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

Optimal. Leaf size=77 \[ -\frac{a^2 \tanh ^3(c+d x)}{3 d}-\frac{a^2 \tanh (c+d x)}{d}+a^2 x+\frac{b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac{b^2 \tanh ^7(c+d x)}{7 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.104635, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4141, 1802, 206} \[ -\frac{a^2 \tanh ^3(c+d x)}{3 d}-\frac{a^2 \tanh (c+d x)}{d}+a^2 x+\frac{b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac{b^2 \tanh ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4141

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

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [B]  time = 1.16638, size = 395, normalized size = 5.13 \[ \frac{\text{sech}(c) \text{sech}^7(c+d x) \left (4480 a^2 \sinh (2 c+d x)-3780 a^2 \sinh (2 c+3 d x)+2100 a^2 \sinh (4 c+3 d x)-1540 a^2 \sinh (4 c+5 d x)+420 a^2 \sinh (6 c+5 d x)-280 a^2 \sinh (6 c+7 d x)+3675 a^2 d x \cosh (2 c+d x)+2205 a^2 d x \cosh (2 c+3 d x)+2205 a^2 d x \cosh (4 c+3 d x)+735 a^2 d x \cosh (4 c+5 d x)+735 a^2 d x \cosh (6 c+5 d x)+105 a^2 d x \cosh (6 c+7 d x)+105 a^2 d x \cosh (8 c+7 d x)-5320 a^2 \sinh (d x)+3675 a^2 d x \cosh (d x)-1260 a b \sinh (2 c+d x)+924 a b \sinh (2 c+3 d x)-840 a b \sinh (4 c+3 d x)+168 a b \sinh (4 c+5 d x)-420 a b \sinh (6 c+5 d x)+84 a b \sinh (6 c+7 d x)+1680 a b \sinh (d x)+420 b^2 \sinh (2 c+d x)-168 b^2 \sinh (2 c+3 d x)-420 b^2 \sinh (4 c+3 d x)+84 b^2 \sinh (4 c+5 d x)+12 b^2 \sinh (6 c+7 d x)+840 b^2 \sinh (d x)\right )}{13440 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sech[c]*Sech[c + d*x]^7*(3675*a^2*d*x*Cosh[d*x] + 3675*a^2*d*x*Cosh[2*c + d*x] + 2205*a^2*d*x*Cosh[2*c + 3*d*
x] + 2205*a^2*d*x*Cosh[4*c + 3*d*x] + 735*a^2*d*x*Cosh[4*c + 5*d*x] + 735*a^2*d*x*Cosh[6*c + 5*d*x] + 105*a^2*
d*x*Cosh[6*c + 7*d*x] + 105*a^2*d*x*Cosh[8*c + 7*d*x] - 5320*a^2*Sinh[d*x] + 1680*a*b*Sinh[d*x] + 840*b^2*Sinh
[d*x] + 4480*a^2*Sinh[2*c + d*x] - 1260*a*b*Sinh[2*c + d*x] + 420*b^2*Sinh[2*c + d*x] - 3780*a^2*Sinh[2*c + 3*
d*x] + 924*a*b*Sinh[2*c + 3*d*x] - 168*b^2*Sinh[2*c + 3*d*x] + 2100*a^2*Sinh[4*c + 3*d*x] - 840*a*b*Sinh[4*c +
3*d*x] - 420*b^2*Sinh[4*c + 3*d*x] - 1540*a^2*Sinh[4*c + 5*d*x] + 168*a*b*Sinh[4*c + 5*d*x] + 84*b^2*Sinh[4*c
+ 5*d*x] + 420*a^2*Sinh[6*c + 5*d*x] - 420*a*b*Sinh[6*c + 5*d*x] - 280*a^2*Sinh[6*c + 7*d*x] + 84*a*b*Sinh[6*
c + 7*d*x] + 12*b^2*Sinh[6*c + 7*d*x]))/(13440*d)

________________________________________________________________________________________

Maple [B]  time = 0.046, size = 181, normalized size = 2.4 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( dx+c-\tanh \left ( dx+c \right ) -{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{3}} \right ) +2\,ab \left ( -1/2\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}-3/8\,{\frac{\sinh \left ( dx+c \right ) }{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+3/8\, \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 ) \right ) +{b}^{2} \left ( -{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{4\, \left ( \cosh \left ( dx+c \right ) \right ) ^{7}}}-{\frac{\sinh \left ( dx+c \right ) }{8\, \left ( \cosh \left ( dx+c \right ) \right ) ^{7}}}+{\frac{\tanh \left ( dx+c \right ) }{8} \left ({\frac{16}{35}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{6}}{7}}+{\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 ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^2*(d*x+c-tanh(d*x+c)-1/3*tanh(d*x+c)^3)+2*a*b*(-1/2*sinh(d*x+c)^3/cosh(d*x+c)^5-3/8*sinh(d*x+c)/cosh(d*
x+c)^5+3/8*(8/15+1/5*sech(d*x+c)^4+4/15*sech(d*x+c)^2)*tanh(d*x+c))+b^2*(-1/4*sinh(d*x+c)^3/cosh(d*x+c)^7-1/8*
sinh(d*x+c)/cosh(d*x+c)^7+1/8*(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)))

________________________________________________________________________________________

Maxima [B]  time = 1.22086, size = 876, normalized size = 11.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*a*b*tanh(d*x + c)^5/d + 1/3*a^2*(3*x + 3*c/d - 4*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + 2)/(d*(3*e^(-2
*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1))) + 4/35*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)) - 14*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 - 14*c) + 1)) +
70*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)) - 35*e^(-8*d*x - 8*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)) + 35*e^(-10*d*x - 10*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 3
5*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)))

________________________________________________________________________________________

Fricas [B]  time = 2.08535, size = 1868, normalized size = 24.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*((105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*cosh(d*x + c)^7 + 7*(105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*c
osh(d*x + c)*sinh(d*x + c)^6 - 2*(70*a^2 - 21*a*b - 3*b^2)*sinh(d*x + c)^7 + 7*(105*a^2*d*x + 140*a^2 - 42*a*b
- 6*b^2)*cosh(d*x + c)^5 - 14*(3*(70*a^2 - 21*a*b - 3*b^2)*cosh(d*x + c)^2 + 40*a^2 + 9*a*b - 3*b^2)*sinh(d*x
+ c)^5 + 35*((105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*cosh(d*x + c)^3 + (105*a^2*d*x + 140*a^2 - 42*a*b - 6*b
^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 21*(105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*cosh(d*x + c)^3 - 14*(5*(70*a
^2 - 21*a*b - 3*b^2)*cosh(d*x + c)^4 + 10*(40*a^2 + 9*a*b - 3*b^2)*cosh(d*x + c)^2 + 60*a^2 - 3*a*b + 21*b^2)*
sinh(d*x + c)^3 + 7*(3*(105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*cosh(d*x + c)^5 + 10*(105*a^2*d*x + 140*a^2 -
42*a*b - 6*b^2)*cosh(d*x + c)^3 + 9*(105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 +
35*(105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*cosh(d*x + c) - 14*((70*a^2 - 21*a*b - 3*b^2)*cosh(d*x + c)^6 + 5*
(40*a^2 + 9*a*b - 3*b^2)*cosh(d*x + c)^4 + 9*(20*a^2 - a*b + 7*b^2)*cosh(d*x + c)^2 + 30*a^2 - 15*a*b - 45*b^2
)*sinh(d*x + c))/(d*cosh(d*x + c)^7 + 7*d*cosh(d*x + c)*sinh(d*x + c)^6 + 7*d*cosh(d*x + c)^5 + 35*(d*cosh(d*x
+ c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^4 + 21*d*cosh(d*x + c)^3 + 7*(3*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c
)^3 + 9*d*cosh(d*x + c))*sinh(d*x + c)^2 + 35*d*cosh(d*x + c))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.32151, size = 371, normalized size = 4.82 \begin{align*} \frac{105 \, a^{2} d x + \frac{4 \,{\left (105 \, a^{2} e^{\left (12 \, d x + 12 \, c\right )} - 105 \, a b e^{\left (12 \, d x + 12 \, c\right )} + 525 \, a^{2} e^{\left (10 \, d x + 10 \, c\right )} - 210 \, a b e^{\left (10 \, d x + 10 \, c\right )} - 105 \, b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1120 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} - 315 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 105 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 1330 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} - 420 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 210 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 945 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 231 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 42 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 385 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 42 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 21 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}}}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(105*a^2*d*x + 4*(105*a^2*e^(12*d*x + 12*c) - 105*a*b*e^(12*d*x + 12*c) + 525*a^2*e^(10*d*x + 10*c) - 21
0*a*b*e^(10*d*x + 10*c) - 105*b^2*e^(10*d*x + 10*c) + 1120*a^2*e^(8*d*x + 8*c) - 315*a*b*e^(8*d*x + 8*c) + 105
*b^2*e^(8*d*x + 8*c) + 1330*a^2*e^(6*d*x + 6*c) - 420*a*b*e^(6*d*x + 6*c) - 210*b^2*e^(6*d*x + 6*c) + 945*a^2*
e^(4*d*x + 4*c) - 231*a*b*e^(4*d*x + 4*c) + 42*b^2*e^(4*d*x + 4*c) + 385*a^2*e^(2*d*x + 2*c) - 42*a*b*e^(2*d*x
+ 2*c) - 21*b^2*e^(2*d*x + 2*c) + 70*a^2 - 21*a*b - 3*b^2)/(e^(2*d*x + 2*c) + 1)^7)/d