∫ (asech4(x))7∕2 dx [45]

Optimal. Leaf size=163 \[ a^3 \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-2 a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh (x)+3 a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^3(x)-\frac {20}{7} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^5(x)+\frac {5}{3} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^7(x)-\frac {6}{11} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^9(x)+\frac {1}{13} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^{11}(x) \]

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

^4)^(1/2)*tanh(x)^3-20/7*a^3*sinh(x)^2*(a*sech(x)^4)^(1/2)*tanh(x)^5+5/3*a^3*sinh(x)^2*(a*sech(x)^4)^(1/2)*tan

h(x)^7-6/11*a^3*sinh(x)^2*(a*sech(x)^4)^(1/2)*tanh(x)^9+1/13*a^3*sinh(x)^2*(a*sech(x)^4)^(1/2)*tanh(x)^11

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Rubi [A]
time = 0.03, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4208, 3852} \begin {gather*} a^3 \sinh (x) \cosh (x) \sqrt {a \text {sech}^4(x)}+\frac {1}{13} a^3 \sinh ^2(x) \tanh ^{11}(x) \sqrt {a \text {sech}^4(x)}-\frac {6}{11} a^3 \sinh ^2(x) \tanh ^9(x) \sqrt {a \text {sech}^4(x)}+\frac {5}{3} a^3 \sinh ^2(x) \tanh ^7(x) \sqrt {a \text {sech}^4(x)}-\frac {20}{7} a^3 \sinh ^2(x) \tanh ^5(x) \sqrt {a \text {sech}^4(x)}+3 a^3 \sinh ^2(x) \tanh ^3(x) \sqrt {a \text {sech}^4(x)}-2 a^3 \sinh ^2(x) \tanh (x) \sqrt {a \text {sech}^4(x)} \end {gather*}

a^3*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x] - 2*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x] + 3*a^3*Sqrt[a*Sech[x]^4]*Si

nh[x]^2*Tanh[x]^3 - (20*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^5)/7 + (5*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh

[x]^7)/3 - (6*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^9)/11 + (a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^11)/13

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]

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},

\begin {align*} \int \left (a \text {sech}^4(x)\right )^{7/2} \, dx &=\left (a^3 \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int \text {sech}^{14}(x) \, dx\\ &=\left (i a^3 \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \text {Subst}\left (\int \left (1+6 x^2+15 x^4+20 x^6+15 x^8+6 x^{10}+x^{12}\right ) \, dx,x,-i \tanh (x)\right )\\ &=a^3 \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-2 a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh (x)+3 a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^3(x)-\frac {20}{7} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^5(x)+\frac {5}{3} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^7(x)-\frac {6}{11} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^9(x)+\frac {1}{13} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^{11}(x)\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 54, normalized size = 0.33 \begin {gather*} \frac {\cosh (x) (2048+2380 \cosh (2 x)+1093 \cosh (4 x)+378 \cosh (6 x)+92 \cosh (8 x)+14 \cosh (10 x)+\cosh (12 x)) \left (a \text {sech}^4(x)\right )^{7/2} \sinh (x)}{6006} \end {gather*}

(Cosh[x]*(2048 + 2380*Cosh[2*x] + 1093*Cosh[4*x] + 378*Cosh[6*x] + 92*Cosh[8*x] + 14*Cosh[10*x] + Cosh[12*x])*

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method	result	size

risch	\(-\frac {2048 a^{3} {\mathrm e}^{-2 x} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1716 \,{\mathrm e}^{12 x}+1287 \,{\mathrm e}^{10 x}+715 \,{\mathrm e}^{8 x}+286 \,{\mathrm e}^{6 x}+78 \,{\mathrm e}^{4 x}+13 \,{\mathrm e}^{2 x}+1\right )}{3003 \left (1+{\mathrm e}^{2 x}\right )^{11}}\)	\(72\)

-2048/3003*a^3*exp(-2*x)/(1+exp(2*x))^11*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*(1716*exp(12*x)+1287*exp(10*x)+715*

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (141) = 282\).
time = 0.54, size = 620, normalized size = 3.80 \begin {gather*} \frac {2048 \, a^{\frac {7}{2}} e^{\left (-2 \, x\right )}}{231 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {4096 \, a^{\frac {7}{2}} e^{\left (-4 \, x\right )}}{77 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {4096 \, a^{\frac {7}{2}} e^{\left (-6 \, x\right )}}{21 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {10240 \, a^{\frac {7}{2}} e^{\left (-8 \, x\right )}}{21 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {6144 \, a^{\frac {7}{2}} e^{\left (-10 \, x\right )}}{7 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {8192 \, a^{\frac {7}{2}} e^{\left (-12 \, x\right )}}{7 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {2048 \, a^{\frac {7}{2}}}{3003 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} \end {gather*}

2048/231*a^(7/2)*e^(-2*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(

-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-2

6*x) + 1) + 4096/77*a^(7/2)*e^(-4*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x)

+ 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24

*x) + e^(-26*x) + 1) + 4096/21*a^(7/2)*e^(-6*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 128

7*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^(-20*x) + 78*e^(-22*x)

+ 13*e^(-24*x) + e^(-26*x) + 1) + 10240/21*a^(7/2)*e^(-8*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^

(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^(-20*x) + 7

8*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) + 6144/7*a^(7/2)*e^(-10*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*

x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^

(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) + 8192/7*a^(7/2)*e^(-12*x)/(13*e^(-2*x) + 78*e^(-4*x) +

286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*

x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) + 2048/3003*a^(7/2)/(13*e^(-2*x) + 78*e^(-4*

x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2804 vs. \(2 (141) = 282\).
time = 0.44, size = 2804, normalized size = 17.20 \begin {gather*} \text {Too large to display} \end {gather*}

-2048/3003*(1716*a^3*cosh(x)^12 + 1287*a^3*cosh(x)^10 + 1716*(a^3*e^(4*x) + 2*a^3*e^(2*x) + a^3)*sinh(x)^12 +

20592*(a^3*cosh(x)*e^(4*x) + 2*a^3*cosh(x)*e^(2*x) + a^3*cosh(x))*sinh(x)^11 + 715*a^3*cosh(x)^8 + 1287*(88*a^

3*cosh(x)^2 + a^3 + (88*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(88*a^3*cosh(x)^2 + a^3)*e^(2*x))*sinh(x)^10 + 4290*(

88*a^3*cosh(x)^3 + 3*a^3*cosh(x) + (88*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(4*x) + 2*(88*a^3*cosh(x)^3 + 3*a^3*co

sh(x))*e^(2*x))*sinh(x)^9 + 286*a^3*cosh(x)^6 + 715*(1188*a^3*cosh(x)^4 + 81*a^3*cosh(x)^2 + a^3 + (1188*a^3*c

osh(x)^4 + 81*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(1188*a^3*cosh(x)^4 + 81*a^3*cosh(x)^2 + a^3)*e^(2*x))*sinh(x)^

8 + 1144*(1188*a^3*cosh(x)^5 + 135*a^3*cosh(x)^3 + 5*a^3*cosh(x) + (1188*a^3*cosh(x)^5 + 135*a^3*cosh(x)^3 + 5

*a^3*cosh(x))*e^(4*x) + 2*(1188*a^3*cosh(x)^5 + 135*a^3*cosh(x)^3 + 5*a^3*cosh(x))*e^(2*x))*sinh(x)^7 + 78*a^3

*cosh(x)^4 + 286*(5544*a^3*cosh(x)^6 + 945*a^3*cosh(x)^4 + 70*a^3*cosh(x)^2 + a^3 + (5544*a^3*cosh(x)^6 + 945*

a^3*cosh(x)^4 + 70*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(5544*a^3*cosh(x)^6 + 945*a^3*cosh(x)^4 + 70*a^3*cosh(x)^2

+ a^3)*e^(2*x))*sinh(x)^6 + 572*(2376*a^3*cosh(x)^7 + 567*a^3*cosh(x)^5 + 70*a^3*cosh(x)^3 + 3*a^3*cosh(x) +

(2376*a^3*cosh(x)^7 + 567*a^3*cosh(x)^5 + 70*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(4*x) + 2*(2376*a^3*cosh(x)^7 +

567*a^3*cosh(x)^5 + 70*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(2*x))*sinh(x)^5 + 13*a^3*cosh(x)^2 + 26*(32670*a^3*co

sh(x)^8 + 10395*a^3*cosh(x)^6 + 1925*a^3*cosh(x)^4 + 165*a^3*cosh(x)^2 + 3*a^3 + (32670*a^3*cosh(x)^8 + 10395*

a^3*cosh(x)^6 + 1925*a^3*cosh(x)^4 + 165*a^3*cosh(x)^2 + 3*a^3)*e^(4*x) + 2*(32670*a^3*cosh(x)^8 + 10395*a^3*c

osh(x)^6 + 1925*a^3*cosh(x)^4 + 165*a^3*cosh(x)^2 + 3*a^3)*e^(2*x))*sinh(x)^4 + 104*(3630*a^3*cosh(x)^9 + 1485

*a^3*cosh(x)^7 + 385*a^3*cosh(x)^5 + 55*a^3*cosh(x)^3 + 3*a^3*cosh(x) + (3630*a^3*cosh(x)^9 + 1485*a^3*cosh(x)

^7 + 385*a^3*cosh(x)^5 + 55*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(4*x) + 2*(3630*a^3*cosh(x)^9 + 1485*a^3*cosh(x)^

7 + 385*a^3*cosh(x)^5 + 55*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(2*x))*sinh(x)^3 + a^3 + 13*(8712*a^3*cosh(x)^10 +

4455*a^3*cosh(x)^8 + 1540*a^3*cosh(x)^6 + 330*a^3*cosh(x)^4 + 36*a^3*cosh(x)^2 + a^3 + (8712*a^3*cosh(x)^10 +

4455*a^3*cosh(x)^8 + 1540*a^3*cosh(x)^6 + 330*a^3*cosh(x)^4 + 36*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(8712*a^3*c

osh(x)^10 + 4455*a^3*cosh(x)^8 + 1540*a^3*cosh(x)^6 + 330*a^3*cosh(x)^4 + 36*a^3*cosh(x)^2 + a^3)*e^(2*x))*sin

h(x)^2 + (1716*a^3*cosh(x)^12 + 1287*a^3*cosh(x)^10 + 715*a^3*cosh(x)^8 + 286*a^3*cosh(x)^6 + 78*a^3*cosh(x)^4

+ 13*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(1716*a^3*cosh(x)^12 + 1287*a^3*cosh(x)^10 + 715*a^3*cosh(x)^8 + 286*a^

3*cosh(x)^6 + 78*a^3*cosh(x)^4 + 13*a^3*cosh(x)^2 + a^3)*e^(2*x) + 26*(792*a^3*cosh(x)^11 + 495*a^3*cosh(x)^9

+ 220*a^3*cosh(x)^7 + 66*a^3*cosh(x)^5 + 12*a^3*cosh(x)^3 + a^3*cosh(x) + (792*a^3*cosh(x)^11 + 495*a^3*cosh(x

)^9 + 220*a^3*cosh(x)^7 + 66*a^3*cosh(x)^5 + 12*a^3*cosh(x)^3 + a^3*cosh(x))*e^(4*x) + 2*(792*a^3*cosh(x)^11 +

495*a^3*cosh(x)^9 + 220*a^3*cosh(x)^7 + 66*a^3*cosh(x)^5 + 12*a^3*cosh(x)^3 + a^3*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)/(26*cosh(x)*e^(2*x)*sinh(x)^25 + e^(2*x)*sin

h(x)^26 + 13*(25*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^24 + 104*(25*cosh(x)^3 + 3*cosh(x))*e^(2*x)*sinh(x)^23 + 26*(5

75*cosh(x)^4 + 138*cosh(x)^2 + 3)*e^(2*x)*sinh(x)^22 + 572*(115*cosh(x)^5 + 46*cosh(x)^3 + 3*cosh(x))*e^(2*x)*

sinh(x)^21 + 286*(805*cosh(x)^6 + 483*cosh(x)^4 + 63*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^20 + 1144*(575*cosh(x)^7 +

483*cosh(x)^5 + 105*cosh(x)^3 + 5*cosh(x))*e^(2*x)*sinh(x)^19 + 143*(10925*cosh(x)^8 + 12236*cosh(x)^6 + 3990

*cosh(x)^4 + 380*cosh(x)^2 + 5)*e^(2*x)*sinh(x)^18 + 286*(10925*cosh(x)^9 + 15732*cosh(x)^7 + 7182*cosh(x)^5 +

1140*cosh(x)^3 + 45*cosh(x))*e^(2*x)*sinh(x)^17 + 143*(37145*cosh(x)^10 + 66861*cosh(x)^8 + 40698*cosh(x)^6 +

9690*cosh(x)^4 + 765*cosh(x)^2 + 9)*e^(2*x)*sinh(x)^16 + 208*(37145*cosh(x)^11 + 81719*cosh(x)^9 + 63954*cosh

(x)^7 + 21318*cosh(x)^5 + 2805*cosh(x)^3 + 99*cosh(x))*e^(2*x)*sinh(x)^15 + 52*(185725*cosh(x)^12 + 490314*cos

h(x)^10 + 479655*cosh(x)^8 + 213180*cosh(x)^6 + 42075*cosh(x)^4 + 2970*cosh(x)^2 + 33)*e^(2*x)*sinh(x)^14 + 8*

(1300075*cosh(x)^13 + 4056234*cosh(x)^11 + 4849845*cosh(x)^9 + 2771340*cosh(x)^7 + 765765*cosh(x)^5 + 90090*co

sh(x)^3 + 3003*cosh(x))*e^(2*x)*sinh(x)^13 + 52*(185725*cosh(x)^14 + 676039*cosh(x)^12 + 969969*cosh(x)^10 + 6

92835*cosh(x)^8 + 255255*cosh(x)^6 + 45045*cosh(x)^4 + 3003*cosh(x)^2 + 33)*e^(2*x)*sinh(x)^12 + 208*(37145*co

sh(x)^15 + 156009*cosh(x)^13 + 264537*cosh(x)^11 + 230945*cosh(x)^9 + 109395*cosh(x)^7 + 27027*cosh(x)^5 + 300

3*cosh(x)^3 + 99*cosh(x))*e^(2*x)*sinh(x)^11 + 143*(37145*cosh(x)^16 + 178296*cosh(x)^14 + 352716*cosh(x)^12 +

369512*cosh(x)^10 + 218790*cosh(x)^8 + 72072*cosh(x)^6 + 12012*cosh(x)^4 + 792*cosh(x)^2 + 9)*e^(2*x)*sinh(x)

^10 + 286*(10925*cosh(x)^17 + 59432*cosh(x)^15 + 135660*cosh(x)^13 + 167960*cosh(x)^11 + 121550*cosh(x)^9 + 51

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Exception raised: SystemError >> excessive stack use: stack is 3062 deep

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Giac [A]
time = 0.39, size = 51, normalized size = 0.31 \begin {gather*} -\frac {2048 \, a^{\frac {7}{2}} {\left (1716 \, e^{\left (12 \, x\right )} + 1287 \, e^{\left (10 \, x\right )} + 715 \, e^{\left (8 \, x\right )} + 286 \, e^{\left (6 \, x\right )} + 78 \, e^{\left (4 \, x\right )} + 13 \, e^{\left (2 \, x\right )} + 1\right )}}{3003 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{13}} \end {gather*}

-2048/3003*a^(7/2)*(1716*e^(12*x) + 1287*e^(10*x) + 715*e^(8*x) + 286*e^(6*x) + 78*e^(4*x) + 13*e^(2*x) + 1)/(

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Mupad [B]
time = 1.45, size = 498, normalized size = 3.06 \begin {gather*} \frac {1536\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^8\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {2048\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{7\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^7\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {10240\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^9\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}+\frac {4096\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^{10}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {30720\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{11\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^{11}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}+\frac {1024\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^{12}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {2048\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{13\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^{13}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \end {gather*}

(1536*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1))/((exp(2*x)

+ 1)^8*(exp(2*x) + 2*exp(4*x) + exp(6*x))) - (2048*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp

(4*x) + 4*exp(6*x) + exp(8*x) + 1))/(7*(exp(2*x) + 1)^7*(exp(2*x) + 2*exp(4*x) + exp(6*x))) - (10240*a^3*(a/(e

xp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1))/(3*(exp(2*x) + 1)^9*(exp(

2*x) + 2*exp(4*x) + exp(6*x))) + (4096*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp

(6*x) + exp(8*x) + 1))/((exp(2*x) + 1)^10*(exp(2*x) + 2*exp(4*x) + exp(6*x))) - (30720*a^3*(a/(exp(-x)/2 + exp

(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1))/(11*(exp(2*x) + 1)^11*(exp(2*x) + 2*exp

(4*x) + exp(6*x))) + (1024*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(

8*x) + 1))/((exp(2*x) + 1)^12*(exp(2*x) + 2*exp(4*x) + exp(6*x))) - (2048*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/

2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1))/(13*(exp(2*x) + 1)^13*(exp(2*x) + 2*exp(4*x) + exp(6

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3.1.45 \(\int (a \text {sech}^4(x))^{7/2} \, dx\) [45]