Integrand size = 15, antiderivative size = 35 \[ \int \frac {\text {sech}^3(x)}{a+a \sinh ^2(x)} \, dx=\frac {3 \arctan (\sinh (x))}{8 a}+\frac {3 \text {sech}(x) \tanh (x)}{8 a}+\frac {\text {sech}^3(x) \tanh (x)}{4 a} \]
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \frac {\text {sech}^3(x)}{a+a \sinh ^2(x)} \, dx=\frac {\frac {3}{8} \arctan (\sinh (x))+\frac {3}{8} \text {sech}(x) \tanh (x)+\frac {1}{4} \text {sech}^3(x) \tanh (x)}{a} \]
Time = 0.36 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3042, 3654, 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.
\(\displaystyle \int \frac {\text {sech}^3(x)}{a \sinh ^2(x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cos (i x)^3 \left (a-a \sin (i x)^2\right )}dx\) |
\(\Big \downarrow \) 3654 |
\(\displaystyle \frac {\int \text {sech}^5(x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \csc \left (i x+\frac {\pi }{2}\right )^5dx}{a}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {\frac {3}{4} \int \text {sech}^3(x)dx+\frac {1}{4} \tanh (x) \text {sech}^3(x)}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{4} \int \csc \left (i x+\frac {\pi }{2}\right )^3dx}{a}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {\frac {3}{4} \left (\frac {\int \text {sech}(x)dx}{2}+\frac {1}{2} \tanh (x) \text {sech}(x)\right )+\frac {1}{4} \tanh (x) \text {sech}^3(x)}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{4} \left (\frac {1}{2} \tanh (x) \text {sech}(x)+\frac {1}{2} \int \csc \left (i x+\frac {\pi }{2}\right )dx\right )}{a}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \arctan (\sinh (x))+\frac {1}{2} \tanh (x) \text {sech}(x)\right )+\frac {1}{4} \tanh (x) \text {sech}^3(x)}{a}\) |
3.3.82.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[ a^p Int[ActivateTrig[u*cos[e + f*x]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]
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 = 80.83 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.60
method | result | size |
default | \(\frac {\frac {2 \left (-\frac {5 \tanh \left (\frac {x}{2}\right )^{7}}{8}+\frac {3 \tanh \left (\frac {x}{2}\right )^{5}}{8}-\frac {3 \tanh \left (\frac {x}{2}\right )^{3}}{8}+\frac {5 \tanh \left (\frac {x}{2}\right )}{8}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )^{4}}+\frac {3 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{4}}{a}\) | \(56\) |
risch | \(\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{6 x}+11 \,{\mathrm e}^{4 x}-11 \,{\mathrm e}^{2 x}-3\right )}{4 \left ({\mathrm e}^{2 x}+1\right )^{4} a}+\frac {3 i \ln \left ({\mathrm e}^{x}+i\right )}{8 a}-\frac {3 i \ln \left ({\mathrm e}^{x}-i\right )}{8 a}\) | \(61\) |
2/a*((-5/8*tanh(1/2*x)^7+3/8*tanh(1/2*x)^5-3/8*tanh(1/2*x)^3+5/8*tanh(1/2* x))/(tanh(1/2*x)^2+1)^4+3/8*arctan(tanh(1/2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 488, normalized size of antiderivative = 13.94 \[ \int \frac {\text {sech}^3(x)}{a+a \sinh ^2(x)} \, dx=\frac {3 \, \cosh \left (x\right )^{7} + 21 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + 3 \, \sinh \left (x\right )^{7} + {\left (63 \, \cosh \left (x\right )^{2} + 11\right )} \sinh \left (x\right )^{5} + 11 \, \cosh \left (x\right )^{5} + 5 \, {\left (21 \, \cosh \left (x\right )^{3} + 11 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + {\left (105 \, \cosh \left (x\right )^{4} + 110 \, \cosh \left (x\right )^{2} - 11\right )} \sinh \left (x\right )^{3} - 11 \, \cosh \left (x\right )^{3} + {\left (63 \, \cosh \left (x\right )^{5} + 110 \, \cosh \left (x\right )^{3} - 33 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 3 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{6} + 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} + 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} + 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} + 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} + 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (21 \, \cosh \left (x\right )^{6} + 55 \, \cosh \left (x\right )^{4} - 33 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right ) - 3 \, \cosh \left (x\right )}{4 \, {\left (a \cosh \left (x\right )^{8} + 8 \, a \cosh \left (x\right ) \sinh \left (x\right )^{7} + a \sinh \left (x\right )^{8} + 4 \, a \cosh \left (x\right )^{6} + 4 \, {\left (7 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{6} + 8 \, {\left (7 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 6 \, a \cosh \left (x\right )^{4} + 2 \, {\left (35 \, a \cosh \left (x\right )^{4} + 30 \, a \cosh \left (x\right )^{2} + 3 \, a\right )} \sinh \left (x\right )^{4} + 8 \, {\left (7 \, a \cosh \left (x\right )^{5} + 10 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, a \cosh \left (x\right )^{2} + 4 \, {\left (7 \, a \cosh \left (x\right )^{6} + 15 \, a \cosh \left (x\right )^{4} + 9 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 8 \, {\left (a \cosh \left (x\right )^{7} + 3 \, a \cosh \left (x\right )^{5} + 3 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a\right )}} \]
1/4*(3*cosh(x)^7 + 21*cosh(x)*sinh(x)^6 + 3*sinh(x)^7 + (63*cosh(x)^2 + 11 )*sinh(x)^5 + 11*cosh(x)^5 + 5*(21*cosh(x)^3 + 11*cosh(x))*sinh(x)^4 + (10 5*cosh(x)^4 + 110*cosh(x)^2 - 11)*sinh(x)^3 - 11*cosh(x)^3 + (63*cosh(x)^5 + 110*cosh(x)^3 - 33*cosh(x))*sinh(x)^2 + 3*(cosh(x)^8 + 8*cosh(x)*sinh(x )^7 + sinh(x)^8 + 4*(7*cosh(x)^2 + 1)*sinh(x)^6 + 4*cosh(x)^6 + 8*(7*cosh( x)^3 + 3*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 + 30*cosh(x)^2 + 3)*sinh(x)^ 4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 + 10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4 *(7*cosh(x)^6 + 15*cosh(x)^4 + 9*cosh(x)^2 + 1)*sinh(x)^2 + 4*cosh(x)^2 + 8*(cosh(x)^7 + 3*cosh(x)^5 + 3*cosh(x)^3 + cosh(x))*sinh(x) + 1)*arctan(co sh(x) + sinh(x)) + (21*cosh(x)^6 + 55*cosh(x)^4 - 33*cosh(x)^2 - 3)*sinh(x ) - 3*cosh(x))/(a*cosh(x)^8 + 8*a*cosh(x)*sinh(x)^7 + a*sinh(x)^8 + 4*a*co sh(x)^6 + 4*(7*a*cosh(x)^2 + a)*sinh(x)^6 + 8*(7*a*cosh(x)^3 + 3*a*cosh(x) )*sinh(x)^5 + 6*a*cosh(x)^4 + 2*(35*a*cosh(x)^4 + 30*a*cosh(x)^2 + 3*a)*si nh(x)^4 + 8*(7*a*cosh(x)^5 + 10*a*cosh(x)^3 + 3*a*cosh(x))*sinh(x)^3 + 4*a *cosh(x)^2 + 4*(7*a*cosh(x)^6 + 15*a*cosh(x)^4 + 9*a*cosh(x)^2 + a)*sinh(x )^2 + 8*(a*cosh(x)^7 + 3*a*cosh(x)^5 + 3*a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)
\[ \int \frac {\text {sech}^3(x)}{a+a \sinh ^2(x)} \, dx=\frac {\int \frac {\operatorname {sech}^{3}{\left (x \right )}}{\sinh ^{2}{\left (x \right )} + 1}\, dx}{a} \]
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (29) = 58\).
Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.97 \[ \int \frac {\text {sech}^3(x)}{a+a \sinh ^2(x)} \, dx=\frac {3 \, e^{\left (-x\right )} + 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a\right )}} - \frac {3 \, \arctan \left (e^{\left (-x\right )}\right )}{4 \, a} \]
1/4*(3*e^(-x) + 11*e^(-3*x) - 11*e^(-5*x) - 3*e^(-7*x))/(4*a*e^(-2*x) + 6* a*e^(-4*x) + 4*a*e^(-6*x) + a*e^(-8*x) + a) - 3/4*arctan(e^(-x))/a
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.91 \[ \int \frac {\text {sech}^3(x)}{a+a \sinh ^2(x)} \, dx=\frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )}}{16 \, a} - \frac {3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 20 \, e^{\left (-x\right )} - 20 \, e^{x}}{4 \, {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2} a} \]
3/16*(pi + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))/a - 1/4*(3*(e^(-x) - e^x)^3 + 20*e^(-x) - 20*e^x)/(((e^(-x) - e^x)^2 + 4)^2*a)
Time = 2.48 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.37 \[ \int \frac {\text {sech}^3(x)}{a+a \sinh ^2(x)} \, dx=\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{4\,\sqrt {a^2}}-\frac {4\,{\mathrm {e}}^{3\,x}}{a\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {2\,{\mathrm {e}}^x}{a\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}+\frac {{\mathrm {e}}^x}{2\,a\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )}+\frac {3\,{\mathrm {e}}^x}{4\,a\,\left ({\mathrm {e}}^{2\,x}+1\right )} \]