\frac {{\mathrm e}^{-4 x}}{160 a^{2} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}\)	\(196\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sech(x)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/160/a^2*exp(6*x)/(1+exp(2*x))/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)+5/96/a^2*exp(4*x)/(1+exp(2*x))/(a*exp(2*x)/(
1+exp(2*x))^2)^(1/2)+5/16/a^2*exp(2*x)/(1+exp(2*x))/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)-5/16/(a*exp(2*x)/(1+exp(
2*x))^2)^(1/2)/(1+exp(2*x))/a^2-5/96/a^2*exp(-2*x)/(1+exp(2*x))/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)-1/160/a^2*ex
p(-4*x)/(1+exp(2*x))/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.47, size = 53, normalized size = 0.96 \begin {gather*} \frac {e^{\left (5 \, x\right )}}{160 \, a^{\frac {5}{2}}} + \frac {5 \, e^{\left (3 \, x\right )}}{96 \, a^{\frac {5}{2}}} - \frac {5 \, e^{\left (-x\right )}}{16 \, a^{\frac {5}{2}}} - \frac {5 \, e^{\left (-3 \, x\right )}}{96 \, a^{\frac {5}{2}}} - \frac {e^{\left (-5 \, x\right )}}{160 \, a^{\frac {5}{2}}} + \frac {5 \, e^{x}}{16 \, a^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)^2)^(5/2),x, algorithm="maxima")

[Out]

1/160*e^(5*x)/a^(5/2) + 5/96*e^(3*x)/a^(5/2) - 5/16*e^(-x)/a^(5/2) - 5/96*e^(-3*x)/a^(5/2) - 1/160*e^(-5*x)/a^
(5/2) + 5/16*e^x/a^(5/2)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (43) = 86\).
time = 0.39, size = 580, normalized size = 10.55 \begin {gather*} \frac {{\left (3 \, {\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{10} + 3 \, \cosh \left (x\right )^{10} + 30 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{9} + 5 \, {\left (27 \, \cosh \left (x\right )^{2} + {\left (27 \, \cosh \left (x\right )^{2} + 5\right )} e^{\left (2 \, x\right )} + 5\right )} \sinh \left (x\right )^{8} + 25 \, \cosh \left (x\right )^{8} + 40 \, {\left (9 \, \cosh \left (x\right )^{3} + {\left (9 \, \cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} + 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{7} + 10 \, {\left (63 \, \cosh \left (x\right )^{4} + 70 \, \cosh \left (x\right )^{2} + {\left (63 \, \cosh \left (x\right )^{4} + 70 \, \cosh \left (x\right )^{2} + 15\right )} e^{\left (2 \, x\right )} + 15\right )} \sinh \left (x\right )^{6} + 150 \, \cosh \left (x\right )^{6} + 4 \, {\left (189 \, \cosh \left (x\right )^{5} + 350 \, \cosh \left (x\right )^{3} + {\left (189 \, \cosh \left (x\right )^{5} + 350 \, \cosh \left (x\right )^{3} + 225 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} + 225 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 10 \, {\left (63 \, \cosh \left (x\right )^{6} + 175 \, \cosh \left (x\right )^{4} + 225 \, \cosh \left (x\right )^{2} + {\left (63 \, \cosh \left (x\right )^{6} + 175 \, \cosh \left (x\right )^{4} + 225 \, \cosh \left (x\right )^{2} - 15\right )} e^{\left (2 \, x\right )} - 15\right )} \sinh \left (x\right )^{4} - 150 \, \cosh \left (x\right )^{4} + 40 \, {\left (9 \, \cosh \left (x\right )^{7} + 35 \, \cosh \left (x\right )^{5} + 75 \, \cosh \left (x\right )^{3} + {\left (9 \, \cosh \left (x\right )^{7} + 35 \, \cosh \left (x\right )^{5} + 75 \, \cosh \left (x\right )^{3} - 15 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} - 15 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 5 \, {\left (27 \, \cosh \left (x\right )^{8} + 140 \, \cosh \left (x\right )^{6} + 450 \, \cosh \left (x\right )^{4} - 180 \, \cosh \left (x\right )^{2} + {\left (27 \, \cosh \left (x\right )^{8} + 140 \, \cosh \left (x\right )^{6} + 450 \, \cosh \left (x\right )^{4} - 180 \, \cosh \left (x\right )^{2} - 5\right )} e^{\left (2 \, x\right )} - 5\right )} \sinh \left (x\right )^{2} - 25 \, \cosh \left (x\right )^{2} + {\left (3 \, \cosh \left (x\right )^{10} + 25 \, \cosh \left (x\right )^{8} + 150 \, \cosh \left (x\right )^{6} - 150 \, \cosh \left (x\right )^{4} - 25 \, \cosh \left (x\right )^{2} - 3\right )} e^{\left (2 \, x\right )} + 10 \, {\left (3 \, \cosh \left (x\right )^{9} + 20 \, \cosh \left (x\right )^{7} + 90 \, \cosh \left (x\right )^{5} - 60 \, \cosh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{9} + 20 \, \cosh \left (x\right )^{7} + 90 \, \cosh \left (x\right )^{5} - 60 \, \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 3\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{480 \, {\left (a^{3} \cosh \left (x\right )^{5} e^{x} + 5 \, a^{3} \cosh \left (x\right )^{4} e^{x} \sinh \left (x\right ) + 10 \, a^{3} \cosh \left (x\right )^{3} e^{x} \sinh \left (x\right )^{2} + 10 \, a^{3} \cosh \left (x\right )^{2} e^{x} \sinh \left (x\right )^{3} + 5 \, a^{3} \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{4} + a^{3} e^{x} \sinh \left (x\right )^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)^2)^(5/2),x, algorithm="fricas")

[Out]

1/480*(3*(e^(2*x) + 1)*sinh(x)^10 + 3*cosh(x)^10 + 30*(cosh(x)*e^(2*x) + cosh(x))*sinh(x)^9 + 5*(27*cosh(x)^2
+ (27*cosh(x)^2 + 5)*e^(2*x) + 5)*sinh(x)^8 + 25*cosh(x)^8 + 40*(9*cosh(x)^3 + (9*cosh(x)^3 + 5*cosh(x))*e^(2*
x) + 5*cosh(x))*sinh(x)^7 + 10*(63*cosh(x)^4 + 70*cosh(x)^2 + (63*cosh(x)^4 + 70*cosh(x)^2 + 15)*e^(2*x) + 15)
*sinh(x)^6 + 150*cosh(x)^6 + 4*(189*cosh(x)^5 + 350*cosh(x)^3 + (189*cosh(x)^5 + 350*cosh(x)^3 + 225*cosh(x))*
e^(2*x) + 225*cosh(x))*sinh(x)^5 + 10*(63*cosh(x)^6 + 175*cosh(x)^4 + 225*cosh(x)^2 + (63*cosh(x)^6 + 175*cosh
(x)^4 + 225*cosh(x)^2 - 15)*e^(2*x) - 15)*sinh(x)^4 - 150*cosh(x)^4 + 40*(9*cosh(x)^7 + 35*cosh(x)^5 + 75*cosh
(x)^3 + (9*cosh(x)^7 + 35*cosh(x)^5 + 75*cosh(x)^3 - 15*cosh(x))*e^(2*x) - 15*cosh(x))*sinh(x)^3 + 5*(27*cosh(
x)^8 + 140*cosh(x)^6 + 450*cosh(x)^4 - 180*cosh(x)^2 + (27*cosh(x)^8 + 140*cosh(x)^6 + 450*cosh(x)^4 - 180*cos
h(x)^2 - 5)*e^(2*x) - 5)*sinh(x)^2 - 25*cosh(x)^2 + (3*cosh(x)^10 + 25*cosh(x)^8 + 150*cosh(x)^6 - 150*cosh(x)
^4 - 25*cosh(x)^2 - 3)*e^(2*x) + 10*(3*cosh(x)^9 + 20*cosh(x)^7 + 90*cosh(x)^5 - 60*cosh(x)^3 + (3*cosh(x)^9 +
20*cosh(x)^7 + 90*cosh(x)^5 - 60*cosh(x)^3 - 5*cosh(x))*e^(2*x) - 5*cosh(x))*sinh(x) - 3)*sqrt(a/(e^(4*x) + 2
*e^(2*x) + 1))*e^x/(a^3*cosh(x)^5*e^x + 5*a^3*cosh(x)^4*e^x*sinh(x) + 10*a^3*cosh(x)^3*e^x*sinh(x)^2 + 10*a^3*
cosh(x)^2*e^x*sinh(x)^3 + 5*a^3*cosh(x)*e^x*sinh(x)^4 + a^3*e^x*sinh(x)^5)

________________________________________________________________________________________

Sympy [A]
time = 1.61, size = 49, normalized size = 0.89 \begin {gather*} \frac {8 \tanh ^{5}{\left (x \right )}}{15 \left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {5}{2}}} - \frac {4 \tanh ^{3}{\left (x \right )}}{3 \left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {5}{2}}} + \frac {\tanh {\left (x \right )}}{\left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)**2)**(5/2),x)

[Out]

8*tanh(x)**5/(15*(a*sech(x)**2)**(5/2)) - 4*tanh(x)**3/(3*(a*sech(x)**2)**(5/2)) + tanh(x)/(a*sech(x)**2)**(5/
2)

________________________________________________________________________________________

Giac [A]
time = 0.39, size = 41, normalized size = 0.75 \begin {gather*} -\frac {{\left (150 \, e^{\left (4 \, x\right )} + 25 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} - 3 \, e^{\left (5 \, x\right )} - 25 \, e^{\left (3 \, x\right )} - 150 \, e^{x}}{480 \, a^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)^2)^(5/2),x, algorithm="giac")

[Out]

-1/480*((150*e^(4*x) + 25*e^(2*x) + 3)*e^(-5*x) - 3*e^(5*x) - 25*e^(3*x) - 150*e^x)/a^(5/2)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^2}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a/cosh(x)^2)^(5/2),x)

[Out]

int(1/(a/cosh(x)^2)^(5/2), x)

________________________________________________________________________________________