3.46 \(\int (a \text{sech}^4(x))^{5/2} \, dx\)

Optimal. Leaf size=117 \[ a^2 \sinh (x) \cosh (x) \sqrt{a \text{sech}^4(x)}+\frac{1}{9} a^2 \sinh ^2(x) \tanh ^7(x) \sqrt{a \text{sech}^4(x)}-\frac{4}{7} a^2 \sinh ^2(x) \tanh ^5(x) \sqrt{a \text{sech}^4(x)}+\frac{6}{5} a^2 \sinh ^2(x) \tanh ^3(x) \sqrt{a \text{sech}^4(x)}-\frac{4}{3} a^2 \sinh ^2(x) \tanh (x) \sqrt{a \text{sech}^4(x)} \]

[Out]

a^2*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x] - (4*a^2*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x])/3 + (6*a^2*Sqrt[a*Sech[x]^
4]*Sinh[x]^2*Tanh[x]^3)/5 - (4*a^2*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^5)/7 + (a^2*Sqrt[a*Sech[x]^4]*Sinh[x]^2
*Tanh[x]^7)/9

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Rubi [A]  time = 0.0351459, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4123, 3767} \[ a^2 \sinh (x) \cosh (x) \sqrt{a \text{sech}^4(x)}+\frac{1}{9} a^2 \sinh ^2(x) \tanh ^7(x) \sqrt{a \text{sech}^4(x)}-\frac{4}{7} a^2 \sinh ^2(x) \tanh ^5(x) \sqrt{a \text{sech}^4(x)}+\frac{6}{5} a^2 \sinh ^2(x) \tanh ^3(x) \sqrt{a \text{sech}^4(x)}-\frac{4}{3} a^2 \sinh ^2(x) \tanh (x) \sqrt{a \text{sech}^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sech[x]^4)^(5/2),x]

[Out]

a^2*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x] - (4*a^2*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x])/3 + (6*a^2*Sqrt[a*Sech[x]^
4]*Sinh[x]^2*Tanh[x]^3)/5 - (4*a^2*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^5)/7 + (a^2*Sqrt[a*Sech[x]^4]*Sinh[x]^2
*Tanh[x]^7)/9

Rule 4123

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]

Rule 3767

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]

Rubi steps

\begin{align*} \int \left (a \text{sech}^4(x)\right )^{5/2} \, dx &=\left (a^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \text{sech}^{10}(x) \, dx\\ &=\left (i a^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \operatorname{Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,-i \tanh (x)\right )\\ &=a^2 \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{4}{3} a^2 \sqrt{a \text{sech}^4(x)} \sinh ^2(x) \tanh (x)+\frac{6}{5} a^2 \sqrt{a \text{sech}^4(x)} \sinh ^2(x) \tanh ^3(x)-\frac{4}{7} a^2 \sqrt{a \text{sech}^4(x)} \sinh ^2(x) \tanh ^5(x)+\frac{1}{9} a^2 \sqrt{a \text{sech}^4(x)} \sinh ^2(x) \tanh ^7(x)\\ \end{align*}

Mathematica [A]  time = 0.0926697, size = 42, normalized size = 0.36 \[ \frac{1}{315} \sinh (x) \cosh (x) (130 \cosh (2 x)+46 \cosh (4 x)+10 \cosh (6 x)+\cosh (8 x)+128) \left (a \text{sech}^4(x)\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sech[x]^4)^(5/2),x]

[Out]

(Cosh[x]*(128 + 130*Cosh[2*x] + 46*Cosh[4*x] + 10*Cosh[6*x] + Cosh[8*x])*(a*Sech[x]^4)^(5/2)*Sinh[x])/315

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Maple [A]  time = 0.07, size = 60, normalized size = 0.5 \begin{align*} -{\frac{256\,{a}^{2}{{\rm e}^{-2\,x}} \left ( 126\,{{\rm e}^{8\,x}}+84\,{{\rm e}^{6\,x}}+36\,{{\rm e}^{4\,x}}+9\,{{\rm e}^{2\,x}}+1 \right ) }{315\, \left ({{\rm e}^{2\,x}}+1 \right ) ^{7}}\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sech(x)^4)^(5/2),x)

[Out]

-256/315*a^2*exp(-2*x)/(exp(2*x)+1)^7*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*(126*exp(8*x)+84*exp(6*x)+36*exp(4*x)+
9*exp(2*x)+1)

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Maxima [B]  time = 1.76867, size = 435, normalized size = 3.72 \begin{align*} \frac{256 \, a^{\frac{5}{2}} e^{\left (-2 \, x\right )}}{35 \,{\left (9 \, e^{\left (-2 \, x\right )} + 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} + 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} + 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} + 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} + 1\right )}} + \frac{1024 \, a^{\frac{5}{2}} e^{\left (-4 \, x\right )}}{35 \,{\left (9 \, e^{\left (-2 \, x\right )} + 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} + 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} + 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} + 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} + 1\right )}} + \frac{1024 \, a^{\frac{5}{2}} e^{\left (-6 \, x\right )}}{15 \,{\left (9 \, e^{\left (-2 \, x\right )} + 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} + 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} + 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} + 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} + 1\right )}} + \frac{512 \, a^{\frac{5}{2}} e^{\left (-8 \, x\right )}}{5 \,{\left (9 \, e^{\left (-2 \, x\right )} + 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} + 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} + 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} + 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} + 1\right )}} + \frac{256 \, a^{\frac{5}{2}}}{315 \,{\left (9 \, e^{\left (-2 \, x\right )} + 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} + 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} + 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} + 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^4)^(5/2),x, algorithm="maxima")

[Out]

256/35*a^(5/2)*e^(-2*x)/(9*e^(-2*x) + 36*e^(-4*x) + 84*e^(-6*x) + 126*e^(-8*x) + 126*e^(-10*x) + 84*e^(-12*x)
+ 36*e^(-14*x) + 9*e^(-16*x) + e^(-18*x) + 1) + 1024/35*a^(5/2)*e^(-4*x)/(9*e^(-2*x) + 36*e^(-4*x) + 84*e^(-6*
x) + 126*e^(-8*x) + 126*e^(-10*x) + 84*e^(-12*x) + 36*e^(-14*x) + 9*e^(-16*x) + e^(-18*x) + 1) + 1024/15*a^(5/
2)*e^(-6*x)/(9*e^(-2*x) + 36*e^(-4*x) + 84*e^(-6*x) + 126*e^(-8*x) + 126*e^(-10*x) + 84*e^(-12*x) + 36*e^(-14*
x) + 9*e^(-16*x) + e^(-18*x) + 1) + 512/5*a^(5/2)*e^(-8*x)/(9*e^(-2*x) + 36*e^(-4*x) + 84*e^(-6*x) + 126*e^(-8
*x) + 126*e^(-10*x) + 84*e^(-12*x) + 36*e^(-14*x) + 9*e^(-16*x) + e^(-18*x) + 1) + 256/315*a^(5/2)/(9*e^(-2*x)
+ 36*e^(-4*x) + 84*e^(-6*x) + 126*e^(-8*x) + 126*e^(-10*x) + 84*e^(-12*x) + 36*e^(-14*x) + 9*e^(-16*x) + e^(-
18*x) + 1)

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Fricas [B]  time = 2.43031, size = 4632, normalized size = 39.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^4)^(5/2),x, algorithm="fricas")

[Out]

-256/315*(126*a^2*cosh(x)^8 + 126*(a^2*e^(4*x) + 2*a^2*e^(2*x) + a^2)*sinh(x)^8 + 84*a^2*cosh(x)^6 + 1008*(a^2
*cosh(x)*e^(4*x) + 2*a^2*cosh(x)*e^(2*x) + a^2*cosh(x))*sinh(x)^7 + 84*(42*a^2*cosh(x)^2 + a^2 + (42*a^2*cosh(
x)^2 + a^2)*e^(4*x) + 2*(42*a^2*cosh(x)^2 + a^2)*e^(2*x))*sinh(x)^6 + 36*a^2*cosh(x)^4 + 504*(14*a^2*cosh(x)^3
+ a^2*cosh(x) + (14*a^2*cosh(x)^3 + a^2*cosh(x))*e^(4*x) + 2*(14*a^2*cosh(x)^3 + a^2*cosh(x))*e^(2*x))*sinh(x
)^5 + 36*(245*a^2*cosh(x)^4 + 35*a^2*cosh(x)^2 + a^2 + (245*a^2*cosh(x)^4 + 35*a^2*cosh(x)^2 + a^2)*e^(4*x) +
2*(245*a^2*cosh(x)^4 + 35*a^2*cosh(x)^2 + a^2)*e^(2*x))*sinh(x)^4 + 9*a^2*cosh(x)^2 + 48*(147*a^2*cosh(x)^5 +
35*a^2*cosh(x)^3 + 3*a^2*cosh(x) + (147*a^2*cosh(x)^5 + 35*a^2*cosh(x)^3 + 3*a^2*cosh(x))*e^(4*x) + 2*(147*a^2
*cosh(x)^5 + 35*a^2*cosh(x)^3 + 3*a^2*cosh(x))*e^(2*x))*sinh(x)^3 + 9*(392*a^2*cosh(x)^6 + 140*a^2*cosh(x)^4 +
24*a^2*cosh(x)^2 + a^2 + (392*a^2*cosh(x)^6 + 140*a^2*cosh(x)^4 + 24*a^2*cosh(x)^2 + a^2)*e^(4*x) + 2*(392*a^
2*cosh(x)^6 + 140*a^2*cosh(x)^4 + 24*a^2*cosh(x)^2 + a^2)*e^(2*x))*sinh(x)^2 + a^2 + (126*a^2*cosh(x)^8 + 84*a
^2*cosh(x)^6 + 36*a^2*cosh(x)^4 + 9*a^2*cosh(x)^2 + a^2)*e^(4*x) + 2*(126*a^2*cosh(x)^8 + 84*a^2*cosh(x)^6 + 3
6*a^2*cosh(x)^4 + 9*a^2*cosh(x)^2 + a^2)*e^(2*x) + 18*(56*a^2*cosh(x)^7 + 28*a^2*cosh(x)^5 + 8*a^2*cosh(x)^3 +
a^2*cosh(x) + (56*a^2*cosh(x)^7 + 28*a^2*cosh(x)^5 + 8*a^2*cosh(x)^3 + a^2*cosh(x))*e^(4*x) + 2*(56*a^2*cosh(
x)^7 + 28*a^2*cosh(x)^5 + 8*a^2*cosh(x)^3 + a^2*cosh(x))*e^(2*x))*sinh(x))*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(
4*x) + 4*e^(2*x) + 1))*e^(2*x)/(18*cosh(x)*e^(2*x)*sinh(x)^17 + e^(2*x)*sinh(x)^18 + 9*(17*cosh(x)^2 + 1)*e^(2
*x)*sinh(x)^16 + 48*(17*cosh(x)^3 + 3*cosh(x))*e^(2*x)*sinh(x)^15 + 36*(85*cosh(x)^4 + 30*cosh(x)^2 + 1)*e^(2*
x)*sinh(x)^14 + 504*(17*cosh(x)^5 + 10*cosh(x)^3 + cosh(x))*e^(2*x)*sinh(x)^13 + 84*(221*cosh(x)^6 + 195*cosh(
x)^4 + 39*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^12 + 144*(221*cosh(x)^7 + 273*cosh(x)^5 + 91*cosh(x)^3 + 7*cosh(x))*e
^(2*x)*sinh(x)^11 + 18*(2431*cosh(x)^8 + 4004*cosh(x)^6 + 2002*cosh(x)^4 + 308*cosh(x)^2 + 7)*e^(2*x)*sinh(x)^
10 + 4*(12155*cosh(x)^9 + 25740*cosh(x)^7 + 18018*cosh(x)^5 + 4620*cosh(x)^3 + 315*cosh(x))*e^(2*x)*sinh(x)^9
+ 18*(2431*cosh(x)^10 + 6435*cosh(x)^8 + 6006*cosh(x)^6 + 2310*cosh(x)^4 + 315*cosh(x)^2 + 7)*e^(2*x)*sinh(x)^
8 + 144*(221*cosh(x)^11 + 715*cosh(x)^9 + 858*cosh(x)^7 + 462*cosh(x)^5 + 105*cosh(x)^3 + 7*cosh(x))*e^(2*x)*s
inh(x)^7 + 84*(221*cosh(x)^12 + 858*cosh(x)^10 + 1287*cosh(x)^8 + 924*cosh(x)^6 + 315*cosh(x)^4 + 42*cosh(x)^2
+ 1)*e^(2*x)*sinh(x)^6 + 504*(17*cosh(x)^13 + 78*cosh(x)^11 + 143*cosh(x)^9 + 132*cosh(x)^7 + 63*cosh(x)^5 +
14*cosh(x)^3 + cosh(x))*e^(2*x)*sinh(x)^5 + 36*(85*cosh(x)^14 + 455*cosh(x)^12 + 1001*cosh(x)^10 + 1155*cosh(x
)^8 + 735*cosh(x)^6 + 245*cosh(x)^4 + 35*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^4 + 48*(17*cosh(x)^15 + 105*cosh(x)^13
+ 273*cosh(x)^11 + 385*cosh(x)^9 + 315*cosh(x)^7 + 147*cosh(x)^5 + 35*cosh(x)^3 + 3*cosh(x))*e^(2*x)*sinh(x)^
3 + 9*(17*cosh(x)^16 + 120*cosh(x)^14 + 364*cosh(x)^12 + 616*cosh(x)^10 + 630*cosh(x)^8 + 392*cosh(x)^6 + 140*
cosh(x)^4 + 24*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^2 + 18*(cosh(x)^17 + 8*cosh(x)^15 + 28*cosh(x)^13 + 56*cosh(x)^1
1 + 70*cosh(x)^9 + 56*cosh(x)^7 + 28*cosh(x)^5 + 8*cosh(x)^3 + cosh(x))*e^(2*x)*sinh(x) + (cosh(x)^18 + 9*cosh
(x)^16 + 36*cosh(x)^14 + 84*cosh(x)^12 + 126*cosh(x)^10 + 126*cosh(x)^8 + 84*cosh(x)^6 + 36*cosh(x)^4 + 9*cosh
(x)^2 + 1)*e^(2*x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)**4)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.1718, size = 53, normalized size = 0.45 \begin{align*} -\frac{256 \, a^{\frac{5}{2}}{\left (126 \, e^{\left (8 \, x\right )} + 84 \, e^{\left (6 \, x\right )} + 36 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} + 1\right )}}{315 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^4)^(5/2),x, algorithm="giac")

[Out]

-256/315*a^(5/2)*(126*e^(8*x) + 84*e^(6*x) + 36*e^(4*x) + 9*e^(2*x) + 1)/(e^(2*x) + 1)^9