∫ sech8(x) tanh6(x) dx [100]

Optimal. Leaf size=33 \[ \frac {\tanh ^7(x)}{7}-\frac {\tanh ^9(x)}{3}+\frac {3 \tanh ^{11}(x)}{11}-\frac {\tanh ^{13}(x)}{13} \]

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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2687, 276} \begin {gather*} -\frac {1}{13} \tanh ^{13}(x)+\frac {3 \tanh ^{11}(x)}{11}-\frac {\tanh ^9(x)}{3}+\frac {\tanh ^7(x)}{7} \end {gather*}

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,

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

\begin {align*} \int \text {sech}^8(x) \tanh ^6(x) \, dx &=i \text {Subst}\left (\int x^6 \left (1+x^2\right )^3 \, dx,x,i \tanh (x)\right )\\ &=i \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(33)=66\).
time = 0.02, size = 67, normalized size = 2.03 \begin {gather*} \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) \end {gather*}

(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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(25)=50\).
time = 0.54, size = 67, normalized size = 2.03


method	result	size

risch	\(-\frac {32 \left (3003 \,{\mathrm e}^{18 x}-9009 \,{\mathrm e}^{16 x}+18018 \,{\mathrm e}^{14 x}-16302 \,{\mathrm e}^{12 x}+10296 \,{\mathrm e}^{10 x}-2288 \,{\mathrm e}^{8 x}+286 \,{\mathrm e}^{6 x}+78 \,{\mathrm e}^{4 x}+13 \,{\mathrm e}^{2 x}+1\right )}{3003 \left (1+{\mathrm e}^{2 x}\right )^{13}}\)	\(67\)

-32/3003*(3003*exp(18*x)-9009*exp(16*x)+18018*exp(14*x)-16302*exp(12*x)+10296*exp(10*x)-2288*exp(8*x)+286*exp(

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (25) = 50\).
time = 0.28, size = 857, normalized size = 25.97 \begin {gather*} \frac {32 \, e^{\left (-2 \, x\right )}}{231 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {64 \, e^{\left (-4 \, x\right )}}{77 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {64 \, e^{\left (-6 \, x\right )}}{21 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} - \frac {512 \, e^{\left (-8 \, x\right )}}{21 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {768 \, e^{\left (-10 \, x\right )}}{7 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} - \frac {1216 \, e^{\left (-12 \, x\right )}}{7 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {192 \, e^{\left (-14 \, x\right )}}{13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1} - \frac {96 \, e^{\left (-16 \, x\right )}}{13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1} + \frac {32 \, e^{\left (-18 \, x\right )}}{13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1} + \frac {32}{3003 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} \end {gather*}

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) + 1

716*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)

+ 64/21*e^(-6*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

) - 512/21*e^(-8*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) + 768/7*e^(-10*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) - 1216/7*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) + 192*e^(-14*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) - 96*e^(-16*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-1

2*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) + 32*e^(-18*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) + 32/3003/(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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 778 vs. \(2 (25) = 50\).
time = 0.35, size = 778, normalized size = 23.58 \begin {gather*} -\frac {64 \, {\left (1502 \, \cosh \left (x\right )^{9} + 13518 \, \cosh \left (x\right ) \sinh \left (x\right )^{8} + 1501 \, \sinh \left (x\right )^{9} + {\left (54036 \, \cosh \left (x\right )^{2} - 4511\right )} \sinh \left (x\right )^{7} - 4498 \, \cosh \left (x\right )^{7} + 14 \, {\left (9012 \, \cosh \left (x\right )^{3} - 2249 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{6} + 3 \, {\left (63042 \, \cosh \left (x\right )^{4} - 31577 \, \cosh \left (x\right )^{2} + 2990\right )} \sinh \left (x\right )^{5} + 9048 \, \cosh \left (x\right )^{5} + 2 \, {\left (94626 \, \cosh \left (x\right )^{5} - 78715 \, \cosh \left (x\right )^{3} + 22620 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + {\left (126084 \, \cosh \left (x\right )^{6} - 157885 \, \cosh \left (x\right )^{4} + 89700 \, \cosh \left (x\right )^{2} - 8294\right )} \sinh \left (x\right )^{3} - 8008 \, \cosh \left (x\right )^{3} + 6 \, {\left (9012 \, \cosh \left (x\right )^{7} - 15743 \, \cosh \left (x\right )^{5} + 15080 \, \cosh \left (x\right )^{3} - 4004 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (13509 \, \cosh \left (x\right )^{8} - 31577 \, \cosh \left (x\right )^{6} + 44850 \, \cosh \left (x\right )^{4} - 24882 \, \cosh \left (x\right )^{2} + 6292\right )} \sinh \left (x\right ) + 4004 \, \cosh \left (x\right )\right )}}{3003 \, {\left (\cosh \left (x\right )^{17} + 17 \, \cosh \left (x\right ) \sinh \left (x\right )^{16} + \sinh \left (x\right )^{17} + {\left (136 \, \cosh \left (x\right )^{2} + 13\right )} \sinh \left (x\right )^{15} + 13 \, \cosh \left (x\right )^{15} + 5 \, {\left (136 \, \cosh \left (x\right )^{3} + 39 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{14} + {\left (2380 \, \cosh \left (x\right )^{4} + 1365 \, \cosh \left (x\right )^{2} + 78\right )} \sinh \left (x\right )^{13} + 78 \, \cosh \left (x\right )^{13} + 13 \, {\left (476 \, \cosh \left (x\right )^{5} + 455 \, \cosh \left (x\right )^{3} + 78 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{12} + 13 \, {\left (952 \, \cosh \left (x\right )^{6} + 1365 \, \cosh \left (x\right )^{4} + 468 \, \cosh \left (x\right )^{2} + 22\right )} \sinh \left (x\right )^{11} + 286 \, \cosh \left (x\right )^{11} + 143 \, {\left (136 \, \cosh \left (x\right )^{7} + 273 \, \cosh \left (x\right )^{5} + 156 \, \cosh \left (x\right )^{3} + 22 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{10} + {\left (24310 \, \cosh \left (x\right )^{8} + 65065 \, \cosh \left (x\right )^{6} + 55770 \, \cosh \left (x\right )^{4} + 15730 \, \cosh \left (x\right )^{2} + 714\right )} \sinh \left (x\right )^{9} + 716 \, \cosh \left (x\right )^{9} + {\left (24310 \, \cosh \left (x\right )^{9} + 83655 \, \cosh \left (x\right )^{7} + 100386 \, \cosh \left (x\right )^{5} + 47190 \, \cosh \left (x\right )^{3} + 6444 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{8} + {\left (19448 \, \cosh \left (x\right )^{10} + 83655 \, \cosh \left (x\right )^{8} + 133848 \, \cosh \left (x\right )^{6} + 94380 \, \cosh \left (x\right )^{4} + 25704 \, \cosh \left (x\right )^{2} + 1274\right )} \sinh \left (x\right )^{7} + 1300 \, \cosh \left (x\right )^{7} + {\left (12376 \, \cosh \left (x\right )^{11} + 65065 \, \cosh \left (x\right )^{9} + 133848 \, \cosh \left (x\right )^{7} + 132132 \, \cosh \left (x\right )^{5} + 60144 \, \cosh \left (x\right )^{3} + 9100 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{6} + {\left (6188 \, \cosh \left (x\right )^{12} + 39039 \, \cosh \left (x\right )^{10} + 100386 \, \cosh \left (x\right )^{8} + 132132 \, \cosh \left (x\right )^{6} + 89964 \, \cosh \left (x\right )^{4} + 26754 \, \cosh \left (x\right )^{2} + 1638\right )} \sinh \left (x\right )^{5} + 1794 \, \cosh \left (x\right )^{5} + {\left (2380 \, \cosh \left (x\right )^{13} + 17745 \, \cosh \left (x\right )^{11} + 55770 \, \cosh \left (x\right )^{9} + 94380 \, \cosh \left (x\right )^{7} + 90216 \, \cosh \left (x\right )^{5} + 45500 \, \cosh \left (x\right )^{3} + 8970 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + {\left (680 \, \cosh \left (x\right )^{14} + 5915 \, \cosh \left (x\right )^{12} + 22308 \, \cosh \left (x\right )^{10} + 47190 \, \cosh \left (x\right )^{8} + 59976 \, \cosh \left (x\right )^{6} + 44590 \, \cosh \left (x\right )^{4} + 16380 \, \cosh \left (x\right )^{2} + 1430\right )} \sinh \left (x\right )^{3} + 2002 \, \cosh \left (x\right )^{3} + {\left (136 \, \cosh \left (x\right )^{15} + 1365 \, \cosh \left (x\right )^{13} + 6084 \, \cosh \left (x\right )^{11} + 15730 \, \cosh \left (x\right )^{9} + 25776 \, \cosh \left (x\right )^{7} + 27300 \, \cosh \left (x\right )^{5} + 17940 \, \cosh \left (x\right )^{3} + 6006 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (17 \, \cosh \left (x\right )^{16} + 195 \, \cosh \left (x\right )^{14} + 1014 \, \cosh \left (x\right )^{12} + 3146 \, \cosh \left (x\right )^{10} + 6426 \, \cosh \left (x\right )^{8} + 8918 \, \cosh \left (x\right )^{6} + 8190 \, \cosh \left (x\right )^{4} + 4290 \, \cosh \left (x\right )^{2} + 572\right )} \sinh \left (x\right ) + 2002 \, \cosh \left (x\right )\right )}} \end {gather*}

-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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \tanh ^{6}{\left (x \right )} \operatorname {sech}^{8}{\left (x \right )}\, dx \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (25) = 50\).
time = 0.39, size = 66, normalized size = 2.00 \begin {gather*} -\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}} \end {gather*}

-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

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Mupad [B]
time = 1.45, size = 820, normalized size = 24.85 \begin {gather*} -\frac {\frac {64\,{\mathrm {e}}^{4\,x}}{143}-\frac {256\,{\mathrm {e}}^{2\,x}}{429}+\frac {80}{429}}{6\,{\mathrm {e}}^{2\,x}+15\,{\mathrm {e}}^{4\,x}+20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}+6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1}-\frac {\frac {64\,{\mathrm {e}}^{2\,x}}{143}-\frac {768\,{\mathrm {e}}^{4\,x}}{143}+\frac {3200\,{\mathrm {e}}^{6\,x}}{143}-\frac {6400\,{\mathrm {e}}^{8\,x}}{143}+\frac {6720\,{\mathrm {e}}^{10\,x}}{143}-\frac {3584\,{\mathrm {e}}^{12\,x}}{143}+\frac {768\,{\mathrm {e}}^{14\,x}}{143}}{11\,{\mathrm {e}}^{2\,x}+55\,{\mathrm {e}}^{4\,x}+165\,{\mathrm {e}}^{6\,x}+330\,{\mathrm {e}}^{8\,x}+462\,{\mathrm {e}}^{10\,x}+462\,{\mathrm {e}}^{12\,x}+330\,{\mathrm {e}}^{14\,x}+165\,{\mathrm {e}}^{16\,x}+55\,{\mathrm {e}}^{18\,x}+11\,{\mathrm {e}}^{20\,x}+{\mathrm {e}}^{22\,x}+1}-\frac {\frac {160\,{\mathrm {e}}^{2\,x}}{143}-\frac {256\,{\mathrm {e}}^{4\,x}}{143}+\frac {128\,{\mathrm {e}}^{6\,x}}{143}-\frac {640}{3003}}{7\,{\mathrm {e}}^{2\,x}+21\,{\mathrm {e}}^{4\,x}+35\,{\mathrm {e}}^{6\,x}+35\,{\mathrm {e}}^{8\,x}+21\,{\mathrm {e}}^{10\,x}+7\,{\mathrm {e}}^{12\,x}+{\mathrm {e}}^{14\,x}+1}-\frac {\frac {128\,{\mathrm {e}}^{6\,x}}{13}-\frac {768\,{\mathrm {e}}^{8\,x}}{13}+\frac {1920\,{\mathrm {e}}^{10\,x}}{13}-\frac {2560\,{\mathrm {e}}^{12\,x}}{13}+\frac {1920\,{\mathrm {e}}^{14\,x}}{13}-\frac {768\,{\mathrm {e}}^{16\,x}}{13}+\frac {128\,{\mathrm {e}}^{18\,x}}{13}}{13\,{\mathrm {e}}^{2\,x}+78\,{\mathrm {e}}^{4\,x}+286\,{\mathrm {e}}^{6\,x}+715\,{\mathrm {e}}^{8\,x}+1287\,{\mathrm {e}}^{10\,x}+1716\,{\mathrm {e}}^{12\,x}+1716\,{\mathrm {e}}^{14\,x}+1287\,{\mathrm {e}}^{16\,x}+715\,{\mathrm {e}}^{18\,x}+286\,{\mathrm {e}}^{20\,x}+78\,{\mathrm {e}}^{22\,x}+13\,{\mathrm {e}}^{24\,x}+{\mathrm {e}}^{26\,x}+1}-\frac {\frac {560\,{\mathrm {e}}^{4\,x}}{143}-\frac {640\,{\mathrm {e}}^{2\,x}}{429}-\frac {1792\,{\mathrm {e}}^{6\,x}}{429}+\frac {224\,{\mathrm {e}}^{8\,x}}{143}+\frac {80}{429}}{8\,{\mathrm {e}}^{2\,x}+28\,{\mathrm {e}}^{4\,x}+56\,{\mathrm {e}}^{6\,x}+70\,{\mathrm {e}}^{8\,x}+56\,{\mathrm {e}}^{10\,x}+28\,{\mathrm {e}}^{12\,x}+8\,{\mathrm {e}}^{14\,x}+{\mathrm {e}}^{16\,x}+1}-\frac {\frac {640\,{\mathrm {e}}^{2\,x}}{429}-\frac {2560\,{\mathrm {e}}^{4\,x}}{429}+\frac {4480\,{\mathrm {e}}^{6\,x}}{429}-\frac {3584\,{\mathrm {e}}^{8\,x}}{429}+\frac {1792\,{\mathrm {e}}^{10\,x}}{715}-\frac {256}{2145}}{9\,{\mathrm {e}}^{2\,x}+36\,{\mathrm {e}}^{4\,x}+84\,{\mathrm {e}}^{6\,x}+126\,{\mathrm {e}}^{8\,x}+126\,{\mathrm {e}}^{10\,x}+84\,{\mathrm {e}}^{12\,x}+36\,{\mathrm {e}}^{14\,x}+9\,{\mathrm {e}}^{16\,x}+{\mathrm {e}}^{18\,x}+1}-\frac {\frac {32\,{\mathrm {e}}^{4\,x}}{13}-\frac {256\,{\mathrm {e}}^{6\,x}}{13}+\frac {800\,{\mathrm {e}}^{8\,x}}{13}-\frac {1280\,{\mathrm {e}}^{10\,x}}{13}+\frac {1120\,{\mathrm {e}}^{12\,x}}{13}-\frac {512\,{\mathrm {e}}^{14\,x}}{13}+\frac {96\,{\mathrm {e}}^{16\,x}}{13}}{12\,{\mathrm {e}}^{2\,x}+66\,{\mathrm {e}}^{4\,x}+220\,{\mathrm {e}}^{6\,x}+495\,{\mathrm {e}}^{8\,x}+792\,{\mathrm {e}}^{10\,x}+924\,{\mathrm {e}}^{12\,x}+792\,{\mathrm {e}}^{14\,x}+495\,{\mathrm {e}}^{16\,x}+220\,{\mathrm {e}}^{18\,x}+66\,{\mathrm {e}}^{20\,x}+12\,{\mathrm {e}}^{22\,x}+{\mathrm {e}}^{24\,x}+1}-\frac {\frac {128\,{\mathrm {e}}^{2\,x}}{715}-\frac {256}{2145}}{5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1}-\frac {32}{715\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {\frac {960\,{\mathrm {e}}^{4\,x}}{143}-\frac {768\,{\mathrm {e}}^{2\,x}}{715}-\frac {2560\,{\mathrm {e}}^{6\,x}}{143}+\frac {3360\,{\mathrm {e}}^{8\,x}}{143}-\frac {10752\,{\mathrm {e}}^{10\,x}}{715}+\frac {2688\,{\mathrm {e}}^{12\,x}}{715}+\frac {32}{715}}{10\,{\mathrm {e}}^{2\,x}+45\,{\mathrm {e}}^{4\,x}+120\,{\mathrm {e}}^{6\,x}+210\,{\mathrm {e}}^{8\,x}+252\,{\mathrm {e}}^{10\,x}+210\,{\mathrm {e}}^{12\,x}+120\,{\mathrm {e}}^{14\,x}+45\,{\mathrm {e}}^{16\,x}+10\,{\mathrm {e}}^{18\,x}+{\mathrm {e}}^{20\,x}+1} \end {gather*}

- ((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

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3.1.100 \(\int \text {sech}^8(x) \tanh ^6(x) \, dx\) [100]