Integrand size = 23, antiderivative size = 70 \[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=-\frac {(a+2 b) \log (\cosh (c+d x))}{b^2 d}+\frac {(a+b)^2 \log \left (b+a \cosh ^2(c+d x)\right )}{2 a b^2 d}-\frac {\text {sech}^2(c+d x)}{2 b d} \]
-(a+2*b)*ln(cosh(d*x+c))/b^2/d+1/2*(a+b)^2*ln(b+a*cosh(d*x+c)^2)/a/b^2/d-1 /2*sech(d*x+c)^2/b/d
Time = 0.33 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.40 \[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=-\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^2(c+d x) \left (2 a (a+2 b) \log (\cosh (c+d x))-(a+b)^2 \log \left (a+b+a \sinh ^2(c+d x)\right )+a b \text {sech}^2(c+d x)\right )}{4 a b^2 d \left (a+b \text {sech}^2(c+d x)\right )} \]
-1/4*((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(2*a*(a + 2*b)*Log[C osh[c + d*x]] - (a + b)^2*Log[a + b + a*Sinh[c + d*x]^2] + a*b*Sech[c + d* x]^2))/(a*b^2*d*(a + b*Sech[c + d*x]^2))
Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.90, 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.
\(\displaystyle \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \tan (i c+i d x)^5}{a+b \sec (i c+i d x)^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\tan (i c+i d x)^5}{b \sec (i c+i d x)^2+a}dx\) |
\(\Big \downarrow \) 4626 |
\(\displaystyle \frac {\int \frac {\left (1-\cosh ^2(c+d x)\right )^2 \text {sech}^3(c+d x)}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {\int \frac {\left (1-\cosh ^2(c+d x)\right )^2 \text {sech}^2(c+d x)}{a \cosh ^2(c+d x)+b}d\cosh ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\int \left (\frac {(a+b)^2}{b^2 \left (a \cosh ^2(c+d x)+b\right )}+\frac {\text {sech}^2(c+d x)}{b}+\frac {(-a-2 b) \text {sech}(c+d x)}{b^2}\right )d\cosh ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {(a+b)^2 \log \left (a \cosh ^2(c+d x)+b\right )}{a b^2}-\frac {(a+2 b) \log \left (\cosh ^2(c+d x)\right )}{b^2}-\frac {\text {sech}(c+d x)}{b}}{2 d}\) |
(-(((a + 2*b)*Log[Cosh[c + d*x]^2])/b^2) + ((a + b)^2*Log[b + a*Cosh[c + d *x]^2])/(a*b^2) - Sech[c + d*x]/b)/(2*d)
3.2.38.3.1 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. \(182\) vs. \(2(66)=132\).
Time = 4.32 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.61
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (\frac {1}{4} a^{2}+\frac {1}{2} a b +\frac {1}{4} b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )}{a \,b^{2}}-\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\left (a +2 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )+\frac {2 b}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )^{2}}-\frac {2 b}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}}{b^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}}{d}\) | \(183\) |
default | \(\frac {\frac {2 \left (\frac {1}{4} a^{2}+\frac {1}{2} a b +\frac {1}{4} b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )}{a \,b^{2}}-\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\left (a +2 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )+\frac {2 b}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )^{2}}-\frac {2 b}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}}{b^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}}{d}\) | \(183\) |
risch | \(-\frac {x}{a}-\frac {2 c}{d a}-\frac {2 \,{\mathrm e}^{2 d x +2 c}}{b d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+1\right ) a}{b^{2} d}-\frac {2 \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{b d}+\frac {a \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 b^{2} d}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{b d}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a d}\) | \(205\) |
1/d*(2/a/b^2*(1/4*a^2+1/2*a*b+1/4*b^2)*ln(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2 *d*x+1/2*c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)-1 /a*ln(1+tanh(1/2*d*x+1/2*c))-1/b^2*((a+2*b)*ln(tanh(1/2*d*x+1/2*c)^2+1)+2* b/(tanh(1/2*d*x+1/2*c)^2+1)^2-2*b/(tanh(1/2*d*x+1/2*c)^2+1))-1/a*ln(tanh(1 /2*d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 736 vs. \(2 (66) = 132\).
Time = 0.33 (sec) , antiderivative size = 736, normalized size of antiderivative = 10.51 \[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=-\frac {2 \, b^{2} d x \cosh \left (d x + c\right )^{4} + 8 \, b^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 2 \, b^{2} d x \sinh \left (d x + c\right )^{4} + 2 \, b^{2} d x + 4 \, {\left (b^{2} d x + a b\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (3 \, b^{2} d x \cosh \left (d x + c\right )^{2} + b^{2} d x + a b\right )} \sinh \left (d x + c\right )^{2} - {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2} + 4 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b + b^{2}\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 ({\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, a b\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 2 \, a b + 4 \, {\left ({\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a 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 (b^{2} d x \cosh \left (d x + c\right )^{3} + {\left (b^{2} d x + a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (a b^{2} d \cosh \left (d x + c\right )^{4} + 4 \, a b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a b^{2} d \sinh \left (d x + c\right )^{4} + 2 \, a b^{2} d \cosh \left (d x + c\right )^{2} + a b^{2} d + 2 \, {\left (3 \, a b^{2} d \cosh \left (d x + c\right )^{2} + a b^{2} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a b^{2} d \cosh \left (d x + c\right )^{3} + a b^{2} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \]
-1/2*(2*b^2*d*x*cosh(d*x + c)^4 + 8*b^2*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 2*b^2*d*x*sinh(d*x + c)^4 + 2*b^2*d*x + 4*(b^2*d*x + a*b)*cosh(d*x + c)^ 2 + 4*(3*b^2*d*x*cosh(d*x + c)^2 + b^2*d*x + a*b)*sinh(d*x + c)^2 - ((a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh( d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b + b^2)*c osh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^2 + a^2 + 2*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*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)*sinh(d*x + c) + sinh(d*x + c)^2)) + 2*((a^2 + 2*a*b)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b)*sinh(d*x + c )^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b)*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 2*a*b + 4*((a^2 + 2*a*b)*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/( cosh(d*x + c) - sinh(d*x + c))) + 8*(b^2*d*x*cosh(d*x + c)^3 + (b^2*d*x + a*b)*cosh(d*x + c))*sinh(d*x + c))/(a*b^2*d*cosh(d*x + c)^4 + 4*a*b^2*d*co sh(d*x + c)*sinh(d*x + c)^3 + a*b^2*d*sinh(d*x + c)^4 + 2*a*b^2*d*cosh(d*x + c)^2 + a*b^2*d + 2*(3*a*b^2*d*cosh(d*x + c)^2 + a*b^2*d)*sinh(d*x + c)^ 2 + 4*(a*b^2*d*cosh(d*x + c)^3 + a*b^2*d*cosh(d*x + c))*sinh(d*x + c))
\[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int \frac {\tanh ^{5}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.87 \[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {d x + c}{a d} - \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (2 \, b e^{\left (-2 \, d x - 2 \, c\right )} + b e^{\left (-4 \, d x - 4 \, c\right )} + b\right )} d} - \frac {{\left (a + 2 \, b\right )} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{b^{2} d} + \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, a b^{2} d} \]
(d*x + c)/(a*d) - 2*e^(-2*d*x - 2*c)/((2*b*e^(-2*d*x - 2*c) + b*e^(-4*d*x - 4*c) + b)*d) - (a + 2*b)*log(e^(-2*d*x - 2*c) + 1)/(b^2*d) + 1/2*(a^2 + 2*a*b + b^2)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a *b^2*d)
\[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int { \frac {\tanh \left (d x + c\right )^{5}}{b \operatorname {sech}\left (d x + c\right )^{2} + a} \,d x } \]
Time = 2.46 (sec) , antiderivative size = 421, normalized size of antiderivative = 6.01 \[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {2}{b\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {2}{b\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {x}{a}-\frac {\ln \left (39\,a\,b^7+243\,a^7\,b+27\,a^8+2\,b^8+289\,a^2\,b^6+1017\,a^3\,b^5+1791\,a^4\,b^4+1701\,a^5\,b^3+891\,a^6\,b^2+27\,a^8\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+2\,b^8\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+39\,a\,b^7\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+243\,a^7\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+289\,a^2\,b^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1017\,a^3\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1791\,a^4\,b^4\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1701\,a^5\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+891\,a^6\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (a+2\,b\right )}{b^2\,d}+\frac {\ln \left (a\,b^2+6\,a^2\,b+3\,a^3+6\,a^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+3\,a^3\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}+4\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+26\,a\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+24\,a^2\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+a\,b^2\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}+6\,a^2\,b\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}\right )\,\left (a^2+2\,a\,b+b^2\right )}{2\,a\,b^2\,d} \]
2/(b*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - 2/(b*d*(exp(2*c + 2* d*x) + 1)) - x/a - (log(39*a*b^7 + 243*a^7*b + 27*a^8 + 2*b^8 + 289*a^2*b^ 6 + 1017*a^3*b^5 + 1791*a^4*b^4 + 1701*a^5*b^3 + 891*a^6*b^2 + 27*a^8*exp( 2*c)*exp(2*d*x) + 2*b^8*exp(2*c)*exp(2*d*x) + 39*a*b^7*exp(2*c)*exp(2*d*x) + 243*a^7*b*exp(2*c)*exp(2*d*x) + 289*a^2*b^6*exp(2*c)*exp(2*d*x) + 1017* a^3*b^5*exp(2*c)*exp(2*d*x) + 1791*a^4*b^4*exp(2*c)*exp(2*d*x) + 1701*a^5* b^3*exp(2*c)*exp(2*d*x) + 891*a^6*b^2*exp(2*c)*exp(2*d*x))*(a + 2*b))/(b^2 *d) + (log(a*b^2 + 6*a^2*b + 3*a^3 + 6*a^3*exp(2*c)*exp(2*d*x) + 3*a^3*exp (4*c)*exp(4*d*x) + 4*b^3*exp(2*c)*exp(2*d*x) + 26*a*b^2*exp(2*c)*exp(2*d*x ) + 24*a^2*b*exp(2*c)*exp(2*d*x) + a*b^2*exp(4*c)*exp(4*d*x) + 6*a^2*b*exp (4*c)*exp(4*d*x))*(2*a*b + a^2 + b^2))/(2*a*b^2*d)