Integrand size = 13, antiderivative size = 36 \[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=\frac {\log (\cosh (x))}{a}+\frac {\text {sech}(x)}{a}+\frac {\text {sech}^2(x)}{2 a}-\frac {\text {sech}^3(x)}{3 a} \]
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=\frac {(2+6 \cosh (2 x)+3 \cosh (3 x) \log (\cosh (x))+\cosh (x) (6+9 \log (\cosh (x)))) \text {sech}^3(x)}{12 a} \]
((2 + 6*Cosh[2*x] + 3*Cosh[3*x]*Log[Cosh[x]] + Cosh[x]*(6 + 9*Log[Cosh[x]] ))*Sech[x]^3)/(12*a)
Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3042, 26, 4367, 27, 84, 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.
\(\displaystyle \int \frac {\tanh ^5(x)}{a \text {sech}(x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \cot \left (\frac {\pi }{2}+i x\right )^5}{a+a \csc \left (\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\cot \left (i x+\frac {\pi }{2}\right )^5}{\csc \left (i x+\frac {\pi }{2}\right ) a+a}dx\) |
\(\Big \downarrow \) 4367 |
\(\displaystyle \frac {\int a^3 (1-\cosh (x))^2 (\cosh (x)+1) \text {sech}^4(x)d\cosh (x)}{a^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int (1-\cosh (x))^2 (\cosh (x)+1) \text {sech}^4(x)d\cosh (x)}{a}\) |
\(\Big \downarrow \) 84 |
\(\displaystyle \frac {\int \left (\text {sech}^4(x)-\text {sech}^3(x)-\text {sech}^2(x)+\text {sech}(x)\right )d\cosh (x)}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{3} \text {sech}^3(x)+\frac {\text {sech}^2(x)}{2}+\text {sech}(x)+\log (\cosh (x))}{a}\) |
3.2.4.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] && !(ILtQ[n + p + 2, 0 ] && GtQ[n + 2*p, 0])
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _.), x_Symbol] :> Simp[1/(a^(m - n - 1)*b^n*d) Subst[Int[(a - b*x)^((m - 1)/2)*((a + b*x)^((m - 1)/2 + n)/x^(m + n)), x], x, Sin[c + d*x]], x] /; Fr eeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && Integer Q[n]
Time = 0.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(-\frac {\frac {\operatorname {sech}\left (x \right )^{3}}{3}-\frac {\operatorname {sech}\left (x \right )^{2}}{2}-\operatorname {sech}\left (x \right )+\ln \left (\operatorname {sech}\left (x \right )\right )}{a}\) | \(26\) |
default | \(-\frac {\frac {\operatorname {sech}\left (x \right )^{3}}{3}-\frac {\operatorname {sech}\left (x \right )^{2}}{2}-\operatorname {sech}\left (x \right )+\ln \left (\operatorname {sech}\left (x \right )\right )}{a}\) | \(26\) |
risch | \(-\frac {x}{a}+\frac {2 \,{\mathrm e}^{x} \left (3 \,{\mathrm e}^{4 x}+3 \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{x}+3\right )}{3 \left (1+{\mathrm e}^{2 x}\right )^{3} a}+\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{a}\) | \(58\) |
Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (32) = 64\).
Time = 0.27 (sec) , antiderivative size = 437, normalized size of antiderivative = 12.14 \[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=-\frac {3 \, x \cosh \left (x\right )^{6} + 3 \, x \sinh \left (x\right )^{6} + 6 \, {\left (3 \, x \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{5} + 3 \, {\left (3 \, x - 2\right )} \cosh \left (x\right )^{4} - 6 \, \cosh \left (x\right )^{5} + 3 \, {\left (15 \, x \cosh \left (x\right )^{2} + 3 \, x - 10 \, \cosh \left (x\right ) - 2\right )} \sinh \left (x\right )^{4} + 4 \, {\left (15 \, x \cosh \left (x\right )^{3} + 3 \, {\left (3 \, x - 2\right )} \cosh \left (x\right ) - 15 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{3} + 3 \, {\left (3 \, x - 2\right )} \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{3} + 3 \, {\left (15 \, x \cosh \left (x\right )^{4} + 6 \, {\left (3 \, x - 2\right )} \cosh \left (x\right )^{2} - 20 \, \cosh \left (x\right )^{3} + 3 \, x - 4 \, \cosh \left (x\right ) - 2\right )} \sinh \left (x\right )^{2} - 3 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 6 \, {\left (3 \, x \cosh \left (x\right )^{5} + 2 \, {\left (3 \, x - 2\right )} \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )^{4} + {\left (3 \, x - 2\right )} \cosh \left (x\right ) - 2 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) + 3 \, x - 6 \, \cosh \left (x\right )}{3 \, {\left (a \cosh \left (x\right )^{6} + 6 \, a \cosh \left (x\right ) \sinh \left (x\right )^{5} + a \sinh \left (x\right )^{6} + 3 \, a \cosh \left (x\right )^{4} + 3 \, {\left (5 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{4} + 4 \, {\left (5 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )^{2} + 3 \, {\left (5 \, a \cosh \left (x\right )^{4} + 6 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 6 \, {\left (a \cosh \left (x\right )^{5} + 2 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a\right )}} \]
-1/3*(3*x*cosh(x)^6 + 3*x*sinh(x)^6 + 6*(3*x*cosh(x) - 1)*sinh(x)^5 + 3*(3 *x - 2)*cosh(x)^4 - 6*cosh(x)^5 + 3*(15*x*cosh(x)^2 + 3*x - 10*cosh(x) - 2 )*sinh(x)^4 + 4*(15*x*cosh(x)^3 + 3*(3*x - 2)*cosh(x) - 15*cosh(x)^2 - 1)* sinh(x)^3 + 3*(3*x - 2)*cosh(x)^2 - 4*cosh(x)^3 + 3*(15*x*cosh(x)^4 + 6*(3 *x - 2)*cosh(x)^2 - 20*cosh(x)^3 + 3*x - 4*cosh(x) - 2)*sinh(x)^2 - 3*(cos h(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 + 1)*sinh(x)^4 + 3*cosh(x)^4 + 4*(5*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 + 6* cosh(x)^2 + 1)*sinh(x)^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 + 2*cosh(x)^3 + cosh (x))*sinh(x) + 1)*log(2*cosh(x)/(cosh(x) - sinh(x))) + 6*(3*x*cosh(x)^5 + 2*(3*x - 2)*cosh(x)^3 - 5*cosh(x)^4 + (3*x - 2)*cosh(x) - 2*cosh(x)^2 - 1) *sinh(x) + 3*x - 6*cosh(x))/(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh( x)^6 + 3*a*cosh(x)^4 + 3*(5*a*cosh(x)^2 + a)*sinh(x)^4 + 4*(5*a*cosh(x)^3 + 3*a*cosh(x))*sinh(x)^3 + 3*a*cosh(x)^2 + 3*(5*a*cosh(x)^4 + 6*a*cosh(x)^ 2 + a)*sinh(x)^2 + 6*(a*cosh(x)^5 + 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) + a )
\[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\tanh ^{5}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (32) = 64\).
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.06 \[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a} + \frac {2 \, {\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + 2 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )}\right )}}{3 \, {\left (3 \, a e^{\left (-2 \, x\right )} + 3 \, a e^{\left (-4 \, x\right )} + a e^{\left (-6 \, x\right )} + a\right )}} + \frac {\log \left (e^{\left (-2 \, x\right )} + 1\right )}{a} \]
x/a + 2/3*(3*e^(-x) + 3*e^(-2*x) + 2*e^(-3*x) + 3*e^(-4*x) + 3*e^(-5*x))/( 3*a*e^(-2*x) + 3*a*e^(-4*x) + a*e^(-6*x) + a) + log(e^(-2*x) + 1)/a
Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.69 \[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=\frac {\log \left (e^{\left (-x\right )} + e^{x}\right )}{a} - \frac {11 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 12 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 12 \, e^{\left (-x\right )} - 12 \, e^{x} + 16}{6 \, a {\left (e^{\left (-x\right )} + e^{x}\right )}^{3}} \]
log(e^(-x) + e^x)/a - 1/6*(11*(e^(-x) + e^x)^3 - 12*(e^(-x) + e^x)^2 - 12* e^(-x) - 12*e^x + 16)/(a*(e^(-x) + e^x)^3)
Time = 2.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.67 \[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )}{a}-\frac {\frac {2}{a}+\frac {8\,{\mathrm {e}}^x}{3\,a}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {x}{a}+\frac {\frac {2}{a}+\frac {2\,{\mathrm {e}}^x}{a}}{{\mathrm {e}}^{2\,x}+1}+\frac {8\,{\mathrm {e}}^x}{3\,a\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )} \]