Integrand size = 9, antiderivative size = 33 \[ \int \text {sech}^8(x) \tanh ^6(x) \, dx=\frac {\tanh ^7(x)}{7}-\frac {\tanh ^9(x)}{3}+\frac {3 \tanh ^{11}(x)}{11}-\frac {\tanh ^{13}(x)}{13} \]
Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(33)=66\).
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.03 \[ \int \text {sech}^8(x) \tanh ^6(x) \, dx=\frac {16 \tanh (x)}{3003}+\frac {8 \text {sech}^2(x) \tanh (x)}{3003}+\frac {2 \text {sech}^4(x) \tanh (x)}{1001}+\frac {5 \text {sech}^6(x) \tanh (x)}{3003}-\frac {53}{429} \text {sech}^8(x) \tanh (x)+\frac {27}{143} \text {sech}^{10}(x) \tanh (x)-\frac {1}{13} \text {sech}^{12}(x) \tanh (x) \]
(16*Tanh[x])/3003 + (8*Sech[x]^2*Tanh[x])/3003 + (2*Sech[x]^4*Tanh[x])/100 1 + (5*Sech[x]^6*Tanh[x])/3003 - (53*Sech[x]^8*Tanh[x])/429 + (27*Sech[x]^ 10*Tanh[x])/143 - (Sech[x]^12*Tanh[x])/13
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 25, 3087, 244, 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.
\(\displaystyle \int \tanh ^6(x) \text {sech}^8(x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (i x)^6 \left (-\sec (i x)^8\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \sec (i x)^8 \tan (i x)^6dx\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle i \int -\tanh ^6(x) \left (1-\tanh ^2(x)\right )^3d(i \tanh (x))\) |
\(\Big \downarrow \) 244 |
\(\displaystyle i \int \left (\tanh ^{12}(x)-3 \tanh ^{10}(x)+3 \tanh ^8(x)-\tanh ^6(x)\right )d(i \tanh (x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i \left (\frac {1}{13} i \tanh ^{13}(x)-\frac {3}{11} i \tanh ^{11}(x)+\frac {1}{3} i \tanh ^9(x)-\frac {1}{7} i \tanh ^7(x)\right )\) |
3.1.100.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79
\[\frac {\tanh \left (x \right )^{7}}{7}-\frac {\tanh \left (x \right )^{9}}{3}+\frac {3 \tanh \left (x \right )^{11}}{11}-\frac {\tanh \left (x \right )^{13}}{13}\]
Leaf count of result is larger than twice the leaf count of optimal. 778 vs. \(2 (25) = 50\).
Time = 0.23 (sec) , antiderivative size = 778, normalized size of antiderivative = 23.58 \[ \int \text {sech}^8(x) \tanh ^6(x) \, dx=\text {Too large to display} \]
-64/3003*(1502*cosh(x)^9 + 13518*cosh(x)*sinh(x)^8 + 1501*sinh(x)^9 + (540 36*cosh(x)^2 - 4511)*sinh(x)^7 - 4498*cosh(x)^7 + 14*(9012*cosh(x)^3 - 224 9*cosh(x))*sinh(x)^6 + 3*(63042*cosh(x)^4 - 31577*cosh(x)^2 + 2990)*sinh(x )^5 + 9048*cosh(x)^5 + 2*(94626*cosh(x)^5 - 78715*cosh(x)^3 + 22620*cosh(x ))*sinh(x)^4 + (126084*cosh(x)^6 - 157885*cosh(x)^4 + 89700*cosh(x)^2 - 82 94)*sinh(x)^3 - 8008*cosh(x)^3 + 6*(9012*cosh(x)^7 - 15743*cosh(x)^5 + 150 80*cosh(x)^3 - 4004*cosh(x))*sinh(x)^2 + (13509*cosh(x)^8 - 31577*cosh(x)^ 6 + 44850*cosh(x)^4 - 24882*cosh(x)^2 + 6292)*sinh(x) + 4004*cosh(x))/(cos h(x)^17 + 17*cosh(x)*sinh(x)^16 + sinh(x)^17 + (136*cosh(x)^2 + 13)*sinh(x )^15 + 13*cosh(x)^15 + 5*(136*cosh(x)^3 + 39*cosh(x))*sinh(x)^14 + (2380*c osh(x)^4 + 1365*cosh(x)^2 + 78)*sinh(x)^13 + 78*cosh(x)^13 + 13*(476*cosh( x)^5 + 455*cosh(x)^3 + 78*cosh(x))*sinh(x)^12 + 13*(952*cosh(x)^6 + 1365*c osh(x)^4 + 468*cosh(x)^2 + 22)*sinh(x)^11 + 286*cosh(x)^11 + 143*(136*cosh (x)^7 + 273*cosh(x)^5 + 156*cosh(x)^3 + 22*cosh(x))*sinh(x)^10 + (24310*co sh(x)^8 + 65065*cosh(x)^6 + 55770*cosh(x)^4 + 15730*cosh(x)^2 + 714)*sinh( x)^9 + 716*cosh(x)^9 + (24310*cosh(x)^9 + 83655*cosh(x)^7 + 100386*cosh(x) ^5 + 47190*cosh(x)^3 + 6444*cosh(x))*sinh(x)^8 + (19448*cosh(x)^10 + 83655 *cosh(x)^8 + 133848*cosh(x)^6 + 94380*cosh(x)^4 + 25704*cosh(x)^2 + 1274)* sinh(x)^7 + 1300*cosh(x)^7 + (12376*cosh(x)^11 + 65065*cosh(x)^9 + 133848* cosh(x)^7 + 132132*cosh(x)^5 + 60144*cosh(x)^3 + 9100*cosh(x))*sinh(x)^...
\[ \int \text {sech}^8(x) \tanh ^6(x) \, dx=\int \tanh ^{6}{\left (x \right )} \operatorname {sech}^{8}{\left (x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (25) = 50\).
Time = 0.21 (sec) , antiderivative size = 857, normalized size of antiderivative = 25.97 \[ \int \text {sech}^8(x) \tanh ^6(x) \, dx=\text {Too large to display} \]
32/231*e^(-2*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e ^(-18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) + 64/77*e^(-4*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^ (-18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) + 6 4/21*e^(-6*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1 287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^( -18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) - 51 2/21*e^(-8*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1 287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^( -18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) + 76 8/7*e^(-10*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1 287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^( -18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) - 12 16/7*e^(-12*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^ (-18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) + 1 92*e^(-14*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 12 87*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e...
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.00 \[ \int \text {sech}^8(x) \tanh ^6(x) \, dx=-\frac {32 \, {\left (3003 \, e^{\left (18 \, x\right )} - 9009 \, e^{\left (16 \, x\right )} + 18018 \, e^{\left (14 \, x\right )} - 16302 \, e^{\left (12 \, x\right )} + 10296 \, e^{\left (10 \, x\right )} - 2288 \, e^{\left (8 \, x\right )} + 286 \, e^{\left (6 \, x\right )} + 78 \, e^{\left (4 \, x\right )} + 13 \, e^{\left (2 \, x\right )} + 1\right )}}{3003 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{13}} \]
-32/3003*(3003*e^(18*x) - 9009*e^(16*x) + 18018*e^(14*x) - 16302*e^(12*x) + 10296*e^(10*x) - 2288*e^(8*x) + 286*e^(6*x) + 78*e^(4*x) + 13*e^(2*x) + 1)/(e^(2*x) + 1)^13
Time = 2.13 (sec) , antiderivative size = 820, normalized size of antiderivative = 24.85 \[ \int \text {sech}^8(x) \tanh ^6(x) \, dx=\text {Too large to display} \]
- ((64*exp(4*x))/143 - (256*exp(2*x))/429 + 80/429)/(6*exp(2*x) + 15*exp(4 *x) + 20*exp(6*x) + 15*exp(8*x) + 6*exp(10*x) + exp(12*x) + 1) - ((64*exp( 2*x))/143 - (768*exp(4*x))/143 + (3200*exp(6*x))/143 - (6400*exp(8*x))/143 + (6720*exp(10*x))/143 - (3584*exp(12*x))/143 + (768*exp(14*x))/143)/(11* exp(2*x) + 55*exp(4*x) + 165*exp(6*x) + 330*exp(8*x) + 462*exp(10*x) + 462 *exp(12*x) + 330*exp(14*x) + 165*exp(16*x) + 55*exp(18*x) + 11*exp(20*x) + exp(22*x) + 1) - ((160*exp(2*x))/143 - (256*exp(4*x))/143 + (128*exp(6*x) )/143 - 640/3003)/(7*exp(2*x) + 21*exp(4*x) + 35*exp(6*x) + 35*exp(8*x) + 21*exp(10*x) + 7*exp(12*x) + exp(14*x) + 1) - ((128*exp(6*x))/13 - (768*ex p(8*x))/13 + (1920*exp(10*x))/13 - (2560*exp(12*x))/13 + (1920*exp(14*x))/ 13 - (768*exp(16*x))/13 + (128*exp(18*x))/13)/(13*exp(2*x) + 78*exp(4*x) + 286*exp(6*x) + 715*exp(8*x) + 1287*exp(10*x) + 1716*exp(12*x) + 1716*exp( 14*x) + 1287*exp(16*x) + 715*exp(18*x) + 286*exp(20*x) + 78*exp(22*x) + 13 *exp(24*x) + exp(26*x) + 1) - ((560*exp(4*x))/143 - (640*exp(2*x))/429 - ( 1792*exp(6*x))/429 + (224*exp(8*x))/143 + 80/429)/(8*exp(2*x) + 28*exp(4*x ) + 56*exp(6*x) + 70*exp(8*x) + 56*exp(10*x) + 28*exp(12*x) + 8*exp(14*x) + exp(16*x) + 1) - ((640*exp(2*x))/429 - (2560*exp(4*x))/429 + (4480*exp(6 *x))/429 - (3584*exp(8*x))/429 + (1792*exp(10*x))/715 - 256/2145)/(9*exp(2 *x) + 36*exp(4*x) + 84*exp(6*x) + 126*exp(8*x) + 126*exp(10*x) + 84*exp(12 *x) + 36*exp(14*x) + 9*exp(16*x) + exp(18*x) + 1) - ((32*exp(4*x))/13 -...