Integrand size = 10, antiderivative size = 86 \[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{3/2}} \, dx=\frac {5 x \text {sech}^2(x)}{16 a \sqrt {a \text {sech}^4(x)}}+\frac {5 \cosh (x) \sinh (x)}{24 a \sqrt {a \text {sech}^4(x)}}+\frac {\cosh ^3(x) \sinh (x)}{6 a \sqrt {a \text {sech}^4(x)}}+\frac {5 \tanh (x)}{16 a \sqrt {a \text {sech}^4(x)}} \] Output:
5/16*x*sech(x)^2/a/(a*sech(x)^4)^(1/2)+5/24*cosh(x)*sinh(x)/a/(a*sech(x)^4 )^(1/2)+1/6*cosh(x)^3*sinh(x)/a/(a*sech(x)^4)^(1/2)+5/16*tanh(x)/a/(a*sech (x)^4)^(1/2)
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{3/2}} \, dx=\frac {\text {sech}^6(x) (60 x+45 \sinh (2 x)+9 \sinh (4 x)+\sinh (6 x))}{192 \left (a \text {sech}^4(x)\right )^{3/2}} \] Input:
Integrate[(a*Sech[x]^4)^(-3/2),x]
Output:
(Sech[x]^6*(60*x + 45*Sinh[2*x] + 9*Sinh[4*x] + Sinh[6*x]))/(192*(a*Sech[x ]^4)^(3/2))
Time = 0.56 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.72, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {3042, 4611, 3042, 3115, 3042, 3115, 3042, 3115, 24}
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.
\(\displaystyle \int \frac {1}{\left (a \text {sech}^4(x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a \sec (i x)^4\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle \frac {\text {sech}^2(x) \int \cosh ^6(x)dx}{a \sqrt {a \text {sech}^4(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\text {sech}^2(x) \int \sin \left (i x+\frac {\pi }{2}\right )^6dx}{a \sqrt {a \text {sech}^4(x)}}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\text {sech}^2(x) \left (\frac {5}{6} \int \cosh ^4(x)dx+\frac {1}{6} \sinh (x) \cosh ^5(x)\right )}{a \sqrt {a \text {sech}^4(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\text {sech}^2(x) \left (\frac {1}{6} \sinh (x) \cosh ^5(x)+\frac {5}{6} \int \sin \left (i x+\frac {\pi }{2}\right )^4dx\right )}{a \sqrt {a \text {sech}^4(x)}}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\text {sech}^2(x) \left (\frac {5}{6} \left (\frac {3}{4} \int \cosh ^2(x)dx+\frac {1}{4} \sinh (x) \cosh ^3(x)\right )+\frac {1}{6} \sinh (x) \cosh ^5(x)\right )}{a \sqrt {a \text {sech}^4(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\text {sech}^2(x) \left (\frac {1}{6} \sinh (x) \cosh ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sinh (x) \cosh ^3(x)+\frac {3}{4} \int \sin \left (i x+\frac {\pi }{2}\right )^2dx\right )\right )}{a \sqrt {a \text {sech}^4(x)}}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\text {sech}^2(x) \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {1}{4} \sinh (x) \cosh ^3(x)\right )+\frac {1}{6} \sinh (x) \cosh ^5(x)\right )}{a \sqrt {a \text {sech}^4(x)}}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\text {sech}^2(x) \left (\frac {1}{6} \sinh (x) \cosh ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sinh (x) \cosh ^3(x)+\frac {3}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\right )\right )}{a \sqrt {a \text {sech}^4(x)}}\) |
Input:
Int[(a*Sech[x]^4)^(-3/2),x]
Output:
(Sech[x]^2*((Cosh[x]^5*Sinh[x])/6 + (5*((Cosh[x]^3*Sinh[x])/4 + (3*(x/2 + (Cosh[x]*Sinh[x])/2))/4))/6))/(a*Sqrt[a*Sech[x]^4])
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(229\) vs. \(2(70)=140\).
Time = 0.12 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.67
method | result | size |
risch | \(\frac {5 \,{\mathrm e}^{2 x} x}{16 a \left ({\mathrm e}^{2 x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}+1\right )^{4}}}}+\frac {{\mathrm e}^{8 x}}{384 a \left ({\mathrm e}^{2 x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}+1\right )^{4}}}}+\frac {3 \,{\mathrm e}^{6 x}}{128 a \left ({\mathrm e}^{2 x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}+1\right )^{4}}}}+\frac {15 \,{\mathrm e}^{4 x}}{128 a \left ({\mathrm e}^{2 x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}+1\right )^{4}}}}-\frac {15}{128 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}+1\right )^{4}}}\, \left ({\mathrm e}^{2 x}+1\right )^{2} a}-\frac {3 \,{\mathrm e}^{-2 x}}{128 a \left ({\mathrm e}^{2 x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}+1\right )^{4}}}}-\frac {{\mathrm e}^{-4 x}}{384 a \left ({\mathrm e}^{2 x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}+1\right )^{4}}}}\) | \(230\) |
Input:
int(1/(a*sech(x)^4)^(3/2),x,method=_RETURNVERBOSE)
Output:
5/16/a*exp(2*x)/(exp(2*x)+1)^2/(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*x+1/384/a *exp(8*x)/(exp(2*x)+1)^2/(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)+3/128/a*exp(6*x )/(exp(2*x)+1)^2/(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)+15/128/a*exp(4*x)/(exp( 2*x)+1)^2/(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)-15/128/(a*exp(4*x)/(exp(2*x)+1 )^4)^(1/2)/(exp(2*x)+1)^2/a-3/128/a*exp(-2*x)/(exp(2*x)+1)^2/(a*exp(4*x)/( exp(2*x)+1)^4)^(1/2)-1/384/a*exp(-4*x)/(exp(2*x)+1)^2/(a*exp(4*x)/(exp(2*x )+1)^4)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 1141 vs. \(2 (70) = 140\).
Time = 0.11 (sec) , antiderivative size = 1141, normalized size of antiderivative = 13.27 \[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(a*sech(x)^4)^(3/2),x, algorithm="fricas")
Output:
1/384*((e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^12 + cosh(x)^12 + 12*(cosh(x)*e^( 4*x) + 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^11 + 3*(22*cosh(x)^2 + (22*cos h(x)^2 + 3)*e^(4*x) + 2*(22*cosh(x)^2 + 3)*e^(2*x) + 3)*sinh(x)^10 + 9*cos h(x)^10 + 10*(22*cosh(x)^3 + (22*cosh(x)^3 + 9*cosh(x))*e^(4*x) + 2*(22*co sh(x)^3 + 9*cosh(x))*e^(2*x) + 9*cosh(x))*sinh(x)^9 + 45*(11*cosh(x)^4 + 9 *cosh(x)^2 + (11*cosh(x)^4 + 9*cosh(x)^2 + 1)*e^(4*x) + 2*(11*cosh(x)^4 + 9*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^8 + 45*cosh(x)^8 + 72*(11*cosh(x)^5 + 15*cosh(x)^3 + (11*cosh(x)^5 + 15*cosh(x)^3 + 5*cosh(x))*e^(4*x) + 2*(11 *cosh(x)^5 + 15*cosh(x)^3 + 5*cosh(x))*e^(2*x) + 5*cosh(x))*sinh(x)^7 + 12 0*x*cosh(x)^6 + 6*(154*cosh(x)^6 + 315*cosh(x)^4 + 210*cosh(x)^2 + (154*co sh(x)^6 + 315*cosh(x)^4 + 210*cosh(x)^2 + 20*x)*e^(4*x) + 2*(154*cosh(x)^6 + 315*cosh(x)^4 + 210*cosh(x)^2 + 20*x)*e^(2*x) + 20*x)*sinh(x)^6 + 36*(2 2*cosh(x)^7 + 63*cosh(x)^5 + 70*cosh(x)^3 + 20*x*cosh(x) + (22*cosh(x)^7 + 63*cosh(x)^5 + 70*cosh(x)^3 + 20*x*cosh(x))*e^(4*x) + 2*(22*cosh(x)^7 + 6 3*cosh(x)^5 + 70*cosh(x)^3 + 20*x*cosh(x))*e^(2*x))*sinh(x)^5 + 45*(11*cos h(x)^8 + 42*cosh(x)^6 + 70*cosh(x)^4 + 40*x*cosh(x)^2 + (11*cosh(x)^8 + 42 *cosh(x)^6 + 70*cosh(x)^4 + 40*x*cosh(x)^2 - 1)*e^(4*x) + 2*(11*cosh(x)^8 + 42*cosh(x)^6 + 70*cosh(x)^4 + 40*x*cosh(x)^2 - 1)*e^(2*x) - 1)*sinh(x)^4 - 45*cosh(x)^4 + 20*(11*cosh(x)^9 + 54*cosh(x)^7 + 126*cosh(x)^5 + 120*x* cosh(x)^3 + (11*cosh(x)^9 + 54*cosh(x)^7 + 126*cosh(x)^5 + 120*x*cosh(x...
\[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \operatorname {sech}^{4}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(a*sech(x)**4)**(3/2),x)
Output:
Integral((a*sech(x)**4)**(-3/2), x)
Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{3/2}} \, dx=\frac {{\left (9 \, \sqrt {a} e^{\left (-2 \, x\right )} + 45 \, \sqrt {a} e^{\left (-4 \, x\right )} - 45 \, \sqrt {a} e^{\left (-8 \, x\right )} - 9 \, \sqrt {a} e^{\left (-10 \, x\right )} - \sqrt {a} e^{\left (-12 \, x\right )} + \sqrt {a}\right )} e^{\left (6 \, x\right )}}{384 \, a^{2}} + \frac {5 \, x}{16 \, a^{\frac {3}{2}}} \] Input:
integrate(1/(a*sech(x)^4)^(3/2),x, algorithm="maxima")
Output:
1/384*(9*sqrt(a)*e^(-2*x) + 45*sqrt(a)*e^(-4*x) - 45*sqrt(a)*e^(-8*x) - 9* sqrt(a)*e^(-10*x) - sqrt(a)*e^(-12*x) + sqrt(a))*e^(6*x)/a^2 + 5/16*x/a^(3 /2)
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.60 \[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{3/2}} \, dx=-\frac {{\left (110 \, e^{\left (6 \, x\right )} + 45 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-6 \, x\right )} - 120 \, x - e^{\left (6 \, x\right )} - 9 \, e^{\left (4 \, x\right )} - 45 \, e^{\left (2 \, x\right )}}{384 \, a^{\frac {3}{2}}} \] Input:
integrate(1/(a*sech(x)^4)^(3/2),x, algorithm="giac")
Output:
-1/384*((110*e^(6*x) + 45*e^(4*x) + 9*e^(2*x) + 1)*e^(-6*x) - 120*x - e^(6 *x) - 9*e^(4*x) - 45*e^(2*x))/a^(3/2)
Timed out. \[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^4}\right )}^{3/2}} \,d x \] Input:
int(1/(a/cosh(x)^4)^(3/2),x)
Output:
int(1/(a/cosh(x)^4)^(3/2), x)
Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{3/2}} \, dx=\frac {\sqrt {a}\, \left (e^{12 x}+9 e^{10 x}+45 e^{8 x}+120 e^{6 x} x -45 e^{4 x}-9 e^{2 x}-1\right )}{384 e^{6 x} a^{2}} \] Input:
int(1/(a*sech(x)^4)^(3/2),x)
Output:
(sqrt(a)*(e**(12*x) + 9*e**(10*x) + 45*e**(8*x) + 120*e**(6*x)*x - 45*e**( 4*x) - 9*e**(2*x) - 1))/(384*e**(6*x)*a**2)