\(\int \frac {1}{(a \text {sech}^2(x))^{7/2}} \, dx\) [38]

Optimal result

Integrand size = 10, antiderivative size = 74 \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=\frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}}+\frac {6 \tanh (x)}{35 a \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {8 \tanh (x)}{35 a^2 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {16 \tanh (x)}{35 a^3 \sqrt {a \text {sech}^2(x)}} \]

[Out]

1/7*tanh(x)/(a*sech(x)^2)^(7/2)+6/35*tanh(x)/a/(a*sech(x)^2)^(5/2)+8/35*tanh(x)/a^2/(a*sech(x)^2)^(3/2)+16/35*
tanh(x)/a^3/(a*sech(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4207, 198, 197} \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=\frac {16 \tanh (x)}{35 a^3 \sqrt {a \text {sech}^2(x)}}+\frac {8 \tanh (x)}{35 a^2 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {6 \tanh (x)}{35 a \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}} \]

[In]

Int[(a*Sech[x]^2)^(-7/2),x]

[Out]

Tanh[x]/(7*(a*Sech[x]^2)^(7/2)) + (6*Tanh[x])/(35*a*(a*Sech[x]^2)^(5/2)) + (8*Tanh[x])/(35*a^2*(a*Sech[x]^2)^(
3/2)) + (16*Tanh[x])/(35*a^3*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*} \text {integral}& = a \text {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{9/2}} \, dx,x,\tanh (x)\right ) \\ & = \frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}}+\frac {6}{7} \text {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{7/2}} \, dx,x,\tanh (x)\right ) \\ & = \frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}}+\frac {6 \tanh (x)}{35 a \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {24 \text {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )}{35 a} \\ & = \frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}}+\frac {6 \tanh (x)}{35 a \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {8 \tanh (x)}{35 a^2 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {16 \text {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )}{35 a^2} \\ & = \frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}}+\frac {6 \tanh (x)}{35 a \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {8 \tanh (x)}{35 a^2 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {16 \tanh (x)}{35 a^3 \sqrt {a \text {sech}^2(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=\frac {\left (35+35 \sinh ^2(x)+21 \sinh ^4(x)+5 \sinh ^6(x)\right ) \tanh (x)}{35 a^3 \sqrt {a \text {sech}^2(x)}} \]

[In]

Integrate[(a*Sech[x]^2)^(-7/2),x]

[Out]

((35 + 35*Sinh[x]^2 + 21*Sinh[x]^4 + 5*Sinh[x]^6)*Tanh[x])/(35*a^3*Sqrt[a*Sech[x]^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(261\) vs. \(2(58)=116\).

Time = 0.14 (sec) , antiderivative size = 262, normalized size of antiderivative = 3.54

method	result	size
risch	\(\frac {{\mathrm e}^{8 x}}{896 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}+\frac {7 \,{\mathrm e}^{6 x}}{640 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}+\frac {7 \,{\mathrm e}^{4 x}}{128 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}+\frac {35 \,{\mathrm e}^{2 x}}{128 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}-\frac {35}{128 \sqrt {\frac {{\mathrm e}^{2 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right ) a^{3}}-\frac {7 \,{\mathrm e}^{-2 x}}{128 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}-\frac {7 \,{\mathrm e}^{-4 x}}{640 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}-\frac {{\mathrm e}^{-6 x}}{896 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}\)	\(262\)

[In]

int(1/(sech(x)^2*a)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/896/a^3*exp(8*x)/(1+exp(2*x))/(exp(2*x)*a/(1+exp(2*x))^2)^(1/2)+7/640/a^3*exp(6*x)/(1+exp(2*x))/(exp(2*x)*a/
(1+exp(2*x))^2)^(1/2)+7/128/a^3*exp(4*x)/(1+exp(2*x))/(exp(2*x)*a/(1+exp(2*x))^2)^(1/2)+35/128/a^3*exp(2*x)/(1
+exp(2*x))/(exp(2*x)*a/(1+exp(2*x))^2)^(1/2)-35/128/(exp(2*x)*a/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))/a^3-7/128/a
^3*exp(-2*x)/(1+exp(2*x))/(exp(2*x)*a/(1+exp(2*x))^2)^(1/2)-7/640/a^3*exp(-4*x)/(1+exp(2*x))/(exp(2*x)*a/(1+ex
p(2*x))^2)^(1/2)-1/896/a^3*exp(-6*x)/(1+exp(2*x))/(exp(2*x)*a/(1+exp(2*x))^2)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 970 vs. \(2 (58) = 116\).

Time = 0.28 (sec) , antiderivative size = 970, normalized size of antiderivative = 13.11 \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a*sech(x)^2)^(7/2),x, algorithm="fricas")

[Out]

1/4480*(5*(e^(2*x) + 1)*sinh(x)^14 + 5*cosh(x)^14 + 70*(cosh(x)*e^(2*x) + cosh(x))*sinh(x)^13 + 7*(65*cosh(x)^
2 + (65*cosh(x)^2 + 7)*e^(2*x) + 7)*sinh(x)^12 + 49*cosh(x)^12 + 28*(65*cosh(x)^3 + (65*cosh(x)^3 + 21*cosh(x)
)*e^(2*x) + 21*cosh(x))*sinh(x)^11 + 7*(715*cosh(x)^4 + 462*cosh(x)^2 + (715*cosh(x)^4 + 462*cosh(x)^2 + 35)*e
^(2*x) + 35)*sinh(x)^10 + 245*cosh(x)^10 + 70*(143*cosh(x)^5 + 154*cosh(x)^3 + (143*cosh(x)^5 + 154*cosh(x)^3
+ 35*cosh(x))*e^(2*x) + 35*cosh(x))*sinh(x)^9 + 35*(429*cosh(x)^6 + 693*cosh(x)^4 + 315*cosh(x)^2 + (429*cosh(
x)^6 + 693*cosh(x)^4 + 315*cosh(x)^2 + 35)*e^(2*x) + 35)*sinh(x)^8 + 1225*cosh(x)^8 + 8*(2145*cosh(x)^7 + 4851
*cosh(x)^5 + 3675*cosh(x)^3 + (2145*cosh(x)^7 + 4851*cosh(x)^5 + 3675*cosh(x)^3 + 1225*cosh(x))*e^(2*x) + 1225
*cosh(x))*sinh(x)^7 + 7*(2145*cosh(x)^8 + 6468*cosh(x)^6 + 7350*cosh(x)^4 + 4900*cosh(x)^2 + (2145*cosh(x)^8 +
6468*cosh(x)^6 + 7350*cosh(x)^4 + 4900*cosh(x)^2 - 175)*e^(2*x) - 175)*sinh(x)^6 - 1225*cosh(x)^6 + 14*(715*c
osh(x)^9 + 2772*cosh(x)^7 + 4410*cosh(x)^5 + 4900*cosh(x)^3 + (715*cosh(x)^9 + 2772*cosh(x)^7 + 4410*cosh(x)^5
+ 4900*cosh(x)^3 - 525*cosh(x))*e^(2*x) - 525*cosh(x))*sinh(x)^5 + 35*(143*cosh(x)^10 + 693*cosh(x)^8 + 1470*
cosh(x)^6 + 2450*cosh(x)^4 - 525*cosh(x)^2 + (143*cosh(x)^10 + 693*cosh(x)^8 + 1470*cosh(x)^6 + 2450*cosh(x)^4
- 525*cosh(x)^2 - 7)*e^(2*x) - 7)*sinh(x)^4 - 245*cosh(x)^4 + 140*(13*cosh(x)^11 + 77*cosh(x)^9 + 210*cosh(x)
^7 + 490*cosh(x)^5 - 175*cosh(x)^3 + (13*cosh(x)^11 + 77*cosh(x)^9 + 210*cosh(x)^7 + 490*cosh(x)^5 - 175*cosh(
x)^3 - 7*cosh(x))*e^(2*x) - 7*cosh(x))*sinh(x)^3 + 7*(65*cosh(x)^12 + 462*cosh(x)^10 + 1575*cosh(x)^8 + 4900*c
osh(x)^6 - 2625*cosh(x)^4 - 210*cosh(x)^2 + (65*cosh(x)^12 + 462*cosh(x)^10 + 1575*cosh(x)^8 + 4900*cosh(x)^6
- 2625*cosh(x)^4 - 210*cosh(x)^2 - 7)*e^(2*x) - 7)*sinh(x)^2 - 49*cosh(x)^2 + (5*cosh(x)^14 + 49*cosh(x)^12 +
245*cosh(x)^10 + 1225*cosh(x)^8 - 1225*cosh(x)^6 - 245*cosh(x)^4 - 49*cosh(x)^2 - 5)*e^(2*x) + 14*(5*cosh(x)^1
3 + 42*cosh(x)^11 + 175*cosh(x)^9 + 700*cosh(x)^7 - 525*cosh(x)^5 - 70*cosh(x)^3 + (5*cosh(x)^13 + 42*cosh(x)^
11 + 175*cosh(x)^9 + 700*cosh(x)^7 - 525*cosh(x)^5 - 70*cosh(x)^3 - 7*cosh(x))*e^(2*x) - 7*cosh(x))*sinh(x) -
5)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x/(a^4*cosh(x)^7*e^x + 7*a^4*cosh(x)^6*e^x*sinh(x) + 21*a^4*cosh(x)^5*e
^x*sinh(x)^2 + 35*a^4*cosh(x)^4*e^x*sinh(x)^3 + 35*a^4*cosh(x)^3*e^x*sinh(x)^4 + 21*a^4*cosh(x)^2*e^x*sinh(x)^
5 + 7*a^4*cosh(x)*e^x*sinh(x)^6 + a^4*e^x*sinh(x)^7)

Sympy [A] (verification not implemented)

Time = 13.43 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=- \frac {16 \tanh ^{7}{\left (x \right )}}{35 \left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {7}{2}}} + \frac {8 \tanh ^{5}{\left (x \right )}}{5 \left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {2 \tanh ^{3}{\left (x \right )}}{\left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {7}{2}}} + \frac {\tanh {\left (x \right )}}{\left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {7}{2}}} \]

[In]

integrate(1/(a*sech(x)**2)**(7/2),x)

[Out]

-16*tanh(x)**7/(35*(a*sech(x)**2)**(7/2)) + 8*tanh(x)**5/(5*(a*sech(x)**2)**(7/2)) - 2*tanh(x)**3/(a*sech(x)**
2)**(7/2) + tanh(x)/(a*sech(x)**2)**(7/2)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=\frac {e^{\left (7 \, x\right )}}{896 \, a^{\frac {7}{2}}} + \frac {7 \, e^{\left (5 \, x\right )}}{640 \, a^{\frac {7}{2}}} + \frac {7 \, e^{\left (3 \, x\right )}}{128 \, a^{\frac {7}{2}}} - \frac {35 \, e^{\left (-x\right )}}{128 \, a^{\frac {7}{2}}} - \frac {7 \, e^{\left (-3 \, x\right )}}{128 \, a^{\frac {7}{2}}} - \frac {7 \, e^{\left (-5 \, x\right )}}{640 \, a^{\frac {7}{2}}} - \frac {e^{\left (-7 \, x\right )}}{896 \, a^{\frac {7}{2}}} + \frac {35 \, e^{x}}{128 \, a^{\frac {7}{2}}} \]

[In]

integrate(1/(a*sech(x)^2)^(7/2),x, algorithm="maxima")

[Out]

1/896*e^(7*x)/a^(7/2) + 7/640*e^(5*x)/a^(7/2) + 7/128*e^(3*x)/a^(7/2) - 35/128*e^(-x)/a^(7/2) - 7/128*e^(-3*x)
/a^(7/2) - 7/640*e^(-5*x)/a^(7/2) - 1/896*e^(-7*x)/a^(7/2) + 35/128*e^x/a^(7/2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=-\frac {{\left (1225 \, e^{\left (6 \, x\right )} + 245 \, e^{\left (4 \, x\right )} + 49 \, e^{\left (2 \, x\right )} + 5\right )} e^{\left (-7 \, x\right )} - 5 \, e^{\left (7 \, x\right )} - 49 \, e^{\left (5 \, x\right )} - 245 \, e^{\left (3 \, x\right )} - 1225 \, e^{x}}{4480 \, a^{\frac {7}{2}}} \]

[In]

integrate(1/(a*sech(x)^2)^(7/2),x, algorithm="giac")

[Out]

-1/4480*((1225*e^(6*x) + 245*e^(4*x) + 49*e^(2*x) + 5)*e^(-7*x) - 5*e^(7*x) - 49*e^(5*x) - 245*e^(3*x) - 1225*
e^x)/a^(7/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^2}\right )}^{7/2}} \,d x \]

[In]

int(1/(a/cosh(x)^2)^(7/2),x)

[Out]

int(1/(a/cosh(x)^2)^(7/2), x)