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]

-1/7*sech(x)^7+2/9*sech(x)^9-1/11*sech(x)^11

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Rubi [A]  time = 0.03, antiderivative size = 25, 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 = {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 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 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])

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*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 1.00 \[ -\frac {1}{11} \text {sech}^{11}(x)+\frac {2 \text {sech}^9(x)}{9}-\frac {\text {sech}^7(x)}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*Sech[x]^7 + (2*Sech[x]^9)/9 - Sech[x]^11/11

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fricas [B]  time = 0.49, size = 634, normalized size = 25.36 \[ \text {result too large to display} \]

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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giac [A]  time = 0.12, size = 35, normalized size = 1.40 \[ -\frac {128 \, {\left (99 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 616 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 1008\right )}}{693 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{11}} \]

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

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maple [A]  time = 0.10, size = 28, normalized size = 1.12 \[ -\frac {\sinh ^{4}\relax (x )}{7 \cosh \relax (x )^{11}}-\frac {4 \left (\sinh ^{2}\relax (x )\right )}{63 \cosh \relax (x )^{11}}-\frac {8}{693 \cosh \relax (x )^{11}} \]

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)^11-4/63*sinh(x)^2/cosh(x)^11-8/693/cosh(x)^11

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maxima [B]  time = 0.35, size = 371, normalized size = 14.84 \[ -\frac {128 \, e^{\left (-7 \, x\right )}}{7 \, {\left (11 \, e^{\left (-2 \, x\right )} + 55 \, e^{\left (-4 \, x\right )} + 165 \, e^{\left (-6 \, x\right )} + 330 \, e^{\left (-8 \, x\right )} + 462 \, e^{\left (-10 \, x\right )} + 462 \, e^{\left (-12 \, x\right )} + 330 \, e^{\left (-14 \, x\right )} + 165 \, e^{\left (-16 \, x\right )} + 55 \, e^{\left (-18 \, x\right )} + 11 \, e^{\left (-20 \, x\right )} + e^{\left (-22 \, x\right )} + 1\right )}} + \frac {2560 \, e^{\left (-9 \, x\right )}}{63 \, {\left (11 \, e^{\left (-2 \, x\right )} + 55 \, e^{\left (-4 \, x\right )} + 165 \, e^{\left (-6 \, x\right )} + 330 \, e^{\left (-8 \, x\right )} + 462 \, e^{\left (-10 \, x\right )} + 462 \, e^{\left (-12 \, x\right )} + 330 \, e^{\left (-14 \, x\right )} + 165 \, e^{\left (-16 \, x\right )} + 55 \, e^{\left (-18 \, x\right )} + 11 \, e^{\left (-20 \, x\right )} + e^{\left (-22 \, x\right )} + 1\right )}} - \frac {47360 \, e^{\left (-11 \, x\right )}}{693 \, {\left (11 \, e^{\left (-2 \, x\right )} + 55 \, e^{\left (-4 \, x\right )} + 165 \, e^{\left (-6 \, x\right )} + 330 \, e^{\left (-8 \, x\right )} + 462 \, e^{\left (-10 \, x\right )} + 462 \, e^{\left (-12 \, x\right )} + 330 \, e^{\left (-14 \, x\right )} + 165 \, e^{\left (-16 \, x\right )} + 55 \, e^{\left (-18 \, x\right )} + 11 \, e^{\left (-20 \, x\right )} + e^{\left (-22 \, x\right )} + 1\right )}} + \frac {2560 \, e^{\left (-13 \, x\right )}}{63 \, {\left (11 \, e^{\left (-2 \, x\right )} + 55 \, e^{\left (-4 \, x\right )} + 165 \, e^{\left (-6 \, x\right )} + 330 \, e^{\left (-8 \, x\right )} + 462 \, e^{\left (-10 \, x\right )} + 462 \, e^{\left (-12 \, x\right )} + 330 \, e^{\left (-14 \, x\right )} + 165 \, e^{\left (-16 \, x\right )} + 55 \, e^{\left (-18 \, x\right )} + 11 \, e^{\left (-20 \, x\right )} + e^{\left (-22 \, x\right )} + 1\right )}} - \frac {128 \, e^{\left (-15 \, x\right )}}{7 \, {\left (11 \, e^{\left (-2 \, x\right )} + 55 \, e^{\left (-4 \, x\right )} + 165 \, e^{\left (-6 \, x\right )} + 330 \, e^{\left (-8 \, x\right )} + 462 \, e^{\left (-10 \, x\right )} + 462 \, e^{\left (-12 \, x\right )} + 330 \, e^{\left (-14 \, x\right )} + 165 \, e^{\left (-16 \, x\right )} + 55 \, e^{\left (-18 \, x\right )} + 11 \, e^{\left (-20 \, x\right )} + e^{\left (-22 \, x\right )} + 1\right )}} \]

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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mupad [B]  time = 1.55, size = 520, normalized size = 20.80 \[ \frac {\frac {64\,{\mathrm {e}}^{5\,x}}{11}-\frac {320\,{\mathrm {e}}^{7\,x}}{11}+\frac {640\,{\mathrm {e}}^{9\,x}}{11}-\frac {640\,{\mathrm {e}}^{11\,x}}{11}+\frac {320\,{\mathrm {e}}^{13\,x}}{11}-\frac {64\,{\mathrm {e}}^{15\,x}}{11}}{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 {38464\,{\mathrm {e}}^x}{693\,\left (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\right )}-\frac {640\,{\mathrm {e}}^x}{33\,\left (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\right )}-\frac {104\,{\mathrm {e}}^x}{21\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {1664\,{\mathrm {e}}^x}{63\,\left (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\right )}+\frac {4096\,{\mathrm {e}}^x}{77\,\left (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\right )}+\frac {\frac {16\,{\mathrm {e}}^{3\,x}}{11}-\frac {112\,{\mathrm {e}}^{5\,x}}{11}+\frac {288\,{\mathrm {e}}^{7\,x}}{11}-32\,{\mathrm {e}}^{9\,x}+\frac {208\,{\mathrm {e}}^{11\,x}}{11}-\frac {48\,{\mathrm {e}}^{13\,x}}{11}}{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}-\frac {\frac {280\,{\mathrm {e}}^{3\,x}}{99}-\frac {112\,{\mathrm {e}}^{5\,x}}{11}+16\,{\mathrm {e}}^{7\,x}-\frac {104\,{\mathrm {e}}^{9\,x}}{9}+\frac {104\,{\mathrm {e}}^{11\,x}}{33}-\frac {8\,{\mathrm {e}}^x}{33}}{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} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^5/cosh(x)^7,x)

[Out]

((64*exp(5*x))/11 - (320*exp(7*x))/11 + (640*exp(9*x))/11 - (640*exp(11*x))/11 + (320*exp(13*x))/11 - (64*exp(
15*x))/11)/(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) - (38464*exp(x))/(693*(6*exp(2*x) + 15*ex
p(4*x) + 20*exp(6*x) + 15*exp(8*x) + 6*exp(10*x) + exp(12*x) + 1)) - (640*exp(x))/(33*(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)) - (104*exp(x))/(21
*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1)) + (1664*exp(x))/(63*(5*exp(2*x) + 10*exp(4*x) + 10*exp
(6*x) + 5*exp(8*x) + exp(10*x) + 1)) + (4096*exp(x))/(77*(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)) + ((16*exp(3*x))/11 - (112*exp(5*x))/11 + (288*exp(7*x))/11 -
32*exp(9*x) + (208*exp(11*x))/11 - (48*exp(13*x))/11)/(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) - ((280*exp(3*
x))/99 - (112*exp(5*x))/11 + 16*exp(7*x) - (104*exp(9*x))/9 + (104*exp(11*x))/33 - (8*exp(x))/33)/(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(1
8*x) + 1)

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sympy [A]  time = 22.41, size = 34, normalized size = 1.36 \[ - \frac {\tanh ^{4}{\relax (x )} \operatorname {sech}^{7}{\relax (x )}}{11} - \frac {4 \tanh ^{2}{\relax (x )} \operatorname {sech}^{7}{\relax (x )}}{99} - \frac {8 \operatorname {sech}^{7}{\relax (x )}}{693} \]

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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