### 3.97 $$\int \text{sech}^7(x) \tanh ^5(x) \, dx$$

Optimal. Leaf size=25 $-\frac{1}{11} \text{sech}^{11}(x)+\frac{2 \text{sech}^9(x)}{9}-\frac{\text{sech}^7(x)}{7}$

[Out]

-Sech[x]^7/7 + (2*Sech[x]^9)/9 - Sech[x]^11/11

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Rubi [A]  time = 0.0286869, antiderivative size = 25, 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 = {2606, 270} $-\frac{1}{11} \text{sech}^{11}(x)+\frac{2 \text{sech}^9(x)}{9}-\frac{\text{sech}^7(x)}{7}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[x]^7*Tanh[x]^5,x]

[Out]

-Sech[x]^7/7 + (2*Sech[x]^9)/9 - Sech[x]^11/11

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 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}^7(x) \tanh ^5(x) \, dx &=-\operatorname{Subst}\left (\int x^6 \left (-1+x^2\right )^2 \, dx,x,\text{sech}(x)\right )\\ &=-\operatorname{Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\text{sech}(x)\right )\\ &=-\frac{1}{7} \text{sech}^7(x)+\frac{2 \text{sech}^9(x)}{9}-\frac{\text{sech}^{11}(x)}{11}\\ \end{align*}

Mathematica [A]  time = 0.0145422, size = 25, normalized size = 1. $-\frac{1}{11} \text{sech}^{11}(x)+\frac{2 \text{sech}^9(x)}{9}-\frac{\text{sech}^7(x)}{7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sech[x]^7*Tanh[x]^5,x]

[Out]

-Sech[x]^7/7 + (2*Sech[x]^9)/9 - Sech[x]^11/11

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Maple [B]  time = 0.014, size = 76, normalized size = 3. \begin{align*} -{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{4}}{7\, \left ( \cosh \left ( x \right ) \right ) ^{11}}}-{\frac{4\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{77\, \left ( \cosh \left ( x \right ) \right ) ^{11}}}+{\frac{8\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{693\, \left ( \cosh \left ( x \right ) \right ) ^{9}}}+{\frac{8\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{693\, \left ( \cosh \left ( x \right ) \right ) ^{7}}}+{\frac{8\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{693\, \left ( \cosh \left ( x \right ) \right ) ^{5}}}+{\frac{8\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{693\, \left ( \cosh \left ( x \right ) \right ) ^{3}}}+{\frac{8\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{693\,\cosh \left ( x \right ) }}-{\frac{8\,\cosh \left ( x \right ) }{693}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^7*tanh(x)^5,x)

[Out]

-1/7*sinh(x)^4/cosh(x)^11-4/77*sinh(x)^2/cosh(x)^11+8/693*sinh(x)^2/cosh(x)^9+8/693*sinh(x)^2/cosh(x)^7+8/693*
sinh(x)^2/cosh(x)^5+8/693*sinh(x)^2/cosh(x)^3+8/693*sinh(x)^2/cosh(x)-8/693*cosh(x)

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Maxima [B]  time = 1.25601, size = 501, normalized size = 20.04 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^7*tanh(x)^5,x, algorithm="maxima")

[Out]

-128/7*e^(-7*x)/(11*e^(-2*x) + 55*e^(-4*x) + 165*e^(-6*x) + 330*e^(-8*x) + 462*e^(-10*x) + 462*e^(-12*x) + 330
*e^(-14*x) + 165*e^(-16*x) + 55*e^(-18*x) + 11*e^(-20*x) + e^(-22*x) + 1) + 2560/63*e^(-9*x)/(11*e^(-2*x) + 55
*e^(-4*x) + 165*e^(-6*x) + 330*e^(-8*x) + 462*e^(-10*x) + 462*e^(-12*x) + 330*e^(-14*x) + 165*e^(-16*x) + 55*e
^(-18*x) + 11*e^(-20*x) + e^(-22*x) + 1) - 47360/693*e^(-11*x)/(11*e^(-2*x) + 55*e^(-4*x) + 165*e^(-6*x) + 330
*e^(-8*x) + 462*e^(-10*x) + 462*e^(-12*x) + 330*e^(-14*x) + 165*e^(-16*x) + 55*e^(-18*x) + 11*e^(-20*x) + e^(-
22*x) + 1) + 2560/63*e^(-13*x)/(11*e^(-2*x) + 55*e^(-4*x) + 165*e^(-6*x) + 330*e^(-8*x) + 462*e^(-10*x) + 462*
e^(-12*x) + 330*e^(-14*x) + 165*e^(-16*x) + 55*e^(-18*x) + 11*e^(-20*x) + e^(-22*x) + 1) - 128/7*e^(-15*x)/(11
*e^(-2*x) + 55*e^(-4*x) + 165*e^(-6*x) + 330*e^(-8*x) + 462*e^(-10*x) + 462*e^(-12*x) + 330*e^(-14*x) + 165*e^
(-16*x) + 55*e^(-18*x) + 11*e^(-20*x) + e^(-22*x) + 1)

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Fricas [B]  time = 1.81562, size = 2333, normalized size = 93.32 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^7*tanh(x)^5,x, algorithm="fricas")

[Out]

-128/693*(99*cosh(x)^8 + 792*cosh(x)*sinh(x)^7 + 99*sinh(x)^8 + 44*(63*cosh(x)^2 - 5)*sinh(x)^6 - 220*cosh(x)^
6 + 264*(21*cosh(x)^3 - 5*cosh(x))*sinh(x)^5 + 10*(693*cosh(x)^4 - 330*cosh(x)^2 + 37)*sinh(x)^4 + 370*cosh(x)
^4 + 8*(693*cosh(x)^5 - 550*cosh(x)^3 + 185*cosh(x))*sinh(x)^3 + 4*(693*cosh(x)^6 - 825*cosh(x)^4 + 555*cosh(x
)^2 - 55)*sinh(x)^2 - 220*cosh(x)^2 + 8*(99*cosh(x)^7 - 165*cosh(x)^5 + 185*cosh(x)^3 - 55*cosh(x))*sinh(x) +
99)/(cosh(x)^15 + 15*cosh(x)*sinh(x)^14 + sinh(x)^15 + (105*cosh(x)^2 + 11)*sinh(x)^13 + 11*cosh(x)^13 + 13*(3
5*cosh(x)^3 + 11*cosh(x))*sinh(x)^12 + (1365*cosh(x)^4 + 858*cosh(x)^2 + 55)*sinh(x)^11 + 55*cosh(x)^11 + 11*(
273*cosh(x)^5 + 286*cosh(x)^3 + 55*cosh(x))*sinh(x)^10 + 55*(91*cosh(x)^6 + 143*cosh(x)^4 + 55*cosh(x)^2 + 3)*
sinh(x)^9 + 165*cosh(x)^9 + 33*(195*cosh(x)^7 + 429*cosh(x)^5 + 275*cosh(x)^3 + 45*cosh(x))*sinh(x)^8 + (6435*
cosh(x)^8 + 18876*cosh(x)^6 + 18150*cosh(x)^4 + 5940*cosh(x)^2 + 329)*sinh(x)^7 + 331*cosh(x)^7 + (5005*cosh(x
)^9 + 18876*cosh(x)^7 + 25410*cosh(x)^5 + 13860*cosh(x)^3 + 2317*cosh(x))*sinh(x)^6 + (3003*cosh(x)^10 + 14157
*cosh(x)^8 + 25410*cosh(x)^6 + 20790*cosh(x)^4 + 6909*cosh(x)^2 + 451)*sinh(x)^5 + 473*cosh(x)^5 + 5*(273*cosh
(x)^11 + 1573*cosh(x)^9 + 3630*cosh(x)^7 + 4158*cosh(x)^5 + 2317*cosh(x)^3 + 473*cosh(x))*sinh(x)^4 + (455*cos
h(x)^12 + 3146*cosh(x)^10 + 9075*cosh(x)^8 + 13860*cosh(x)^6 + 11515*cosh(x)^4 + 4510*cosh(x)^2 + 407)*sinh(x)
^3 + 517*cosh(x)^3 + (105*cosh(x)^13 + 858*cosh(x)^11 + 3025*cosh(x)^9 + 5940*cosh(x)^7 + 6951*cosh(x)^5 + 473
0*cosh(x)^3 + 1551*cosh(x))*sinh(x)^2 + (15*cosh(x)^14 + 143*cosh(x)^12 + 605*cosh(x)^10 + 1485*cosh(x)^8 + 23
03*cosh(x)^6 + 2255*cosh(x)^4 + 1221*cosh(x)^2 + 165)*sinh(x) + 495*cosh(x))

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Sympy [A]  time = 41.6912, size = 34, normalized size = 1.36 \begin{align*} - \frac{\tanh ^{4}{\left (x \right )} \operatorname{sech}^{7}{\left (x \right )}}{11} - \frac{4 \tanh ^{2}{\left (x \right )} \operatorname{sech}^{7}{\left (x \right )}}{99} - \frac{8 \operatorname{sech}^{7}{\left (x \right )}}{693} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**7*tanh(x)**5,x)

[Out]

-tanh(x)**4*sech(x)**7/11 - 4*tanh(x)**2*sech(x)**7/99 - 8*sech(x)**7/693

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Giac [A]  time = 1.21379, size = 47, normalized size = 1.88 \begin{align*} -\frac{128 \,{\left (99 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 616 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 1008\right )}}{693 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{11}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^7*tanh(x)^5,x, algorithm="giac")

[Out]

-128/693*(99*(e^(-x) + e^x)^4 - 616*(e^(-x) + e^x)^2 + 1008)/(e^(-x) + e^x)^11