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

Optimal. Leaf size=74 \[ \frac{3 b^2 (a+b) \tanh ^5(c+d x)}{5 d}-\frac{b (a+b)^2 \tanh ^3(c+d x)}{d}+\frac{(a+b)^3 \tanh (c+d x)}{d}-\frac{b^3 \tanh ^7(c+d x)}{7 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.079839, antiderivative size = 74, 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, 194} \[ \frac{3 b^2 (a+b) \tanh ^5(c+d x)}{5 d}-\frac{b (a+b)^2 \tanh ^3(c+d x)}{d}+\frac{(a+b)^3 \tanh (c+d x)}{d}-\frac{b^3 \tanh ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b)^3*Tanh[c + d*x])/d - (b*(a + b)^2*Tanh[c + d*x]^3)/d + (3*b^2*(a + b)*Tanh[c + d*x]^5)/(5*d) - (b^3*T
anh[c + d*x]^7)/(7*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 194

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

Rubi steps

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

Mathematica [B]  time = 1.41505, size = 319, normalized size = 4.31 \[ \frac{\text{sech}(c) \text{sech}(c+d x) \left (-35 a \left (15 a^2+26 a b+16 b^2\right ) \sinh (2 c+d x)+140 \left (11 a^2 b+5 a^3+10 a b^2+4 b^3\right ) \sinh (d x)+1260 a^2 b \sinh (2 c+3 d x)-210 a^2 b \sinh (4 c+3 d x)+490 a^2 b \sinh (4 c+5 d x)+70 a^2 b \sinh (6 c+7 d x)+525 a^3 \sinh (2 c+3 d x)-210 a^3 \sinh (4 c+3 d x)+210 a^3 \sinh (4 c+5 d x)-35 a^3 \sinh (6 c+5 d x)+35 a^3 \sinh (6 c+7 d x)+1176 a b^2 \sinh (2 c+3 d x)+392 a b^2 \sinh (4 c+5 d x)+56 a b^2 \sinh (6 c+7 d x)+336 b^3 \sinh (2 c+3 d x)+112 b^3 \sinh (4 c+5 d x)+16 b^3 \sinh (6 c+7 d x)\right ) \left (a+b \text{sech}^2(c+d x)\right )^3}{280 d (a \cosh (2 (c+d x))+a+2 b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sech[c]*Sech[c + d*x]*(a + b*Sech[c + d*x]^2)^3*(140*(5*a^3 + 11*a^2*b + 10*a*b^2 + 4*b^3)*Sinh[d*x] - 35*a*(
15*a^2 + 26*a*b + 16*b^2)*Sinh[2*c + d*x] + 525*a^3*Sinh[2*c + 3*d*x] + 1260*a^2*b*Sinh[2*c + 3*d*x] + 1176*a*
b^2*Sinh[2*c + 3*d*x] + 336*b^3*Sinh[2*c + 3*d*x] - 210*a^3*Sinh[4*c + 3*d*x] - 210*a^2*b*Sinh[4*c + 3*d*x] +
210*a^3*Sinh[4*c + 5*d*x] + 490*a^2*b*Sinh[4*c + 5*d*x] + 392*a*b^2*Sinh[4*c + 5*d*x] + 112*b^3*Sinh[4*c + 5*d
*x] - 35*a^3*Sinh[6*c + 5*d*x] + 35*a^3*Sinh[6*c + 7*d*x] + 70*a^2*b*Sinh[6*c + 7*d*x] + 56*a*b^2*Sinh[6*c + 7
*d*x] + 16*b^3*Sinh[6*c + 7*d*x]))/(280*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)

________________________________________________________________________________________

Maple [A]  time = 0.026, size = 116, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ({a}^{3}\tanh \left ( dx+c \right ) +3\,{a}^{2}b \left ( 2/3+1/3\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) \tanh \left ( dx+c \right ) +3\,a{b}^{2} \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 ) +{b}^{3} \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 ) \tanh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^3*tanh(d*x+c)+3*a^2*b*(2/3+1/3*sech(d*x+c)^2)*tanh(d*x+c)+3*a*b^2*(8/15+1/5*sech(d*x+c)^4+4/15*sech(d*x
+c)^2)*tanh(d*x+c)+b^3*(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 = 1.23289, size = 938, normalized size = 12.68 \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)^2*(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

32/35*b^3*(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 - 14*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*b^2*(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*a^2*b*(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))) + 2*a^3/(d*
(e^(-2*d*x - 2*c) + 1))

________________________________________________________________________________________

Fricas [B]  time = 2.05481, size = 2103, normalized size = 28.42 \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)^2*(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-4/35*((35*a^3 + 35*a^2*b + 28*a*b^2 + 8*b^3)*cosh(d*x + c)^6 - 6*(35*a^2*b + 28*a*b^2 + 8*b^3)*cosh(d*x + c)*
sinh(d*x + c)^5 + (35*a^3 + 35*a^2*b + 28*a*b^2 + 8*b^3)*sinh(d*x + c)^6 + 14*(15*a^3 + 25*a^2*b + 14*a*b^2 +
4*b^3)*cosh(d*x + c)^4 + (210*a^3 + 350*a^2*b + 196*a*b^2 + 56*b^3 + 15*(35*a^3 + 35*a^2*b + 28*a*b^2 + 8*b^3)
*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 4*(5*(35*a^2*b + 28*a*b^2 + 8*b^3)*cosh(d*x + c)^3 + 28*(5*a^2*b + 7*a*b^2
+ 2*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 350*a^3 + 770*a^2*b + 700*a*b^2 + 280*b^3 + 7*(75*a^3 + 155*a^2*b +
124*a*b^2 + 24*b^3)*cosh(d*x + c)^2 + (15*(35*a^3 + 35*a^2*b + 28*a*b^2 + 8*b^3)*cosh(d*x + c)^4 + 525*a^3 +
1085*a^2*b + 868*a*b^2 + 168*b^3 + 84*(15*a^3 + 25*a^2*b + 14*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2
- 2*(3*(35*a^2*b + 28*a*b^2 + 8*b^3)*cosh(d*x + c)^5 + 56*(5*a^2*b + 7*a*b^2 + 2*b^3)*cosh(d*x + c)^3 + 7*(25*
a^2*b + 44*a*b^2 + 24*b^3)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^8 + 8*d*cosh(d*x + c)*sinh(d*x + c)^
7 + d*sinh(d*x + c)^8 + 8*d*cosh(d*x + c)^6 + 4*(7*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^6 + 4*(14*d*cosh(d*x
+ c)^3 + 9*d*cosh(d*x + c))*sinh(d*x + c)^5 + 28*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4 + 60*d*cosh(d*x
+ c)^2 + 14*d)*sinh(d*x + c)^4 + 8*(7*d*cosh(d*x + c)^5 + 15*d*cosh(d*x + c)^3 + 7*d*cosh(d*x + c))*sinh(d*x +
c)^3 + 56*d*cosh(d*x + c)^2 + 4*(7*d*cosh(d*x + c)^6 + 30*d*cosh(d*x + c)^4 + 42*d*cosh(d*x + c)^2 + 14*d)*si
nh(d*x + c)^2 + 4*(2*d*cosh(d*x + c)^7 + 9*d*cosh(d*x + c)^5 + 14*d*cosh(d*x + c)^3 + 7*d*cosh(d*x + c))*sinh(
d*x + c) + 35*d)

________________________________________________________________________________________

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}^{2}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.16792, size = 408, normalized size = 5.51 \begin{align*} -\frac{2 \,{\left (35 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 210 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 210 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 525 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 910 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 560 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 700 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1540 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 1400 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 560 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 525 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 1260 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 1176 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 336 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 210 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 490 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 392 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 112 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 35 \, a^{3} + 70 \, a^{2} b + 56 \, a b^{2} + 16 \, b^{3}\right )}}{35 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(35*a^3*e^(12*d*x + 12*c) + 210*a^3*e^(10*d*x + 10*c) + 210*a^2*b*e^(10*d*x + 10*c) + 525*a^3*e^(8*d*x +
8*c) + 910*a^2*b*e^(8*d*x + 8*c) + 560*a*b^2*e^(8*d*x + 8*c) + 700*a^3*e^(6*d*x + 6*c) + 1540*a^2*b*e^(6*d*x
+ 6*c) + 1400*a*b^2*e^(6*d*x + 6*c) + 560*b^3*e^(6*d*x + 6*c) + 525*a^3*e^(4*d*x + 4*c) + 1260*a^2*b*e^(4*d*x
+ 4*c) + 1176*a*b^2*e^(4*d*x + 4*c) + 336*b^3*e^(4*d*x + 4*c) + 210*a^3*e^(2*d*x + 2*c) + 490*a^2*b*e^(2*d*x +
2*c) + 392*a*b^2*e^(2*d*x + 2*c) + 112*b^3*e^(2*d*x + 2*c) + 35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)/(d*(e^(2*
d*x + 2*c) + 1)^7)