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

Optimal. Leaf size=111 \[ -\frac{b^2 \left (6 a^2+8 a b+3 b^2\right ) \tanh ^3(c+d x)}{3 d}+\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tanh (c+d x)}{d}+a^4 x+\frac{b^3 (4 a+3 b) \tanh ^5(c+d x)}{5 d}-\frac{b^4 \tanh ^7(c+d x)}{7 d} \]

[Out]

a^4*x + (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tanh[c + d*x])/d - (b^2*(6*a^2 + 8*a*b + 3*b^2)*Tanh[c + d*x]^3)/(3
*d) + (b^3*(4*a + 3*b)*Tanh[c + d*x]^5)/(5*d) - (b^4*Tanh[c + d*x]^7)/(7*d)

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Rubi [A]  time = 0.0700309, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4128, 390, 206} \[ -\frac{b^2 \left (6 a^2+8 a b+3 b^2\right ) \tanh ^3(c+d x)}{3 d}+\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tanh (c+d x)}{d}+a^4 x+\frac{b^3 (4 a+3 b) \tanh ^5(c+d x)}{5 d}-\frac{b^4 \tanh ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sech[c + d*x]^2)^4,x]

[Out]

a^4*x + (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tanh[c + d*x])/d - (b^2*(6*a^2 + 8*a*b + 3*b^2)*Tanh[c + d*x]^3)/(3
*d) + (b^3*(4*a + 3*b)*Tanh[c + d*x]^5)/(5*d) - (b^4*Tanh[c + d*x]^7)/(7*d)

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rule 390

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

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

Mathematica [B]  time = 1.59919, size = 455, normalized size = 4.1 \[ \frac{\text{sech}(c) \text{sech}^7(c+d x) \left (-10920 a^2 b^2 \sinh (2 c+d x)+15120 a^2 b^2 \sinh (2 c+3 d x)-2520 a^2 b^2 \sinh (4 c+3 d x)+5880 a^2 b^2 \sinh (4 c+5 d x)+840 a^2 b^2 \sinh (6 c+7 d x)+18480 a^2 b^2 \sinh (d x)-12600 a^3 b \sinh (2 c+d x)+12600 a^3 b \sinh (2 c+3 d x)-5040 a^3 b \sinh (4 c+3 d x)+5040 a^3 b \sinh (4 c+5 d x)-840 a^3 b \sinh (6 c+5 d x)+840 a^3 b \sinh (6 c+7 d x)+16800 a^3 b \sinh (d x)+3675 a^4 d x \cosh (2 c+d x)+2205 a^4 d x \cosh (2 c+3 d x)+2205 a^4 d x \cosh (4 c+3 d x)+735 a^4 d x \cosh (4 c+5 d x)+735 a^4 d x \cosh (6 c+5 d x)+105 a^4 d x \cosh (6 c+7 d x)+105 a^4 d x \cosh (8 c+7 d x)+3675 a^4 d x \cosh (d x)-4480 a b^3 \sinh (2 c+d x)+9408 a b^3 \sinh (2 c+3 d x)+3136 a b^3 \sinh (4 c+5 d x)+448 a b^3 \sinh (6 c+7 d x)+11200 a b^3 \sinh (d x)+2016 b^4 \sinh (2 c+3 d x)+672 b^4 \sinh (4 c+5 d x)+96 b^4 \sinh (6 c+7 d x)+3360 b^4 \sinh (d x)\right )}{13440 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sech[c]*Sech[c + d*x]^7*(3675*a^4*d*x*Cosh[d*x] + 3675*a^4*d*x*Cosh[2*c + d*x] + 2205*a^4*d*x*Cosh[2*c + 3*d*
x] + 2205*a^4*d*x*Cosh[4*c + 3*d*x] + 735*a^4*d*x*Cosh[4*c + 5*d*x] + 735*a^4*d*x*Cosh[6*c + 5*d*x] + 105*a^4*
d*x*Cosh[6*c + 7*d*x] + 105*a^4*d*x*Cosh[8*c + 7*d*x] + 16800*a^3*b*Sinh[d*x] + 18480*a^2*b^2*Sinh[d*x] + 1120
0*a*b^3*Sinh[d*x] + 3360*b^4*Sinh[d*x] - 12600*a^3*b*Sinh[2*c + d*x] - 10920*a^2*b^2*Sinh[2*c + d*x] - 4480*a*
b^3*Sinh[2*c + d*x] + 12600*a^3*b*Sinh[2*c + 3*d*x] + 15120*a^2*b^2*Sinh[2*c + 3*d*x] + 9408*a*b^3*Sinh[2*c +
3*d*x] + 2016*b^4*Sinh[2*c + 3*d*x] - 5040*a^3*b*Sinh[4*c + 3*d*x] - 2520*a^2*b^2*Sinh[4*c + 3*d*x] + 5040*a^3
*b*Sinh[4*c + 5*d*x] + 5880*a^2*b^2*Sinh[4*c + 5*d*x] + 3136*a*b^3*Sinh[4*c + 5*d*x] + 672*b^4*Sinh[4*c + 5*d*
x] - 840*a^3*b*Sinh[6*c + 5*d*x] + 840*a^3*b*Sinh[6*c + 7*d*x] + 840*a^2*b^2*Sinh[6*c + 7*d*x] + 448*a*b^3*Sin
h[6*c + 7*d*x] + 96*b^4*Sinh[6*c + 7*d*x]))/(13440*d)

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Maple [A]  time = 0.032, size = 129, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{4} \left ( dx+c \right ) +4\,{a}^{3}b\tanh \left ( dx+c \right ) +6\,{a}^{2}{b}^{2} \left ( 2/3+1/3\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) \tanh \left ( dx+c \right ) +4\,a{b}^{3} \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}^{4} \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((a+b*sech(d*x+c)^2)^4,x)

[Out]

1/d*(a^4*(d*x+c)+4*a^3*b*tanh(d*x+c)+6*a^2*b^2*(2/3+1/3*sech(d*x+c)^2)*tanh(d*x+c)+4*a*b^3*(8/15+1/5*sech(d*x+
c)^4+4/15*sech(d*x+c)^2)*tanh(d*x+c)+b^4*(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))

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Maxima [B]  time = 1.46072, size = 949, normalized size = 8.55 \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)^4,x, algorithm="maxima")

[Out]

a^4*x + 32/35*b^4*(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))) + 64/15*a*b^3*(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))) + 8*a^2*b^2*(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)))
+ 8*a^3*b/(d*(e^(-2*d*x - 2*c) + 1))

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Fricas [B]  time = 2.17041, size = 2410, normalized size = 21.71 \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)^4,x, algorithm="fricas")

[Out]

1/105*((105*a^4*d*x - 420*a^3*b - 420*a^2*b^2 - 224*a*b^3 - 48*b^4)*cosh(d*x + c)^7 + 7*(105*a^4*d*x - 420*a^3
*b - 420*a^2*b^2 - 224*a*b^3 - 48*b^4)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(105*a^3*b + 105*a^2*b^2 + 56*a*b^3 +
12*b^4)*sinh(d*x + c)^7 + 7*(105*a^4*d*x - 420*a^3*b - 420*a^2*b^2 - 224*a*b^3 - 48*b^4)*cosh(d*x + c)^5 + 28
*(75*a^3*b + 105*a^2*b^2 + 56*a*b^3 + 12*b^4 + 3*(105*a^3*b + 105*a^2*b^2 + 56*a*b^3 + 12*b^4)*cosh(d*x + c)^2
)*sinh(d*x + c)^5 + 35*((105*a^4*d*x - 420*a^3*b - 420*a^2*b^2 - 224*a*b^3 - 48*b^4)*cosh(d*x + c)^3 + (105*a^
4*d*x - 420*a^3*b - 420*a^2*b^2 - 224*a*b^3 - 48*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 + 21*(105*a^4*d*x - 420*a
^3*b - 420*a^2*b^2 - 224*a*b^3 - 48*b^4)*cosh(d*x + c)^3 + 28*(5*(105*a^3*b + 105*a^2*b^2 + 56*a*b^3 + 12*b^4)
*cosh(d*x + c)^4 + 135*a^3*b + 225*a^2*b^2 + 168*a*b^3 + 36*b^4 + 10*(75*a^3*b + 105*a^2*b^2 + 56*a*b^3 + 12*b
^4)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 7*(3*(105*a^4*d*x - 420*a^3*b - 420*a^2*b^2 - 224*a*b^3 - 48*b^4)*cosh(
d*x + c)^5 + 10*(105*a^4*d*x - 420*a^3*b - 420*a^2*b^2 - 224*a*b^3 - 48*b^4)*cosh(d*x + c)^3 + 9*(105*a^4*d*x
- 420*a^3*b - 420*a^2*b^2 - 224*a*b^3 - 48*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + 35*(105*a^4*d*x - 420*a^3*b -
420*a^2*b^2 - 224*a*b^3 - 48*b^4)*cosh(d*x + c) + 28*((105*a^3*b + 105*a^2*b^2 + 56*a*b^3 + 12*b^4)*cosh(d*x
+ c)^6 + 5*(75*a^3*b + 105*a^2*b^2 + 56*a*b^3 + 12*b^4)*cosh(d*x + c)^4 + 75*a^3*b + 135*a^2*b^2 + 120*a*b^3 +
60*b^4 + 9*(45*a^3*b + 75*a^2*b^2 + 56*a*b^3 + 12*b^4)*cosh(d*x + c)^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))

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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 )^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.15895, size = 451, normalized size = 4.06 \begin{align*} \frac{{\left (d x + c\right )} a^{4}}{d} - \frac{8 \,{\left (105 \, a^{3} b e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{3} b e^{\left (10 \, d x + 10 \, c\right )} + 315 \, a^{2} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} b e^{\left (8 \, d x + 8 \, c\right )} + 1365 \, a^{2} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 560 \, a b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 2100 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 2310 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1400 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 420 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 1890 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 1176 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 252 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 630 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 735 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 392 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 84 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )}}{105 \, 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((a+b*sech(d*x+c)^2)^4,x, algorithm="giac")

[Out]

(d*x + c)*a^4/d - 8/105*(105*a^3*b*e^(12*d*x + 12*c) + 630*a^3*b*e^(10*d*x + 10*c) + 315*a^2*b^2*e^(10*d*x + 1
0*c) + 1575*a^3*b*e^(8*d*x + 8*c) + 1365*a^2*b^2*e^(8*d*x + 8*c) + 560*a*b^3*e^(8*d*x + 8*c) + 2100*a^3*b*e^(6
*d*x + 6*c) + 2310*a^2*b^2*e^(6*d*x + 6*c) + 1400*a*b^3*e^(6*d*x + 6*c) + 420*b^4*e^(6*d*x + 6*c) + 1575*a^3*b
*e^(4*d*x + 4*c) + 1890*a^2*b^2*e^(4*d*x + 4*c) + 1176*a*b^3*e^(4*d*x + 4*c) + 252*b^4*e^(4*d*x + 4*c) + 630*a
^3*b*e^(2*d*x + 2*c) + 735*a^2*b^2*e^(2*d*x + 2*c) + 392*a*b^3*e^(2*d*x + 2*c) + 84*b^4*e^(2*d*x + 2*c) + 105*
a^3*b + 105*a^2*b^2 + 56*a*b^3 + 12*b^4)/(d*(e^(2*d*x + 2*c) + 1)^7)