### 3.100 $$\int \text{sech}^8(x) \tanh ^6(x) \, dx$$

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]

Tanh[x]^7/7 - Tanh[x]^9/3 + (3*Tanh[x]^11)/11 - Tanh[x]^13/13

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Rubi [A]  time = 0.0303592, antiderivative size = 33, normalized size of antiderivative = 1., 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 = {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]^8*Tanh[x]^6,x]

[Out]

Tanh[x]^7/7 - Tanh[x]^9/3 + (3*Tanh[x]^11)/11 - Tanh[x]^13/13

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 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]

Rubi steps

\begin{align*} \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 = 0.0254378, 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]^8*Tanh[x]^6,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

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Maple [B]  time = 0.084, size = 72, normalized size = 2.2 \begin{align*} -{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{5}}{8\, \left ( \cosh \left ( x \right ) \right ) ^{13}}}-{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{3}}{16\, \left ( \cosh \left ( x \right ) \right ) ^{13}}}-{\frac{\sinh \left ( x \right ) }{64\, \left ( \cosh \left ( x \right ) \right ) ^{13}}}+{\frac{\tanh \left ( x \right ) }{64} \left ({\frac{1024}{3003}}+{\frac{ \left ({\rm sech} \left (x\right ) \right ) ^{12}}{13}}+{\frac{12\, \left ({\rm sech} \left (x\right ) \right ) ^{10}}{143}}+{\frac{40\, \left ({\rm sech} \left (x\right ) \right ) ^{8}}{429}}+{\frac{320\, \left ({\rm sech} \left (x\right ) \right ) ^{6}}{3003}}+{\frac{128\, \left ({\rm sech} \left (x\right ) \right ) ^{4}}{1001}}+{\frac{512\, \left ({\rm sech} \left (x\right ) \right ) ^{2}}{3003}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^8*tanh(x)^6,x)

[Out]

-1/8*sinh(x)^5/cosh(x)^13-1/16*sinh(x)^3/cosh(x)^13-1/64*sinh(x)/cosh(x)^13+1/64*(1024/3003+1/13*sech(x)^12+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)

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Maxima [B]  time = 1.06547, size = 1157, normalized size = 35.06 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^8*tanh(x)^6,x, algorithm="maxima")

[Out]

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]  time = 1.86253, size = 3032, normalized size = 91.88 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^8*tanh(x)^6,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))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**8*tanh(x)**6,x)

[Out]

Integral(tanh(x)**6*sech(x)**8, x)

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Giac [B]  time = 1.19937, size = 89, normalized size = 2.7 \begin{align*} -\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{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^8*tanh(x)^6,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