Integrand size = 10, antiderivative size = 117 \[ \int \left (a \text {sech}^4(x)\right )^{5/2} \, dx=a^2 \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {4}{3} a^2 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh (x)+\frac {6}{5} a^2 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^3(x)-\frac {4}{7} a^2 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^5(x)+\frac {1}{9} a^2 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^7(x) \]
a^2*cosh(x)*sinh(x)*(a*sech(x)^4)^(1/2)-4/3*a^2*sinh(x)^2*(a*sech(x)^4)^(1 /2)*tanh(x)+6/5*a^2*sinh(x)^2*(a*sech(x)^4)^(1/2)*tanh(x)^3-4/7*a^2*sinh(x )^2*(a*sech(x)^4)^(1/2)*tanh(x)^5+1/9*a^2*sinh(x)^2*(a*sech(x)^4)^(1/2)*ta nh(x)^7
Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.36 \[ \int \left (a \text {sech}^4(x)\right )^{5/2} \, dx=\frac {1}{315} \cosh (x) (128+130 \cosh (2 x)+46 \cosh (4 x)+10 \cosh (6 x)+\cosh (8 x)) \left (a \text {sech}^4(x)\right )^{5/2} \sinh (x) \]
(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
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.58, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4611, 3042, 4254, 2009}
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.
\(\displaystyle \int \left (a \text {sech}^4(x)\right )^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sec (i x)^4\right )^{5/2}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle a^2 \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \int \text {sech}^{10}(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^2 \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \int \csc \left (i x+\frac {\pi }{2}\right )^{10}dx\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle i a^2 \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \int \left (\tanh ^8(x)-4 \tanh ^6(x)+6 \tanh ^4(x)-4 \tanh ^2(x)+1\right )d(-i \tanh (x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i a^2 \cosh ^2(x) \left (-\frac {1}{9} i \tanh ^9(x)+\frac {4}{7} i \tanh ^7(x)-\frac {6}{5} i \tanh ^5(x)+\frac {4}{3} i \tanh ^3(x)-i \tanh (x)\right ) \sqrt {a \text {sech}^4(x)}\) |
I*a^2*Cosh[x]^2*Sqrt[a*Sech[x]^4]*((-I)*Tanh[x] + ((4*I)/3)*Tanh[x]^3 - (( 6*I)/5)*Tanh[x]^5 + ((4*I)/7)*Tanh[x]^7 - (I/9)*Tanh[x]^9)
3.1.46.3.1 Defintions of rubi rules used
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
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]
Time = 144.83 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.51
method | result | size |
risch | \(-\frac {256 a^{2} {\mathrm e}^{-2 x} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (126 \,{\mathrm e}^{8 x}+84 \,{\mathrm e}^{6 x}+36 \,{\mathrm e}^{4 x}+9 \,{\mathrm e}^{2 x}+1\right )}{315 \left (1+{\mathrm e}^{2 x}\right )^{7}}\) | \(60\) |
-256/315*a^2*exp(-2*x)/(1+exp(2*x))^7*(exp(4*x)*a/(1+exp(2*x))^4)^(1/2)*(1 26*exp(8*x)+84*exp(6*x)+36*exp(4*x)+9*exp(2*x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 1475 vs. \(2 (99) = 198\).
Time = 0.28 (sec) , antiderivative size = 1475, normalized size of antiderivative = 12.61 \[ \int \left (a \text {sech}^4(x)\right )^{5/2} \, dx=\text {Too large to display} \]
-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 + 36*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...
\[ \int \left (a \text {sech}^4(x)\right )^{5/2} \, dx=\int \left (a \operatorname {sech}^{4}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (99) = 198\).
Time = 0.28 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.75 \[ \int \left (a \text {sech}^4(x)\right )^{5/2} \, dx=\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 )}} \]
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)
Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.33 \[ \int \left (a \text {sech}^4(x)\right )^{5/2} \, dx=-\frac {256 \, a^{\frac {5}{2}} {\left (126 \, e^{\left (8 \, x\right )} + 84 \, e^{\left (6 \, x\right )} + 36 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} + 1\right )}}{315 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{9}} \]
Time = 0.10 (sec) , antiderivative size = 356, normalized size of antiderivative = 3.04 \[ \int \left (a \text {sech}^4(x)\right )^{5/2} \, dx=\frac {256\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^6\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {128\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{5\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^5\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {768\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{7\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^7\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}+\frac {64\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^8\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {128\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{9\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^9\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \]
(256*a^2*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*e xp(6*x) + exp(8*x) + 1))/(3*(exp(2*x) + 1)^6*(exp(2*x) + 2*exp(4*x) + exp( 6*x))) - (128*a^2*(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))/(5*(exp(2*x) + 1)^5*(exp(2*x) + 2*exp(4* x) + exp(6*x))) - (768*a^2*(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))) + (64*a^2*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*e xp(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))) - (128*a^2*(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))/(9*(exp(2*x) + 1 )^9*(exp(2*x) + 2*exp(4*x) + exp(6*x)))