Integrand size = 13, antiderivative size = 72 \[ \int \frac {\tanh ^5(x)}{a+b \text {sech}(x)} \, dx=\frac {\log (\cosh (x))}{a}+\frac {\left (a^2-b^2\right )^2 \log (a+b \text {sech}(x))}{a b^4}-\frac {\left (a^2-2 b^2\right ) \text {sech}(x)}{b^3}+\frac {a \text {sech}^2(x)}{2 b^2}-\frac {\text {sech}^3(x)}{3 b} \]
ln(cosh(x))/a+(a^2-b^2)^2*ln(a+b*sech(x))/a/b^4-(a^2-2*b^2)*sech(x)/b^3+1/ 2*a*sech(x)^2/b^2-1/3*sech(x)^3/b
Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.06 \[ \int \frac {\tanh ^5(x)}{a+b \text {sech}(x)} \, dx=-\frac {-\frac {b^4 \log (\cosh (x))}{a}-\frac {\left (a^2-b^2\right )^2 \log (a+b \text {sech}(x))}{a}+b \left (a^2-2 b^2\right ) \text {sech}(x)-\frac {1}{2} a b^2 \text {sech}^2(x)+\frac {1}{3} b^3 \text {sech}^3(x)}{b^4} \]
-((-((b^4*Log[Cosh[x]])/a) - ((a^2 - b^2)^2*Log[a + b*Sech[x]])/a + b*(a^2 - 2*b^2)*Sech[x] - (a*b^2*Sech[x]^2)/2 + (b^3*Sech[x]^3)/3)/b^4)
Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 26, 4373, 522, 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.
\(\displaystyle \int \frac {\tanh ^5(x)}{a+b \text {sech}(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \cot \left (\frac {\pi }{2}+i x\right )^5}{a+b \csc \left (\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\cot \left (i x+\frac {\pi }{2}\right )^5}{a+b \csc \left (i x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4373 |
\(\displaystyle -\frac {\int \frac {\cosh (x) \left (b^2-b^2 \text {sech}^2(x)\right )^2}{b (a+b \text {sech}(x))}d(b \text {sech}(x))}{b^4}\) |
\(\Big \downarrow \) 522 |
\(\displaystyle -\frac {\int \left (\frac {\cosh (x) b^3}{a}+\text {sech}^2(x) b^2-a \text {sech}(x) b+a^2 \left (1-\frac {2 b^2}{a^2}\right )-\frac {\left (a^2-b^2\right )^2}{a (a+b \text {sech}(x))}\right )d(b \text {sech}(x))}{b^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b \left (a^2-2 b^2\right ) \text {sech}(x)-\frac {\left (a^2-b^2\right )^2 \log (a+b \text {sech}(x))}{a}+\frac {b^4 \log (b \text {sech}(x))}{a}-\frac {1}{2} a b^2 \text {sech}^2(x)+\frac {1}{3} b^3 \text {sech}^3(x)}{b^4}\) |
-(((b^4*Log[b*Sech[x]])/a - ((a^2 - b^2)^2*Log[a + b*Sech[x]])/a + b*(a^2 - 2*b^2)*Sech[x] - (a*b^2*Sech[x]^2)/2 + (b^3*Sech[x]^3)/3)/b^4)
3.2.15.3.1 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[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(-1)^((m - 1)/2)/(d*b^(m - 1)) Subst[Int[(b^2 - x^ 2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(68)=136\).
Time = 0.94 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.19
method | result | size |
default | \(-\frac {-\frac {2 b^{2} \left (a +2 b \right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}+\frac {8 b^{3}}{3 \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{3}}+a \left (a^{2}-2 b^{2}\right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )+\frac {2 b \left (a^{2}+a b -b^{2}\right )}{1+\tanh \left (\frac {x}{2}\right )^{2}}}{b^{4}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\left (a -b \right )^{2} \left (a^{2}+2 a b +b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b +a +b \right )}{a \,b^{4}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}\) | \(158\) |
risch | \(-\frac {x}{a}-\frac {2 \,{\mathrm e}^{x} \left (3 a^{2} {\mathrm e}^{4 x}-6 b^{2} {\mathrm e}^{4 x}-3 a b \,{\mathrm e}^{3 x}+6 a^{2} {\mathrm e}^{2 x}-8 b^{2} {\mathrm e}^{2 x}-3 b \,{\mathrm e}^{x} a +3 a^{2}-6 b^{2}\right )}{3 b^{3} \left (1+{\mathrm e}^{2 x}\right )^{3}}-\frac {a^{3} \ln \left (1+{\mathrm e}^{2 x}\right )}{b^{4}}+\frac {2 a \ln \left (1+{\mathrm e}^{2 x}\right )}{b^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{b^{4}}-\frac {2 a \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{b^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{a}\) | \(174\) |
-1/b^4*(-2*b^2*(a+2*b)/(1+tanh(1/2*x)^2)^2+8/3*b^3/(1+tanh(1/2*x)^2)^3+a*( a^2-2*b^2)*ln(1+tanh(1/2*x)^2)+2*b*(a^2+a*b-b^2)/(1+tanh(1/2*x)^2))-1/a*ln (tanh(1/2*x)-1)+(a-b)^2*(a^2+2*a*b+b^2)/a/b^4*ln(tanh(1/2*x)^2*a-tanh(1/2* x)^2*b+a+b)-1/a*ln(tanh(1/2*x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 1280 vs. \(2 (68) = 136\).
Time = 0.29 (sec) , antiderivative size = 1280, normalized size of antiderivative = 17.78 \[ \int \frac {\tanh ^5(x)}{a+b \text {sech}(x)} \, dx=\text {Too large to display} \]
-1/3*(3*b^4*x*cosh(x)^6 + 3*b^4*x*sinh(x)^6 + 6*(a^3*b - 2*a*b^3)*cosh(x)^ 5 + 6*(3*b^4*x*cosh(x) + a^3*b - 2*a*b^3)*sinh(x)^5 + 3*b^4*x + 3*(3*b^4*x - 2*a^2*b^2)*cosh(x)^4 + 3*(15*b^4*x*cosh(x)^2 + 3*b^4*x - 2*a^2*b^2 + 10 *(a^3*b - 2*a*b^3)*cosh(x))*sinh(x)^4 + 4*(3*a^3*b - 4*a*b^3)*cosh(x)^3 + 4*(15*b^4*x*cosh(x)^3 + 3*a^3*b - 4*a*b^3 + 15*(a^3*b - 2*a*b^3)*cosh(x)^2 + 3*(3*b^4*x - 2*a^2*b^2)*cosh(x))*sinh(x)^3 + 3*(3*b^4*x - 2*a^2*b^2)*co sh(x)^2 + 3*(15*b^4*x*cosh(x)^4 + 3*b^4*x - 2*a^2*b^2 + 20*(a^3*b - 2*a*b^ 3)*cosh(x)^3 + 6*(3*b^4*x - 2*a^2*b^2)*cosh(x)^2 + 4*(3*a^3*b - 4*a*b^3)*c osh(x))*sinh(x)^2 + 6*(a^3*b - 2*a*b^3)*cosh(x) - 3*((a^4 - 2*a^2*b^2 + b^ 4)*cosh(x)^6 + 6*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^5 + (a^4 - 2*a^2* b^2 + b^4)*sinh(x)^6 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + 3*(a^4 - 2*a^ 2*b^2 + b^4 + 5*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^4 + a^4 - 2*a^2 *b^2 + b^4 + 4*(5*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^3 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x))*sinh(x)^3 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2 + 3*(5*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 6*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^2 + 6*((a^4 - 2*a^2*b^2 + b^4)*cosh(x)^5 + 2*(a ^4 - 2*a^2*b^2 + b^4)*cosh(x)^3 + (a^4 - 2*a^2*b^2 + b^4)*cosh(x))*sinh(x) )*log(2*(a*cosh(x) + b)/(cosh(x) - sinh(x))) + 3*((a^4 - 2*a^2*b^2)*cosh(x )^6 + 6*(a^4 - 2*a^2*b^2)*cosh(x)*sinh(x)^5 + (a^4 - 2*a^2*b^2)*sinh(x)^6 + 3*(a^4 - 2*a^2*b^2)*cosh(x)^4 + 3*(a^4 - 2*a^2*b^2 + 5*(a^4 - 2*a^2*b...
\[ \int \frac {\tanh ^5(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\tanh ^{5}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (68) = 136\).
Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.28 \[ \int \frac {\tanh ^5(x)}{a+b \text {sech}(x)} \, dx=\frac {2 \, {\left (3 \, a b e^{\left (-2 \, x\right )} + 3 \, a b e^{\left (-4 \, x\right )} - 3 \, {\left (a^{2} - 2 \, b^{2}\right )} e^{\left (-x\right )} - 2 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} e^{\left (-3 \, x\right )} - 3 \, {\left (a^{2} - 2 \, b^{2}\right )} e^{\left (-5 \, x\right )}\right )}}{3 \, {\left (3 \, b^{3} e^{\left (-2 \, x\right )} + 3 \, b^{3} e^{\left (-4 \, x\right )} + b^{3} e^{\left (-6 \, x\right )} + b^{3}\right )}} + \frac {x}{a} - \frac {{\left (a^{3} - 2 \, a b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{4}} + \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a b^{4}} \]
2/3*(3*a*b*e^(-2*x) + 3*a*b*e^(-4*x) - 3*(a^2 - 2*b^2)*e^(-x) - 2*(3*a^2 - 4*b^2)*e^(-3*x) - 3*(a^2 - 2*b^2)*e^(-5*x))/(3*b^3*e^(-2*x) + 3*b^3*e^(-4 *x) + b^3*e^(-6*x) + b^3) + x/a - (a^3 - 2*a*b^2)*log(e^(-2*x) + 1)/b^4 + (a^4 - 2*a^2*b^2 + b^4)*log(2*b*e^(-x) + a*e^(-2*x) + a)/(a*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (68) = 136\).
Time = 0.30 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.11 \[ \int \frac {\tanh ^5(x)}{a+b \text {sech}(x)} \, dx=-\frac {{\left (a^{3} - 2 \, a b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x}\right )}{b^{4}} + \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a b^{4}} + \frac {11 \, a^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 22 \, a b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 12 \, a^{2} b {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 24 \, b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 12 \, a b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )} - 16 \, b^{3}}{6 \, b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3}} \]
-(a^3 - 2*a*b^2)*log(e^(-x) + e^x)/b^4 + (a^4 - 2*a^2*b^2 + b^4)*log(abs(a *(e^(-x) + e^x) + 2*b))/(a*b^4) + 1/6*(11*a^3*(e^(-x) + e^x)^3 - 22*a*b^2* (e^(-x) + e^x)^3 - 12*a^2*b*(e^(-x) + e^x)^2 + 24*b^3*(e^(-x) + e^x)^2 + 1 2*a*b^2*(e^(-x) + e^x) - 16*b^3)/(b^4*(e^(-x) + e^x)^3)
Time = 2.40 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.15 \[ \int \frac {\tanh ^5(x)}{a+b \text {sech}(x)} \, dx=\frac {\frac {2\,a}{b^2}-\frac {2\,{\mathrm {e}}^x\,\left (a^2-2\,b^2\right )}{b^3}}{{\mathrm {e}}^{2\,x}+1}-\frac {x}{a}-\frac {\frac {2\,a}{b^2}+\frac {8\,{\mathrm {e}}^x}{3\,b}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )\,\left (2\,a\,b^2-a^3\right )}{b^4}+\frac {8\,{\mathrm {e}}^x}{3\,b\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}+\frac {\ln \left (a+2\,b\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}{a\,b^4} \]
((2*a)/b^2 - (2*exp(x)*(a^2 - 2*b^2))/b^3)/(exp(2*x) + 1) - x/a - ((2*a)/b ^2 + (8*exp(x))/(3*b))/(2*exp(2*x) + exp(4*x) + 1) + (log(exp(2*x) + 1)*(2 *a*b^2 - a^3))/b^4 + (8*exp(x))/(3*b*(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1)) + (log(a + 2*b*exp(x) + a*exp(2*x))*(a^4 + b^4 - 2*a^2*b^2))/(a*b^4)