3.72 \(\int \text{sech}^4(c+d x) (a+b \text{sech}^2(c+d x))^3 \, dx\)

Optimal. Leaf size=108 \[ -\frac{b^2 (3 a+4 b) \tanh ^7(c+d x)}{7 d}+\frac{3 b (a+b) (a+2 b) \tanh ^5(c+d x)}{5 d}-\frac{(a+b)^2 (a+4 b) \tanh ^3(c+d x)}{3 d}+\frac{(a+b)^3 \tanh (c+d x)}{d}+\frac{b^3 \tanh ^9(c+d x)}{9 d} \]

[Out]

((a + b)^3*Tanh[c + d*x])/d - ((a + b)^2*(a + 4*b)*Tanh[c + d*x]^3)/(3*d) + (3*b*(a + b)*(a + 2*b)*Tanh[c + d*
x]^5)/(5*d) - (b^2*(3*a + 4*b)*Tanh[c + d*x]^7)/(7*d) + (b^3*Tanh[c + d*x]^9)/(9*d)

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Rubi [A]  time = 0.0937486, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {4146, 373} \[ -\frac{b^2 (3 a+4 b) \tanh ^7(c+d x)}{7 d}+\frac{3 b (a+b) (a+2 b) \tanh ^5(c+d x)}{5 d}-\frac{(a+b)^2 (a+4 b) \tanh ^3(c+d x)}{3 d}+\frac{(a+b)^3 \tanh (c+d x)}{d}+\frac{b^3 \tanh ^9(c+d x)}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^4*(a + b*Sech[c + d*x]^2)^3,x]

[Out]

((a + b)^3*Tanh[c + d*x])/d - ((a + b)^2*(a + 4*b)*Tanh[c + d*x]^3)/(3*d) + (3*b*(a + b)*(a + 2*b)*Tanh[c + d*
x]^5)/(5*d) - (b^2*(3*a + 4*b)*Tanh[c + d*x]^7)/(7*d) + (b^3*Tanh[c + d*x]^9)/(9*d)

Rule 4146

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fre
eFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/
2), x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]

Rule 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \text{sech}^4(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (a+b-b x^2\right )^3 \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left ((a+b)^3-(a+b)^2 (a+4 b) x^2+3 b (a+b) (a+2 b) x^4-b^2 (3 a+4 b) x^6+b^3 x^8\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a+b)^3 \tanh (c+d x)}{d}-\frac{(a+b)^2 (a+4 b) \tanh ^3(c+d x)}{3 d}+\frac{3 b (a+b) (a+2 b) \tanh ^5(c+d x)}{5 d}-\frac{b^2 (3 a+4 b) \tanh ^7(c+d x)}{7 d}+\frac{b^3 \tanh ^9(c+d x)}{9 d}\\ \end{align*}

Mathematica [B]  time = 1.70855, size = 348, normalized size = 3.22 \[ \frac{\text{sech}(c) \text{sech}^9(c+d x) \left (-315 a \left (17 a^2+36 a b+24 b^2\right ) \sinh (2 c+d x)+63 \left (324 a^2 b+125 a^3+312 a b^2+128 b^3\right ) \sinh (d x)+18648 a^2 b \sinh (2 c+3 d x)-2520 a^2 b \sinh (4 c+3 d x)+9072 a^2 b \sinh (4 c+5 d x)+2268 a^2 b \sinh (6 c+7 d x)+252 a^2 b \sinh (8 c+9 d x)+6825 a^3 \sinh (2 c+3 d x)-1995 a^3 \sinh (4 c+3 d x)+3465 a^3 \sinh (4 c+5 d x)-315 a^3 \sinh (6 c+5 d x)+945 a^3 \sinh (6 c+7 d x)+105 a^3 \sinh (8 c+9 d x)+18144 a b^2 \sinh (2 c+3 d x)+7776 a b^2 \sinh (4 c+5 d x)+1944 a b^2 \sinh (6 c+7 d x)+216 a b^2 \sinh (8 c+9 d x)+5376 b^3 \sinh (2 c+3 d x)+2304 b^3 \sinh (4 c+5 d x)+576 b^3 \sinh (6 c+7 d x)+64 b^3 \sinh (8 c+9 d x)\right )}{40320 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^4*(a + b*Sech[c + d*x]^2)^3,x]

[Out]

(Sech[c]*Sech[c + d*x]^9*(63*(125*a^3 + 324*a^2*b + 312*a*b^2 + 128*b^3)*Sinh[d*x] - 315*a*(17*a^2 + 36*a*b +
24*b^2)*Sinh[2*c + d*x] + 6825*a^3*Sinh[2*c + 3*d*x] + 18648*a^2*b*Sinh[2*c + 3*d*x] + 18144*a*b^2*Sinh[2*c +
3*d*x] + 5376*b^3*Sinh[2*c + 3*d*x] - 1995*a^3*Sinh[4*c + 3*d*x] - 2520*a^2*b*Sinh[4*c + 3*d*x] + 3465*a^3*Sin
h[4*c + 5*d*x] + 9072*a^2*b*Sinh[4*c + 5*d*x] + 7776*a*b^2*Sinh[4*c + 5*d*x] + 2304*b^3*Sinh[4*c + 5*d*x] - 31
5*a^3*Sinh[6*c + 5*d*x] + 945*a^3*Sinh[6*c + 7*d*x] + 2268*a^2*b*Sinh[6*c + 7*d*x] + 1944*a*b^2*Sinh[6*c + 7*d
*x] + 576*b^3*Sinh[6*c + 7*d*x] + 105*a^3*Sinh[8*c + 9*d*x] + 252*a^2*b*Sinh[8*c + 9*d*x] + 216*a*b^2*Sinh[8*c
+ 9*d*x] + 64*b^3*Sinh[8*c + 9*d*x]))/(40320*d)

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Maple [A]  time = 0.029, size = 158, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{3}} \right ) \tanh \left ( dx+c \right ) +3\,{a}^{2}b \left ({\frac{8}{15}}+1/5\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}+{\frac{4\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{15}} \right ) \tanh \left ( dx+c \right ) +3\,a{b}^{2} \left ({\frac{16}{35}}+1/7\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{6}+{\frac{6\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}}{35}}+{\frac{8\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{35}} \right ) \tanh \left ( dx+c \right ) +{b}^{3} \left ({\frac{128}{315}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{8}}{9}}+{\frac{8\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{6}}{63}}+{\frac{16\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}}{105}}+{\frac{64\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{315}} \right ) \tanh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^4*(a+b*sech(d*x+c)^2)^3,x)

[Out]

1/d*(a^3*(2/3+1/3*sech(d*x+c)^2)*tanh(d*x+c)+3*a^2*b*(8/15+1/5*sech(d*x+c)^4+4/15*sech(d*x+c)^2)*tanh(d*x+c)+3
*a*b^2*(16/35+1/7*sech(d*x+c)^6+6/35*sech(d*x+c)^4+8/35*sech(d*x+c)^2)*tanh(d*x+c)+b^3*(128/315+1/9*sech(d*x+c
)^8+8/63*sech(d*x+c)^6+16/105*sech(d*x+c)^4+64/315*sech(d*x+c)^2)*tanh(d*x+c))

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Maxima [B]  time = 1.1259, size = 1681, normalized size = 15.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^4*(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

256/315*b^3*(9*e^(-2*d*x - 2*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8
*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) +
e^(-18*d*x - 18*c) + 1)) + 36*e^(-4*d*x - 4*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6
*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16
*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) + 84*e^(-6*d*x - 6*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 8
4*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 1
4*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) + 126*e^(-8*d*x - 8*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-
4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 3
6*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) + 1/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d
*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e
^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1))) + 96/35*a*b^2*(7*e^(-2*d*x - 2*c)/(d*(7*e
^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*
e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 21*e^(-4*d*x - 4*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*
c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 1
4*c) + 1)) + 35*e^(-6*d*x - 6*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8
*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 1/(d*(7*e^(-2*d*x - 2*
c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x -
12*c) + e^(-14*d*x - 14*c) + 1))) + 16/5*a^2*b*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c
) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) + 10*e^(-4*d*x - 4*c)/(d*(5*e^(-2*d*x
- 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) + 1/(d*(5*e
^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1))) +
4/3*a^3*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1)) + 1/(d*(3*e^
(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1)))

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Fricas [B]  time = 2.39069, size = 3222, normalized size = 29.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^4*(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-8/315*(2*(105*a^3 + 63*a^2*b + 54*a*b^2 + 16*b^3)*cosh(d*x + c)^7 + 14*(105*a^3 + 63*a^2*b + 54*a*b^2 + 16*b^
3)*cosh(d*x + c)*sinh(d*x + c)^6 + (105*a^3 - 126*a^2*b - 108*a*b^2 - 32*b^3)*sinh(d*x + c)^7 + 6*(245*a^3 + 3
99*a^2*b + 162*a*b^2 + 48*b^3)*cosh(d*x + c)^5 + 3*(175*a^3 + 42*a^2*b - 324*a*b^2 - 96*b^3 + 7*(105*a^3 - 126
*a^2*b - 108*a*b^2 - 32*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(105*a^3 + 63*a^2*b + 54*a*b^2 + 16*b^3)
*cosh(d*x + c)^3 + 3*(245*a^3 + 399*a^2*b + 162*a*b^2 + 48*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 18*(245*a^3 +
567*a^2*b + 426*a*b^2 + 64*b^3)*cosh(d*x + c)^3 + (35*(105*a^3 - 126*a^2*b - 108*a*b^2 - 32*b^3)*cosh(d*x + c
)^4 + 945*a^3 + 1134*a^2*b - 108*a*b^2 - 1152*b^3 + 30*(175*a^3 + 42*a^2*b - 324*a*b^2 - 96*b^3)*cosh(d*x + c)
^2)*sinh(d*x + c)^3 + 6*(7*(105*a^3 + 63*a^2*b + 54*a*b^2 + 16*b^3)*cosh(d*x + c)^5 + 10*(245*a^3 + 399*a^2*b
+ 162*a*b^2 + 48*b^3)*cosh(d*x + c)^3 + 9*(245*a^3 + 567*a^2*b + 426*a*b^2 + 64*b^3)*cosh(d*x + c))*sinh(d*x +
c)^2 + 210*(35*a^3 + 93*a^2*b + 90*a*b^2 + 32*b^3)*cosh(d*x + c) + (7*(105*a^3 - 126*a^2*b - 108*a*b^2 - 32*b
^3)*cosh(d*x + c)^6 + 15*(175*a^3 + 42*a^2*b - 324*a*b^2 - 96*b^3)*cosh(d*x + c)^4 + 525*a^3 + 882*a^2*b + 756
*a*b^2 + 1344*b^3 + 27*(105*a^3 + 126*a^2*b - 12*a*b^2 - 128*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x
+ c)^11 + 11*d*cosh(d*x + c)*sinh(d*x + c)^10 + d*sinh(d*x + c)^11 + 9*d*cosh(d*x + c)^9 + (55*d*cosh(d*x + c)
^2 + 9*d)*sinh(d*x + c)^9 + 3*(55*d*cosh(d*x + c)^3 + 27*d*cosh(d*x + c))*sinh(d*x + c)^8 + 37*d*cosh(d*x + c)
^7 + (330*d*cosh(d*x + c)^4 + 324*d*cosh(d*x + c)^2 + 35*d)*sinh(d*x + c)^7 + 7*(66*d*cosh(d*x + c)^5 + 108*d*
cosh(d*x + c)^3 + 37*d*cosh(d*x + c))*sinh(d*x + c)^6 + 93*d*cosh(d*x + c)^5 + 3*(154*d*cosh(d*x + c)^6 + 378*
d*cosh(d*x + c)^4 + 245*d*cosh(d*x + c)^2 + 25*d)*sinh(d*x + c)^5 + (330*d*cosh(d*x + c)^7 + 1134*d*cosh(d*x +
c)^5 + 1295*d*cosh(d*x + c)^3 + 465*d*cosh(d*x + c))*sinh(d*x + c)^4 + 162*d*cosh(d*x + c)^3 + (165*d*cosh(d*
x + c)^8 + 756*d*cosh(d*x + c)^6 + 1225*d*cosh(d*x + c)^4 + 750*d*cosh(d*x + c)^2 + 90*d)*sinh(d*x + c)^3 + (5
5*d*cosh(d*x + c)^9 + 324*d*cosh(d*x + c)^7 + 777*d*cosh(d*x + c)^5 + 930*d*cosh(d*x + c)^3 + 486*d*cosh(d*x +
c))*sinh(d*x + c)^2 + 210*d*cosh(d*x + c) + (11*d*cosh(d*x + c)^10 + 81*d*cosh(d*x + c)^8 + 245*d*cosh(d*x +
c)^6 + 375*d*cosh(d*x + c)^4 + 270*d*cosh(d*x + c)^2 + 42*d)*sinh(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{sech}^{2}{\left (c + d x \right )}\right )^{3} \operatorname{sech}^{4}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**4*(a+b*sech(d*x+c)**2)**3,x)

[Out]

Integral((a + b*sech(c + d*x)**2)**3*sech(c + d*x)**4, x)

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Giac [B]  time = 1.17785, size = 486, normalized size = 4.5 \begin{align*} -\frac{4 \,{\left (315 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} + 1995 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 2520 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 5355 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 11340 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 7560 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 7875 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 20412 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 19656 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 8064 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 6825 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 18648 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 18144 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 5376 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 3465 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 9072 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 7776 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2304 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 945 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2268 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 1944 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 576 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} + 252 \, a^{2} b + 216 \, a b^{2} + 64 \, b^{3}\right )}}{315 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^4*(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-4/315*(315*a^3*e^(14*d*x + 14*c) + 1995*a^3*e^(12*d*x + 12*c) + 2520*a^2*b*e^(12*d*x + 12*c) + 5355*a^3*e^(10
*d*x + 10*c) + 11340*a^2*b*e^(10*d*x + 10*c) + 7560*a*b^2*e^(10*d*x + 10*c) + 7875*a^3*e^(8*d*x + 8*c) + 20412
*a^2*b*e^(8*d*x + 8*c) + 19656*a*b^2*e^(8*d*x + 8*c) + 8064*b^3*e^(8*d*x + 8*c) + 6825*a^3*e^(6*d*x + 6*c) + 1
8648*a^2*b*e^(6*d*x + 6*c) + 18144*a*b^2*e^(6*d*x + 6*c) + 5376*b^3*e^(6*d*x + 6*c) + 3465*a^3*e^(4*d*x + 4*c)
+ 9072*a^2*b*e^(4*d*x + 4*c) + 7776*a*b^2*e^(4*d*x + 4*c) + 2304*b^3*e^(4*d*x + 4*c) + 945*a^3*e^(2*d*x + 2*c
) + 2268*a^2*b*e^(2*d*x + 2*c) + 1944*a*b^2*e^(2*d*x + 2*c) + 576*b^3*e^(2*d*x + 2*c) + 105*a^3 + 252*a^2*b +
216*a*b^2 + 64*b^3)/(d*(e^(2*d*x + 2*c) + 1)^9)