3.99 \(\int \text {sech}^5(x) \tanh ^2(x) \, dx\)

Optimal. Leaf size=36 \[ \frac {1}{16} \tan ^{-1}(\sinh (x))-\frac {1}{6} \tanh (x) \text {sech}^5(x)+\frac {1}{24} \tanh (x) \text {sech}^3(x)+\frac {1}{16} \tanh (x) \text {sech}(x) \]

[Out]

1/16*arctan(sinh(x))+1/16*sech(x)*tanh(x)+1/24*sech(x)^3*tanh(x)-1/6*sech(x)^5*tanh(x)

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Rubi [A]  time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2611, 3768, 3770} \[ \frac {1}{16} \tan ^{-1}(\sinh (x))-\frac {1}{6} \tanh (x) \text {sech}^5(x)+\frac {1}{24} \tanh (x) \text {sech}^3(x)+\frac {1}{16} \tanh (x) \text {sech}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^5*Tanh[x]^2,x]

[Out]

ArcTan[Sinh[x]]/16 + (Sech[x]*Tanh[x])/16 + (Sech[x]^3*Tanh[x])/24 - (Sech[x]^5*Tanh[x])/6

Rule 2611

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] - Dist[(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] && Integers
Q[2*m, 2*n]

Rule 3768

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] + Dist[(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]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \text {sech}^5(x) \tanh ^2(x) \, dx &=-\frac {1}{6} \text {sech}^5(x) \tanh (x)+\frac {1}{6} \int \text {sech}^5(x) \, dx\\ &=\frac {1}{24} \text {sech}^3(x) \tanh (x)-\frac {1}{6} \text {sech}^5(x) \tanh (x)+\frac {1}{8} \int \text {sech}^3(x) \, dx\\ &=\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)+\frac {1}{16} \int \text {sech}(x) \, dx\\ &=\frac {1}{16} \tan ^{-1}(\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)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 36, normalized size = 1.00 \[ \frac {1}{16} \tan ^{-1}(\sinh (x))-\frac {1}{6} \tanh (x) \text {sech}^5(x)+\frac {1}{24} \tanh (x) \text {sech}^3(x)+\frac {1}{16} \tanh (x) \text {sech}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^5*Tanh[x]^2,x]

[Out]

ArcTan[Sinh[x]]/16 + (Sech[x]*Tanh[x])/16 + (Sech[x]^3*Tanh[x])/24 - (Sech[x]^5*Tanh[x])/6

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fricas [B]  time = 0.41, size = 925, normalized size = 25.69 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^5*tanh(x)^2,x, algorithm="fricas")

[Out]

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*
(33*cosh(x)^5 + 34*cosh(x)^3 - 19*cosh(x))*sinh(x)^6 + 6*(231*cosh(x)^6 + 357*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*co
sh(x)^8 + 1428*cosh(x)^6 - 3990*cosh(x)^4 + 1140*cosh(x)^2 - 17)*sinh(x)^3 - 17*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)*sinh(x)^4 + 15*cosh(x)^4 + 20*(11*cosh(x)^9 + 36*cosh(x)^7 + 42*cosh(x)^5 + 20*cosh(x)^3 + 3*cosh(x))*s
inh(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*cos
h(x)^2 + 12*(cosh(x)^11 + 5*cosh(x)^9 + 10*cosh(x)^7 + 10*cosh(x)^5 + 5*cosh(x)^3 + cosh(x))*sinh(x) + 1)*arct
an(cosh(x) + sinh(x)) + 3*(11*cosh(x)^10 + 51*cosh(x)^8 - 266*cosh(x)^6 + 190*cosh(x)^4 - 17*cosh(x)^2 - 1)*si
nh(x) - 3*cosh(x))/(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*(3
3*cosh(x)^8 + 84*cosh(x)^6 + 70*cosh(x)^4 + 20*cosh(x)^2 + 1)*sinh(x)^4 + 15*cosh(x)^4 + 20*(11*cosh(x)^9 + 36
*cosh(x)^7 + 42*cosh(x)^5 + 20*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 + cosh(x))*sinh(x) + 1)

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giac [B]  time = 0.13, size = 73, normalized size = 2.03 \[ \frac {1}{32} \, \pi - \frac {3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{5} + 32 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - 48 \, e^{\left (-x\right )} + 48 \, e^{x}}{24 \, {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{3}} + \frac {1}{16} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^5*tanh(x)^2,x, algorithm="giac")

[Out]

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))

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maple [A]  time = 0.33, size = 36, normalized size = 1.00 \[ -\frac {\sinh \relax (x )}{5 \cosh \relax (x )^{6}}+\frac {\left (\frac {\mathrm {sech}\relax (x )^{5}}{6}+\frac {5 \mathrm {sech}\relax (x )^{3}}{24}+\frac {5 \,\mathrm {sech}\relax (x )}{16}\right ) \tanh \relax (x )}{5}+\frac {\arctan \left ({\mathrm e}^{x}\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^5*tanh(x)^2,x)

[Out]

-1/5*sinh(x)/cosh(x)^6+1/5*(1/6*sech(x)^5+5/24*sech(x)^3+5/16*sech(x))*tanh(x)+1/8*arctan(exp(x))

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maxima [B]  time = 0.44, size = 85, normalized size = 2.36 \[ \frac {3 \, e^{\left (-x\right )} + 17 \, e^{\left (-3 \, x\right )} - 114 \, e^{\left (-5 \, x\right )} + 114 \, e^{\left (-7 \, x\right )} - 17 \, e^{\left (-9 \, x\right )} - 3 \, e^{\left (-11 \, x\right )}}{24 \, {\left (6 \, e^{\left (-2 \, x\right )} + 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} + e^{\left (-12 \, x\right )} + 1\right )}} - \frac {1}{8} \, \arctan \left (e^{\left (-x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^5*tanh(x)^2,x, algorithm="maxima")

[Out]

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))

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mupad [B]  time = 1.44, size = 206, normalized size = 5.72 \[ \frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{8}+\frac {34\,{\mathrm {e}}^x}{15\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {{\mathrm {e}}^x}{8\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {9\,{\mathrm {e}}^x}{5\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}-\frac {\frac {8\,{\mathrm {e}}^{3\,x}}{3}-\frac {16\,{\mathrm {e}}^{5\,x}}{3}+\frac {8\,{\mathrm {e}}^{7\,x}}{3}}{6\,{\mathrm {e}}^{2\,x}+15\,{\mathrm {e}}^{4\,x}+20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}+6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1}-\frac {\frac {28\,{\mathrm {e}}^{5\,x}}{15}-\frac {8\,{\mathrm {e}}^{3\,x}}{3}+\frac {4\,{\mathrm {e}}^x}{5}}{5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1}+\frac {{\mathrm {e}}^x}{12\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^2/cosh(x)^5,x)

[Out]

atan(exp(x))/8 + (34*exp(x))/(15*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(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*exp(2*x) + 10*exp(4*x) + 10*exp(6*x) + 5*exp(8*x) + exp(10*x) + 1) + ex
p(x)/(12*(2*exp(2*x) + exp(4*x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh ^{2}{\relax (x )} \operatorname {sech}^{5}{\relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**5*tanh(x)**2,x)

[Out]

Integral(tanh(x)**2*sech(x)**5, x)

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