3.994 \(\int \text {sech}^2(x) \tanh ^6(x) (1-\tanh ^2(x))^3 \, dx\)
Optimal. Leaf size=33 \[ -\frac {1}{13} \tanh ^{13}(x)+\frac {3 \tanh ^{11}(x)}{11}-\frac {\tanh ^9(x)}{3}+\frac {\tanh ^7(x)}{7} \]
[Out]
1/7*tanh(x)^7-1/3*tanh(x)^9+3/11*tanh(x)^11-1/13*tanh(x)^13
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 33, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used =
{3657, 2607, 270} \[ -\frac {1}{13} \tanh ^{13}(x)+\frac {3 \tanh ^{11}(x)}{11}-\frac {\tanh ^9(x)}{3}+\frac {\tanh ^7(x)}{7} \]
Antiderivative was successfully verified.
[In]
Int[Sech[x]^2*Tanh[x]^6*(1 - Tanh[x]^2)^3,x]
[Out]
Tanh[x]^7/7 - Tanh[x]^9/3 + (3*Tanh[x]^11)/11 - Tanh[x]^13/13
Rule 270
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Rule 2607
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[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])
Rule 3657
Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]
Rubi steps
\begin {align*} \int \text {sech}^2(x) \tanh ^6(x) \left (1-\tanh ^2(x)\right )^3 \, dx &=\int \text {sech}^8(x) \tanh ^6(x) \, dx\\ &=i \operatorname {Subst}\left (\int x^6 \left (1+x^2\right )^3 \, dx,x,i \tanh (x)\right )\\ &=i \operatorname {Subst}\left (\int \left (x^6+3 x^8+3 x^{10}+x^{12}\right ) \, dx,x,i \tanh (x)\right )\\ &=\frac {\tanh ^7(x)}{7}-\frac {\tanh ^9(x)}{3}+\frac {3 \tanh ^{11}(x)}{11}-\frac {\tanh ^{13}(x)}{13}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.03, size = 67, normalized size = 2.03 \[ \frac {16 \tanh (x)}{3003}-\frac {1}{13} \tanh (x) \text {sech}^{12}(x)+\frac {27}{143} \tanh (x) \text {sech}^{10}(x)-\frac {53}{429} \tanh (x) \text {sech}^8(x)+\frac {5 \tanh (x) \text {sech}^6(x)}{3003}+\frac {2 \tanh (x) \text {sech}^4(x)}{1001}+\frac {8 \tanh (x) \text {sech}^2(x)}{3003} \]
Antiderivative was successfully verified.
[In]
Integrate[Sech[x]^2*Tanh[x]^6*(1 - Tanh[x]^2)^3,x]
[Out]
(16*Tanh[x])/3003 + (8*Sech[x]^2*Tanh[x])/3003 + (2*Sech[x]^4*Tanh[x])/1001 + (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
________________________________________________________________________________________
fricas [B] time = 0.44, size = 778, normalized size = 23.58 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(x)^2*tanh(x)^6*(1-tanh(x)^2)^3,x, algorithm="fricas")
[Out]
-64/3003*(1502*cosh(x)^9 + 13518*cosh(x)*sinh(x)^8 + 1501*sinh(x)^9 + (54036*cosh(x)^2 - 4511)*sinh(x)^7 - 449
8*cosh(x)^7 + 14*(9012*cosh(x)^3 - 2249*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 - 8294)*sinh(x)^3 - 8008*cosh(x)^3 + 6*(9012*cosh(x)^7 - 15743*cosh(x)^5 +
15080*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))/(cosh(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*cosh(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*cosh(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*cosh(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 + 644
4*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)^6 + (6188*cosh(x)^12 + 39039*cosh(x)^10 + 100386*cosh(x)^8 + 13
2132*cosh(x)^6 + 89964*cosh(x)^4 + 26754*cosh(x)^2 + 1638)*sinh(x)^5 + 1794*cosh(x)^5 + (2380*cosh(x)^13 + 177
45*cosh(x)^11 + 55770*cosh(x)^9 + 94380*cosh(x)^7 + 90216*cosh(x)^5 + 45500*cosh(x)^3 + 8970*cosh(x))*sinh(x)^
4 + (680*cosh(x)^14 + 5915*cosh(x)^12 + 22308*cosh(x)^10 + 47190*cosh(x)^8 + 59976*cosh(x)^6 + 44590*cosh(x)^4
+ 16380*cosh(x)^2 + 1430)*sinh(x)^3 + 2002*cosh(x)^3 + (136*cosh(x)^15 + 1365*cosh(x)^13 + 6084*cosh(x)^11 +
15730*cosh(x)^9 + 25776*cosh(x)^7 + 27300*cosh(x)^5 + 17940*cosh(x)^3 + 6006*cosh(x))*sinh(x)^2 + (17*cosh(x)^
16 + 195*cosh(x)^14 + 1014*cosh(x)^12 + 3146*cosh(x)^10 + 6426*cosh(x)^8 + 8918*cosh(x)^6 + 8190*cosh(x)^4 + 4
290*cosh(x)^2 + 572)*sinh(x) + 2002*cosh(x))
________________________________________________________________________________________
giac [B] time = 0.16, size = 66, normalized size = 2.00 \[ -\frac {32 \, {\left (3003 \, e^{\left (18 \, x\right )} - 9009 \, e^{\left (16 \, x\right )} + 18018 \, e^{\left (14 \, x\right )} - 16302 \, e^{\left (12 \, x\right )} + 10296 \, e^{\left (10 \, x\right )} - 2288 \, e^{\left (8 \, x\right )} + 286 \, e^{\left (6 \, x\right )} + 78 \, e^{\left (4 \, x\right )} + 13 \, e^{\left (2 \, x\right )} + 1\right )}}{3003 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(x)^2*tanh(x)^6*(1-tanh(x)^2)^3,x, algorithm="giac")
[Out]
-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) + 28
6*e^(6*x) + 78*e^(4*x) + 13*e^(2*x) + 1)/(e^(2*x) + 1)^13
________________________________________________________________________________________
maple [B] time = 0.64, size = 306, normalized size = 9.27 \[ -\frac {\sinh ^{5}\relax (x )}{2 \cosh \relax (x )^{7}}-\frac {5 \left (\sinh ^{3}\relax (x )\right )}{8 \cosh \relax (x )^{7}}-\frac {5 \sinh \relax (x )}{16 \cosh \relax (x )^{7}}+\frac {5 \left (\frac {16}{35}+\frac {\mathrm {sech}\relax (x )^{6}}{7}+\frac {6 \mathrm {sech}\relax (x )^{4}}{35}+\frac {8 \mathrm {sech}\relax (x )^{2}}{35}\right ) \tanh \relax (x )}{16}+\frac {3 \left (\sinh ^{7}\relax (x )\right )}{2 \cosh \relax (x )^{9}}+\frac {21 \left (\sinh ^{5}\relax (x )\right )}{8 \cosh \relax (x )^{9}}+\frac {35 \left (\sinh ^{3}\relax (x )\right )}{16 \cosh \relax (x )^{9}}+\frac {105 \sinh \relax (x )}{128 \cosh \relax (x )^{9}}-\frac {105 \left (\frac {128}{315}+\frac {\mathrm {sech}\relax (x )^{8}}{9}+\frac {8 \mathrm {sech}\relax (x )^{6}}{63}+\frac {16 \mathrm {sech}\relax (x )^{4}}{105}+\frac {64 \mathrm {sech}\relax (x )^{2}}{315}\right ) \tanh \relax (x )}{128}-\frac {3 \left (\sinh ^{9}\relax (x )\right )}{2 \cosh \relax (x )^{11}}-\frac {27 \left (\sinh ^{7}\relax (x )\right )}{8 \cosh \relax (x )^{11}}-\frac {63 \left (\sinh ^{5}\relax (x )\right )}{16 \cosh \relax (x )^{11}}-\frac {315 \left (\sinh ^{3}\relax (x )\right )}{128 \cosh \relax (x )^{11}}-\frac {189 \sinh \relax (x )}{256 \cosh \relax (x )^{11}}+\frac {189 \left (\frac {256}{693}+\frac {\mathrm {sech}\relax (x )^{10}}{11}+\frac {10 \mathrm {sech}\relax (x )^{8}}{99}+\frac {80 \mathrm {sech}\relax (x )^{6}}{693}+\frac {32 \mathrm {sech}\relax (x )^{4}}{231}+\frac {128 \mathrm {sech}\relax (x )^{2}}{693}\right ) \tanh \relax (x )}{256}+\frac {\sinh ^{11}\relax (x )}{2 \cosh \relax (x )^{13}}+\frac {11 \left (\sinh ^{9}\relax (x )\right )}{8 \cosh \relax (x )^{13}}+\frac {33 \left (\sinh ^{7}\relax (x )\right )}{16 \cosh \relax (x )^{13}}+\frac {231 \left (\sinh ^{5}\relax (x )\right )}{128 \cosh \relax (x )^{13}}+\frac {231 \left (\sinh ^{3}\relax (x )\right )}{256 \cosh \relax (x )^{13}}+\frac {231 \sinh \relax (x )}{1024 \cosh \relax (x )^{13}}-\frac {231 \left (\frac {1024}{3003}+\frac {\mathrm {sech}\relax (x )^{12}}{13}+\frac {12 \mathrm {sech}\relax (x )^{10}}{143}+\frac {40 \mathrm {sech}\relax (x )^{8}}{429}+\frac {320 \mathrm {sech}\relax (x )^{6}}{3003}+\frac {128 \mathrm {sech}\relax (x )^{4}}{1001}+\frac {512 \mathrm {sech}\relax (x )^{2}}{3003}\right ) \tanh \relax (x )}{1024} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sech(x)^2*tanh(x)^6*(1-tanh(x)^2)^3,x)
[Out]
-1/2*sinh(x)^5/cosh(x)^7-5/8*sinh(x)^3/cosh(x)^7-5/16*sinh(x)/cosh(x)^7+5/16*(16/35+1/7*sech(x)^6+6/35*sech(x)
^4+8/35*sech(x)^2)*tanh(x)+3/2*sinh(x)^7/cosh(x)^9+21/8*sinh(x)^5/cosh(x)^9+35/16*sinh(x)^3/cosh(x)^9+105/128*
sinh(x)/cosh(x)^9-105/128*(128/315+1/9*sech(x)^8+8/63*sech(x)^6+16/105*sech(x)^4+64/315*sech(x)^2)*tanh(x)-3/2
*sinh(x)^9/cosh(x)^11-27/8*sinh(x)^7/cosh(x)^11-63/16*sinh(x)^5/cosh(x)^11-315/128*sinh(x)^3/cosh(x)^11-189/25
6*sinh(x)/cosh(x)^11+189/256*(256/693+1/11*sech(x)^10+10/99*sech(x)^8+80/693*sech(x)^6+32/231*sech(x)^4+128/69
3*sech(x)^2)*tanh(x)+1/2*sinh(x)^11/cosh(x)^13+11/8*sinh(x)^9/cosh(x)^13+33/16*sinh(x)^7/cosh(x)^13+231/128*si
nh(x)^5/cosh(x)^13+231/256*sinh(x)^3/cosh(x)^13+231/1024*sinh(x)/cosh(x)^13-231/1024*(1024/3003+1/13*sech(x)^1
2+12/143*sech(x)^10+40/429*sech(x)^8+320/3003*sech(x)^6+128/1001*sech(x)^4+512/3003*sech(x)^2)*tanh(x)
________________________________________________________________________________________
maxima [A] time = 0.30, size = 25, normalized size = 0.76 \[ -\frac {1}{13} \, \tanh \relax (x)^{13} + \frac {3}{11} \, \tanh \relax (x)^{11} - \frac {1}{3} \, \tanh \relax (x)^{9} + \frac {1}{7} \, \tanh \relax (x)^{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(x)^2*tanh(x)^6*(1-tanh(x)^2)^3,x, algorithm="maxima")
[Out]
-1/13*tanh(x)^13 + 3/11*tanh(x)^11 - 1/3*tanh(x)^9 + 1/7*tanh(x)^7
________________________________________________________________________________________
mupad [B] time = 1.74, size = 820, normalized size = 24.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(tanh(x)^6*(tanh(x)^2 - 1)^3)/cosh(x)^2,x)
[Out]
- ((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))/1
43 + (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*exp(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) + 1
716*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*e
xp(2*x))/429 - (2560*exp(4*x))/429 + (4480*exp(6*x))/429 - (3584*exp(8*x))/429 + (1792*exp(10*x))/715 - 256/21
45)/(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*e
xp(16*x) + exp(18*x) + 1) - ((32*exp(4*x))/13 - (256*exp(6*x))/13 + (800*exp(8*x))/13 - (1280*exp(10*x))/13 +
(1120*exp(12*x))/13 - (512*exp(14*x))/13 + (96*exp(16*x))/13)/(12*exp(2*x) + 66*exp(4*x) + 220*exp(6*x) + 495*
exp(8*x) + 792*exp(10*x) + 924*exp(12*x) + 792*exp(14*x) + 495*exp(16*x) + 220*exp(18*x) + 66*exp(20*x) + 12*e
xp(22*x) + exp(24*x) + 1) - ((128*exp(2*x))/715 - 256/2145)/(5*exp(2*x) + 10*exp(4*x) + 10*exp(6*x) + 5*exp(8*
x) + exp(10*x) + 1) - 32/(715*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1)) - ((960*exp(4*x))/143 - (
768*exp(2*x))/715 - (2560*exp(6*x))/143 + (3360*exp(8*x))/143 - (10752*exp(10*x))/715 + (2688*exp(12*x))/715 +
32/715)/(10*exp(2*x) + 45*exp(4*x) + 120*exp(6*x) + 210*exp(8*x) + 252*exp(10*x) + 210*exp(12*x) + 120*exp(14
*x) + 45*exp(16*x) + 10*exp(18*x) + exp(20*x) + 1)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \tanh ^{6}{\relax (x )} \operatorname {sech}^{2}{\relax (x )}\right )\, dx - \int 3 \tanh ^{8}{\relax (x )} \operatorname {sech}^{2}{\relax (x )}\, dx - \int \left (- 3 \tanh ^{10}{\relax (x )} \operatorname {sech}^{2}{\relax (x )}\right )\, dx - \int \tanh ^{12}{\relax (x )} \operatorname {sech}^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(x)**2*tanh(x)**6*(1-tanh(x)**2)**3,x)
[Out]
-Integral(-tanh(x)**6*sech(x)**2, x) - Integral(3*tanh(x)**8*sech(x)**2, x) - Integral(-3*tanh(x)**10*sech(x)*
*2, x) - Integral(tanh(x)**12*sech(x)**2, x)
________________________________________________________________________________________