Integrand size = 10, antiderivative size = 74 \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=\frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}}+\frac {6 \tanh (x)}{35 a \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {8 \tanh (x)}{35 a^2 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {16 \tanh (x)}{35 a^3 \sqrt {a \text {sech}^2(x)}} \] Output:
1/7*tanh(x)/(a*sech(x)^2)^(7/2)+6/35*tanh(x)/a/(a*sech(x)^2)^(5/2)+8/35*ta nh(x)/a^2/(a*sech(x)^2)^(3/2)+16/35*tanh(x)/a^3/(a*sech(x)^2)^(1/2)
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=\frac {\left (35+35 \sinh ^2(x)+21 \sinh ^4(x)+5 \sinh ^6(x)\right ) \tanh (x)}{35 a^3 \sqrt {a \text {sech}^2(x)}} \] Input:
Integrate[(a*Sech[x]^2)^(-7/2),x]
Output:
((35 + 35*Sinh[x]^2 + 21*Sinh[x]^4 + 5*Sinh[x]^6)*Tanh[x])/(35*a^3*Sqrt[a* Sech[x]^2])
Time = 0.38 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.45, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4610, 209, 209, 209, 208}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a \sec (i x)^2\right )^{7/2}}dx\) |
\(\Big \downarrow \) 4610 |
\(\displaystyle a \int \frac {1}{\left (a-a \tanh ^2(x)\right )^{9/2}}d\tanh (x)\) |
\(\Big \downarrow \) 209 |
\(\displaystyle a \left (\frac {6 \int \frac {1}{\left (a-a \tanh ^2(x)\right )^{7/2}}d\tanh (x)}{7 a}+\frac {\tanh (x)}{7 a \left (a-a \tanh ^2(x)\right )^{7/2}}\right )\) |
\(\Big \downarrow \) 209 |
\(\displaystyle a \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (a-a \tanh ^2(x)\right )^{5/2}}d\tanh (x)}{5 a}+\frac {\tanh (x)}{5 a \left (a-a \tanh ^2(x)\right )^{5/2}}\right )}{7 a}+\frac {\tanh (x)}{7 a \left (a-a \tanh ^2(x)\right )^{7/2}}\right )\) |
\(\Big \downarrow \) 209 |
\(\displaystyle a \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (a-a \tanh ^2(x)\right )^{3/2}}d\tanh (x)}{3 a}+\frac {\tanh (x)}{3 a \left (a-a \tanh ^2(x)\right )^{3/2}}\right )}{5 a}+\frac {\tanh (x)}{5 a \left (a-a \tanh ^2(x)\right )^{5/2}}\right )}{7 a}+\frac {\tanh (x)}{7 a \left (a-a \tanh ^2(x)\right )^{7/2}}\right )\) |
\(\Big \downarrow \) 208 |
\(\displaystyle a \left (\frac {6 \left (\frac {4 \left (\frac {2 \tanh (x)}{3 a^2 \sqrt {a-a \tanh ^2(x)}}+\frac {\tanh (x)}{3 a \left (a-a \tanh ^2(x)\right )^{3/2}}\right )}{5 a}+\frac {\tanh (x)}{5 a \left (a-a \tanh ^2(x)\right )^{5/2}}\right )}{7 a}+\frac {\tanh (x)}{7 a \left (a-a \tanh ^2(x)\right )^{7/2}}\right )\) |
Input:
Int[(a*Sech[x]^2)^(-7/2),x]
Output:
a*(Tanh[x]/(7*a*(a - a*Tanh[x]^2)^(7/2)) + (6*(Tanh[x]/(5*a*(a - a*Tanh[x] ^2)^(5/2)) + (4*(Tanh[x]/(3*a*(a - a*Tanh[x]^2)^(3/2)) + (2*Tanh[x])/(3*a^ 2*Sqrt[a - a*Tanh[x]^2])))/(5*a)))/(7*a))
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFac tors[Tan[e + f*x], x]}, Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(261\) vs. \(2(58)=116\).
Time = 0.09 (sec) , antiderivative size = 262, normalized size of antiderivative = 3.54
method | result | size |
risch | \(\frac {{\mathrm e}^{8 x}}{896 a^{3} \left ({\mathrm e}^{2 x}+1\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}}+\frac {7 \,{\mathrm e}^{6 x}}{640 a^{3} \left ({\mathrm e}^{2 x}+1\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}}+\frac {7 \,{\mathrm e}^{4 x}}{128 a^{3} \left ({\mathrm e}^{2 x}+1\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}}+\frac {35 \,{\mathrm e}^{2 x}}{128 a^{3} \left ({\mathrm e}^{2 x}+1\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}}-\frac {35}{128 \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 x}+1\right ) a^{3}}-\frac {7 \,{\mathrm e}^{-2 x}}{128 a^{3} \left ({\mathrm e}^{2 x}+1\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}}-\frac {7 \,{\mathrm e}^{-4 x}}{640 a^{3} \left ({\mathrm e}^{2 x}+1\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}}-\frac {{\mathrm e}^{-6 x}}{896 a^{3} \left ({\mathrm e}^{2 x}+1\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}}\) | \(262\) |
Input:
int(1/(sech(x)^2*a)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/896/a^3*exp(8*x)/(exp(2*x)+1)/(exp(2*x)*a/(exp(2*x)+1)^2)^(1/2)+7/640/a^ 3*exp(6*x)/(exp(2*x)+1)/(exp(2*x)*a/(exp(2*x)+1)^2)^(1/2)+7/128/a^3*exp(4* x)/(exp(2*x)+1)/(exp(2*x)*a/(exp(2*x)+1)^2)^(1/2)+35/128/a^3*exp(2*x)/(exp (2*x)+1)/(exp(2*x)*a/(exp(2*x)+1)^2)^(1/2)-35/128/(exp(2*x)*a/(exp(2*x)+1) ^2)^(1/2)/(exp(2*x)+1)/a^3-7/128/a^3*exp(-2*x)/(exp(2*x)+1)/(exp(2*x)*a/(e xp(2*x)+1)^2)^(1/2)-7/640/a^3*exp(-4*x)/(exp(2*x)+1)/(exp(2*x)*a/(exp(2*x) +1)^2)^(1/2)-1/896/a^3*exp(-6*x)/(exp(2*x)+1)/(exp(2*x)*a/(exp(2*x)+1)^2)^ (1/2)
Leaf count of result is larger than twice the leaf count of optimal. 970 vs. \(2 (58) = 116\).
Time = 0.13 (sec) , antiderivative size = 970, normalized size of antiderivative = 13.11 \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(a*sech(x)^2)^(7/2),x, algorithm="fricas")
Output:
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)*si nh(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 + 64 68*cosh(x)^6 + 7350*cosh(x)^4 + 4900*cosh(x)^2 + (2145*cosh(x)^8 + 6468*co sh(x)^6 + 7350*cosh(x)^4 + 4900*cosh(x)^2 - 175)*e^(2*x) - 175)*sinh(x)^6 - 1225*cosh(x)^6 + 14*(715*cosh(x)^9 + 2772*cosh(x)^7 + 4410*cosh(x)^5 + 4 900*cosh(x)^3 + (715*cosh(x)^9 + 2772*cosh(x)^7 + 4410*cosh(x)^5 + 4900*co sh(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 + (14 3*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...
Time = 10.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=- \frac {16 \tanh ^{7}{\left (x \right )}}{35 \left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {7}{2}}} + \frac {8 \tanh ^{5}{\left (x \right )}}{5 \left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {2 \tanh ^{3}{\left (x \right )}}{\left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {7}{2}}} + \frac {\tanh {\left (x \right )}}{\left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {7}{2}}} \] Input:
integrate(1/(a*sech(x)**2)**(7/2),x)
Output:
-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)
Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=\frac {e^{\left (7 \, x\right )}}{896 \, a^{\frac {7}{2}}} + \frac {7 \, e^{\left (5 \, x\right )}}{640 \, a^{\frac {7}{2}}} + \frac {7 \, e^{\left (3 \, x\right )}}{128 \, a^{\frac {7}{2}}} - \frac {35 \, e^{\left (-x\right )}}{128 \, a^{\frac {7}{2}}} - \frac {7 \, e^{\left (-3 \, x\right )}}{128 \, a^{\frac {7}{2}}} - \frac {7 \, e^{\left (-5 \, x\right )}}{640 \, a^{\frac {7}{2}}} - \frac {e^{\left (-7 \, x\right )}}{896 \, a^{\frac {7}{2}}} + \frac {35 \, e^{x}}{128 \, a^{\frac {7}{2}}} \] Input:
integrate(1/(a*sech(x)^2)^(7/2),x, algorithm="maxima")
Output:
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)
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=-\frac {{\left (1225 \, e^{\left (6 \, x\right )} + 245 \, e^{\left (4 \, x\right )} + 49 \, e^{\left (2 \, x\right )} + 5\right )} e^{\left (-7 \, x\right )} - 5 \, e^{\left (7 \, x\right )} - 49 \, e^{\left (5 \, x\right )} - 245 \, e^{\left (3 \, x\right )} - 1225 \, e^{x}}{4480 \, a^{\frac {7}{2}}} \] Input:
integrate(1/(a*sech(x)^2)^(7/2),x, algorithm="giac")
Output:
-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)
Timed out. \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^2}\right )}^{7/2}} \,d x \] Input:
int(1/(a/cosh(x)^2)^(7/2),x)
Output:
int(1/(a/cosh(x)^2)^(7/2), x)
Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx=\frac {\sqrt {a}\, \left (5 e^{14 x}+49 e^{12 x}+245 e^{10 x}+1225 e^{8 x}-1225 e^{6 x}-245 e^{4 x}-49 e^{2 x}-5\right )}{4480 e^{7 x} a^{4}} \] Input:
int(1/(a*sech(x)^2)^(7/2),x)
Output:
(sqrt(a)*(5*e**(14*x) + 49*e**(12*x) + 245*e**(10*x) + 1225*e**(8*x) - 122 5*e**(6*x) - 245*e**(4*x) - 49*e**(2*x) - 5))/(4480*e**(7*x)*a**4)