Integrand size = 9, antiderivative size = 36 \[ \int \text {sech}^5(x) \tanh ^2(x) \, dx=\frac {1}{16} \arctan (\sinh (x))+\frac {1}{16} \text {sech}(x) \tanh (x)+\frac {1}{24} \text {sech}^3(x) \tanh (x)-\frac {1}{6} \text {sech}^5(x) \tanh (x) \]
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \text {sech}^5(x) \tanh ^2(x) \, dx=\frac {1}{16} \arctan (\sinh (x))+\frac {1}{16} \text {sech}(x) \tanh (x)+\frac {1}{24} \text {sech}^3(x) \tanh (x)-\frac {1}{6} \text {sech}^5(x) \tanh (x) \]
Time = 0.36 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.28, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 25, 3091, 3042, 4255, 3042, 4255, 3042, 4257}
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.
\(\displaystyle \int \tanh ^2(x) \text {sech}^5(x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (i x)^2 \left (-\sec (i x)^5\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \sec (i x)^5 \tan (i x)^2dx\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle \frac {1}{6} \int \text {sech}^5(x)dx-\frac {1}{6} \tanh (x) \text {sech}^5(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{6} \tanh (x) \text {sech}^5(x)+\frac {1}{6} \int \csc \left (i x+\frac {\pi }{2}\right )^5dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {1}{6} \left (\frac {3}{4} \int \text {sech}^3(x)dx+\frac {1}{4} \tanh (x) \text {sech}^3(x)\right )-\frac {1}{6} \tanh (x) \text {sech}^5(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{6} \tanh (x) \text {sech}^5(x)+\frac {1}{6} \left (\frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{4} \int \csc \left (i x+\frac {\pi }{2}\right )^3dx\right )\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {1}{6} \left (\frac {3}{4} \left (\frac {\int \text {sech}(x)dx}{2}+\frac {1}{2} \tanh (x) \text {sech}(x)\right )+\frac {1}{4} \tanh (x) \text {sech}^3(x)\right )-\frac {1}{6} \tanh (x) \text {sech}^5(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{6} \tanh (x) \text {sech}^5(x)+\frac {1}{6} \left (\frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{4} \left (\frac {1}{2} \tanh (x) \text {sech}(x)+\frac {1}{2} \int \csc \left (i x+\frac {\pi }{2}\right )dx\right )\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {1}{6} \left (\frac {3}{4} \left (\frac {1}{2} \arctan (\sinh (x))+\frac {1}{2} \tanh (x) \text {sech}(x)\right )+\frac {1}{4} \tanh (x) \text {sech}^3(x)\right )-\frac {1}{6} \tanh (x) \text {sech}^5(x)\) |
-1/6*(Sech[x]^5*Tanh[x]) + ((Sech[x]^3*Tanh[x])/4 + (3*(ArcTan[Sinh[x]]/2 + (Sech[x]*Tanh[x])/2))/4)/6
3.1.99.3.1 Defintions of rubi rules used
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 169.34 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {\sinh \left (x \right )}{5 \cosh \left (x \right )^{6}}+\frac {\left (\frac {\operatorname {sech}\left (x \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (x \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (x \right )}{16}\right ) \tanh \left (x \right )}{5}+\frac {\arctan \left ({\mathrm e}^{x}\right )}{8}\) | \(36\) |
risch | \(\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{10 x}+17 \,{\mathrm e}^{8 x}-114 \,{\mathrm e}^{6 x}+114 \,{\mathrm e}^{4 x}-17 \,{\mathrm e}^{2 x}-3\right )}{24 \left (1+{\mathrm e}^{2 x}\right )^{6}}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{16}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{16}\) | \(64\) |
-1/5*sinh(x)/cosh(x)^6+1/5*(1/6*sech(x)^5+5/24*sech(x)^3+5/16*sech(x))*tan h(x)+1/8*arctan(exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 925, normalized size of antiderivative = 25.69 \[ \int \text {sech}^5(x) \tanh ^2(x) \, dx=\text {Too large to display} \]
1/24*(3*cosh(x)^11 + 33*cosh(x)*sinh(x)^10 + 3*sinh(x)^11 + (165*cosh(x)^2 + 17)*sinh(x)^9 + 17*cosh(x)^9 + 9*(55*cosh(x)^3 + 17*cosh(x))*sinh(x)^8 + 6*(165*cosh(x)^4 + 102*cosh(x)^2 - 19)*sinh(x)^7 - 114*cosh(x)^7 + 42*(3 3*cosh(x)^5 + 34*cosh(x)^3 - 19*cosh(x))*sinh(x)^6 + 6*(231*cosh(x)^6 + 35 7*cosh(x)^4 - 399*cosh(x)^2 + 19)*sinh(x)^5 + 114*cosh(x)^5 + 6*(165*cosh( x)^7 + 357*cosh(x)^5 - 665*cosh(x)^3 + 95*cosh(x))*sinh(x)^4 + (495*cosh(x )^8 + 1428*cosh(x)^6 - 3990*cosh(x)^4 + 1140*cosh(x)^2 - 17)*sinh(x)^3 - 1 7*cosh(x)^3 + 3*(55*cosh(x)^9 + 204*cosh(x)^7 - 798*cosh(x)^5 + 380*cosh(x )^3 - 17*cosh(x))*sinh(x)^2 + 3*(cosh(x)^12 + 12*cosh(x)*sinh(x)^11 + sinh (x)^12 + 6*(11*cosh(x)^2 + 1)*sinh(x)^10 + 6*cosh(x)^10 + 20*(11*cosh(x)^3 + 3*cosh(x))*sinh(x)^9 + 15*(33*cosh(x)^4 + 18*cosh(x)^2 + 1)*sinh(x)^8 + 15*cosh(x)^8 + 24*(33*cosh(x)^5 + 30*cosh(x)^3 + 5*cosh(x))*sinh(x)^7 + 4 *(231*cosh(x)^6 + 315*cosh(x)^4 + 105*cosh(x)^2 + 5)*sinh(x)^6 + 20*cosh(x )^6 + 24*(33*cosh(x)^7 + 63*cosh(x)^5 + 35*cosh(x)^3 + 5*cosh(x))*sinh(x)^ 5 + 15*(33*cosh(x)^8 + 84*cosh(x)^6 + 70*cosh(x)^4 + 20*cosh(x)^2 + 1)*sin h(x)^4 + 15*cosh(x)^4 + 20*(11*cosh(x)^9 + 36*cosh(x)^7 + 42*cosh(x)^5 + 2 0*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 6*(11*cosh(x)^10 + 45*cosh(x)^8 + 70* cosh(x)^6 + 50*cosh(x)^4 + 15*cosh(x)^2 + 1)*sinh(x)^2 + 6*cosh(x)^2 + 12* (cosh(x)^11 + 5*cosh(x)^9 + 10*cosh(x)^7 + 10*cosh(x)^5 + 5*cosh(x)^3 + co sh(x))*sinh(x) + 1)*arctan(cosh(x) + sinh(x)) + 3*(11*cosh(x)^10 + 51*c...
\[ \int \text {sech}^5(x) \tanh ^2(x) \, dx=\int \tanh ^{2}{\left (x \right )} \operatorname {sech}^{5}{\left (x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (28) = 56\).
Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.36 \[ \int \text {sech}^5(x) \tanh ^2(x) \, dx=\frac {3 \, e^{\left (-x\right )} + 17 \, e^{\left (-3 \, x\right )} - 114 \, e^{\left (-5 \, x\right )} + 114 \, e^{\left (-7 \, x\right )} - 17 \, e^{\left (-9 \, x\right )} - 3 \, e^{\left (-11 \, x\right )}}{24 \, {\left (6 \, e^{\left (-2 \, x\right )} + 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} + e^{\left (-12 \, x\right )} + 1\right )}} - \frac {1}{8} \, \arctan \left (e^{\left (-x\right )}\right ) \]
1/24*(3*e^(-x) + 17*e^(-3*x) - 114*e^(-5*x) + 114*e^(-7*x) - 17*e^(-9*x) - 3*e^(-11*x))/(6*e^(-2*x) + 15*e^(-4*x) + 20*e^(-6*x) + 15*e^(-8*x) + 6*e^ (-10*x) + e^(-12*x) + 1) - 1/8*arctan(e^(-x))
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (28) = 56\).
Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.03 \[ \int \text {sech}^5(x) \tanh ^2(x) \, dx=\frac {1}{32} \, \pi - \frac {3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{5} + 32 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - 48 \, e^{\left (-x\right )} + 48 \, e^{x}}{24 \, {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{3}} + \frac {1}{16} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \]
1/32*pi - 1/24*(3*(e^(-x) - e^x)^5 + 32*(e^(-x) - e^x)^3 - 48*e^(-x) + 48* e^x)/((e^(-x) - e^x)^2 + 4)^3 + 1/16*arctan(1/2*(e^(2*x) - 1)*e^(-x))
Time = 2.17 (sec) , antiderivative size = 206, normalized size of antiderivative = 5.72 \[ \int \text {sech}^5(x) \tanh ^2(x) \, dx=\frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{8}+\frac {34\,{\mathrm {e}}^x}{15\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {{\mathrm {e}}^x}{8\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {9\,{\mathrm {e}}^x}{5\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}-\frac {\frac {8\,{\mathrm {e}}^{3\,x}}{3}-\frac {16\,{\mathrm {e}}^{5\,x}}{3}+\frac {8\,{\mathrm {e}}^{7\,x}}{3}}{6\,{\mathrm {e}}^{2\,x}+15\,{\mathrm {e}}^{4\,x}+20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}+6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1}-\frac {\frac {28\,{\mathrm {e}}^{5\,x}}{15}-\frac {8\,{\mathrm {e}}^{3\,x}}{3}+\frac {4\,{\mathrm {e}}^x}{5}}{5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1}+\frac {{\mathrm {e}}^x}{12\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \]
atan(exp(x))/8 + (34*exp(x))/(15*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + e xp(8*x) + 1)) + exp(x)/(8*(exp(2*x) + 1)) - (9*exp(x))/(5*(3*exp(2*x) + 3* exp(4*x) + exp(6*x) + 1)) - ((8*exp(3*x))/3 - (16*exp(5*x))/3 + (8*exp(7*x ))/3)/(6*exp(2*x) + 15*exp(4*x) + 20*exp(6*x) + 15*exp(8*x) + 6*exp(10*x) + exp(12*x) + 1) - ((28*exp(5*x))/15 - (8*exp(3*x))/3 + (4*exp(x))/5)/(5*e xp(2*x) + 10*exp(4*x) + 10*exp(6*x) + 5*exp(8*x) + exp(10*x) + 1) + exp(x) /(12*(2*exp(2*x) + exp(4*x) + 1))