Integrand size = 23, antiderivative size = 76 \[ \int \frac {\tanh ^5(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {(a+b)^2}{2 a^2 b d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\log (\cosh (c+d x))}{b^2 d}+\frac {\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \log \left (b+a \cosh ^2(c+d x)\right )}{2 d} \]
1/2*(a+b)^2/a^2/b/d/(b+a*cosh(d*x+c)^2)+ln(cosh(d*x+c))/b^2/d+1/2*(1/a^2-1 /b^2)*ln(b+a*cosh(d*x+c)^2)/d
Time = 0.49 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.43 \[ \int \frac {\tanh ^5(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {(a+2 b+a \cosh (2 c+2 d x))^2 \left (\frac {(a+b)^2}{a^2 b \left (b+a \cosh ^2(c+d x)\right )}+\frac {2 \log (\cosh (c+d x))}{b^2}+\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \log \left (b+a \cosh ^2(c+d x)\right )\right ) \text {sech}^4(c+d x)}{8 d \left (a+b \text {sech}^2(c+d x)\right )^2} \]
((a + 2*b + a*Cosh[2*c + 2*d*x])^2*((a + b)^2/(a^2*b*(b + a*Cosh[c + d*x]^ 2)) + (2*Log[Cosh[c + d*x]])/b^2 + (a^(-2) - b^(-2))*Log[b + a*Cosh[c + d* x]^2])*Sech[c + d*x]^4)/(8*d*(a + b*Sech[c + d*x]^2)^2)
Time = 0.32 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 26, 4626, 354, 99, 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.
| \(\displaystyle \int \frac {\tanh ^5(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \tan (i c+i d x)^5}{\left (a+b \sec (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\tan (i c+i d x)^5}{\left (b \sec (i c+i d x)^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 4626 |
| \(\displaystyle \frac {\int \frac {\left (1-\cosh ^2(c+d x)\right )^2 \text {sech}(c+d x)}{\left (a \cosh ^2(c+d x)+b\right )^2}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\left (1-\cosh ^2(c+d x)\right )^2 \text {sech}(c+d x)}{\left (a \cosh ^2(c+d x)+b\right )^2}d\cosh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (-\frac {(a+b)^2}{a b \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {\text {sech}(c+d x)}{b^2}+\frac {b^2-a^2}{a b^2 \left (a \cosh ^2(c+d x)+b\right )}\right )d\cosh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \log \left (a \cosh ^2(c+d x)+b\right )+\frac {(a+b)^2}{a^2 b \left (a \cosh ^2(c+d x)+b\right )}+\frac {\log \left (\cosh ^2(c+d x)\right )}{b^2}}{2 d}\) |
((a + b)^2/(a^2*b*(b + a*Cosh[c + d*x]^2)) + Log[Cosh[c + d*x]^2]/b^2 + (a ^(-2) - b^(-2))*Log[b + a*Cosh[c + d*x]^2])/(2*d)
3.2.48.3.1 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_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> Module[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-(f *ff^(m + n*p - 1))^(-1) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff* x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n} , x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(183\) vs. \(2(72)=144\).
Time = 22.78 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.42
| method | result | size |
| risch | \(-\frac {x}{a^{2}}-\frac {2 c}{a^{2} d}+\frac {2 \left (a^{2}+2 a b +b^{2}\right ) {\mathrm e}^{2 d x +2 c}}{a^{2} b d \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{b^{2} d}-\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 d \,a^{2}}\) | \(184\) |
| derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )}{b^{2}}-\frac {\left (a +b \right ) \left (\frac {2 a b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (a -b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )}{2}\right )}{a^{2} b^{2}}}{d}\) | \(209\) |
| default | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )}{b^{2}}-\frac {\left (a +b \right ) \left (\frac {2 a b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (a -b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )}{2}\right )}{a^{2} b^{2}}}{d}\) | \(209\) |
-x/a^2-2/a^2/d*c+2/a^2/b*(a^2+2*a*b+b^2)/d*exp(2*d*x+2*c)/(a*exp(4*d*x+4*c )+2*exp(2*d*x+2*c)*a+4*b*exp(2*d*x+2*c)+a)+1/b^2/d*ln(exp(2*d*x+2*c)+1)-1/ 2/d/b^2*ln(exp(4*d*x+4*c)+2*(a+2*b)/a*exp(2*d*x+2*c)+1)+1/2/d/a^2*ln(exp(4 *d*x+4*c)+2*(a+2*b)/a*exp(2*d*x+2*c)+1)
Leaf count of result is larger than twice the leaf count of optimal. 853 vs. \(2 (72) = 144\).
Time = 0.32 (sec) , antiderivative size = 853, normalized size of antiderivative = 11.22 \[ \int \frac {\tanh ^5(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {2 \, a b^{2} d x \cosh \left (d x + c\right )^{4} + 8 \, a b^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 2 \, a b^{2} d x \sinh \left (d x + c\right )^{4} + 2 \, a b^{2} d x - 4 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3} - {\left (a b^{2} + 2 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (3 \, a b^{2} d x \cosh \left (d x + c\right )^{2} - a^{2} b - 2 \, a b^{2} - b^{3} + {\left (a b^{2} + 2 \, b^{3}\right )} d x\right )} \sinh \left (d x + c\right )^{2} + {\left ({\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{3} - a b^{2}\right )} \sinh \left (d x + c\right )^{4} + a^{3} - a b^{2} + 2 \, {\left (a^{3} + 2 \, a^{2} b - a b^{2} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{3} + 2 \, a^{2} b - a b^{2} - 2 \, b^{3} + 3 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{3} + 2 \, a^{2} b - a b^{2} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + a + 2 \, b\right )}}{\cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}}\right ) - 2 \, {\left (a^{3} \cosh \left (d x + c\right )^{4} + 4 \, a^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} \sinh \left (d x + c\right )^{4} + a^{3} + 2 \, {\left (a^{3} + 2 \, a^{2} b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} \cosh \left (d x + c\right )^{2} + a^{3} + 2 \, a^{2} b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} \cosh \left (d x + c\right )^{3} + {\left (a^{3} + 2 \, a^{2} b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 8 \, {\left (a b^{2} d x \cosh \left (d x + c\right )^{3} - {\left (a^{2} b + 2 \, a b^{2} + b^{3} - {\left (a b^{2} + 2 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (a^{3} b^{2} d \cosh \left (d x + c\right )^{4} + 4 \, a^{3} b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} b^{2} d \sinh \left (d x + c\right )^{4} + a^{3} b^{2} d + 2 \, {\left (a^{3} b^{2} + 2 \, a^{2} b^{3}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} b^{2} d \cosh \left (d x + c\right )^{2} + {\left (a^{3} b^{2} + 2 \, a^{2} b^{3}\right )} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} b^{2} d \cosh \left (d x + c\right )^{3} + {\left (a^{3} b^{2} + 2 \, a^{2} b^{3}\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \]
-1/2*(2*a*b^2*d*x*cosh(d*x + c)^4 + 8*a*b^2*d*x*cosh(d*x + c)*sinh(d*x + c )^3 + 2*a*b^2*d*x*sinh(d*x + c)^4 + 2*a*b^2*d*x - 4*(a^2*b + 2*a*b^2 + b^3 - (a*b^2 + 2*b^3)*d*x)*cosh(d*x + c)^2 + 4*(3*a*b^2*d*x*cosh(d*x + c)^2 - a^2*b - 2*a*b^2 - b^3 + (a*b^2 + 2*b^3)*d*x)*sinh(d*x + c)^2 + ((a^3 - a* b^2)*cosh(d*x + c)^4 + 4*(a^3 - a*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^ 3 - a*b^2)*sinh(d*x + c)^4 + a^3 - a*b^2 + 2*(a^3 + 2*a^2*b - a*b^2 - 2*b^ 3)*cosh(d*x + c)^2 + 2*(a^3 + 2*a^2*b - a*b^2 - 2*b^3 + 3*(a^3 - a*b^2)*co sh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^3 - a*b^2)*cosh(d*x + c)^3 + (a^3 + 2*a^2*b - a*b^2 - 2*b^3)*cosh(d*x + c))*sinh(d*x + c))*log(2*(a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*s inh(d*x + c) + sinh(d*x + c)^2)) - 2*(a^3*cosh(d*x + c)^4 + 4*a^3*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*sinh(d*x + c)^4 + a^3 + 2*(a^3 + 2*a^2*b)*cosh (d*x + c)^2 + 2*(3*a^3*cosh(d*x + c)^2 + a^3 + 2*a^2*b)*sinh(d*x + c)^2 + 4*(a^3*cosh(d*x + c)^3 + (a^3 + 2*a^2*b)*cosh(d*x + c))*sinh(d*x + c))*log (2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 8*(a*b^2*d*x*cosh(d*x + c)^3 - (a^2*b + 2*a*b^2 + b^3 - (a*b^2 + 2*b^3)*d*x)*cosh(d*x + c))*sinh (d*x + c))/(a^3*b^2*d*cosh(d*x + c)^4 + 4*a^3*b^2*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*b^2*d*sinh(d*x + c)^4 + a^3*b^2*d + 2*(a^3*b^2 + 2*a^2*b^3)* d*cosh(d*x + c)^2 + 2*(3*a^3*b^2*d*cosh(d*x + c)^2 + (a^3*b^2 + 2*a^2*b^3) *d)*sinh(d*x + c)^2 + 4*(a^3*b^2*d*cosh(d*x + c)^3 + (a^3*b^2 + 2*a^2*b...
\[ \int \frac {\tanh ^5(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {\tanh ^{5}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (72) = 144\).
Time = 0.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.03 \[ \int \frac {\tanh ^5(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{3} b e^{\left (-4 \, d x - 4 \, c\right )} + a^{3} b + 2 \, {\left (a^{3} b + 2 \, a^{2} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} + \frac {d x + c}{a^{2} d} + \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{b^{2} d} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, a^{2} b^{2} d} \]
2*(a^2 + 2*a*b + b^2)*e^(-2*d*x - 2*c)/((a^3*b*e^(-4*d*x - 4*c) + a^3*b + 2*(a^3*b + 2*a^2*b^2)*e^(-2*d*x - 2*c))*d) + (d*x + c)/(a^2*d) + log(e^(-2 *d*x - 2*c) + 1)/(b^2*d) - 1/2*(a^2 - b^2)*log(2*(a + 2*b)*e^(-2*d*x - 2*c ) + a*e^(-4*d*x - 4*c) + a)/(a^2*b^2*d)
\[ \int \frac {\tanh ^5(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\tanh \left (d x + c\right )^{5}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\tanh ^5(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4\,{\mathrm {tanh}\left (c+d\,x\right )}^5}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \]