3.2.36 \(\int (a+b \text {sech}^2(c+d x))^4 \, dx\) [136]

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

[Out]

a^4*x+b*(2*a+b)*(2*a^2+2*a*b+b^2)*tanh(d*x+c)/d-1/3*b^2*(6*a^2+8*a*b+3*b^2)*tanh(d*x+c)^3/d+1/5*b^3*(4*a+3*b)*
tanh(d*x+c)^5/d-1/7*b^4*tanh(d*x+c)^7/d

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Rubi [A]
time = 0.05, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 = {4213, 398, 212} \begin {gather*} a^4 x-\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}+\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 {gather*}

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 212

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

Rule 398

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 4213

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]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(455\) vs. \(2(111)=222\).
time = 1.11, size = 455, normalized size = 4.10 \begin {gather*} \frac {\text {sech}(c) \text {sech}^7(c+d x) \left (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)+11200 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 \sinh (6 c+7 d x)+96 b^4 \sinh (6 c+7 d x)\right )}{13440 d} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(309\) vs. \(2(105)=210\).
time = 1.86, size = 310, normalized size = 2.79

method	result	size
risch	\(x \,a^{4}-\frac {8 b \left (105 a^{3} {\mathrm e}^{12 d x +12 c}+630 a^{3} {\mathrm e}^{10 d x +10 c}+315 a^{2} b \,{\mathrm e}^{10 d x +10 c}+1575 a^{3} {\mathrm e}^{8 d x +8 c}+1365 a^{2} b \,{\mathrm e}^{8 d x +8 c}+560 a \,b^{2} {\mathrm e}^{8 d x +8 c}+2100 a^{3} {\mathrm e}^{6 d x +6 c}+2310 a^{2} b \,{\mathrm e}^{6 d x +6 c}+1400 a \,b^{2} {\mathrm e}^{6 d x +6 c}+420 b^{3} {\mathrm e}^{6 d x +6 c}+1575 a^{3} {\mathrm e}^{4 d x +4 c}+1890 a^{2} b \,{\mathrm e}^{4 d x +4 c}+1176 a \,b^{2} {\mathrm e}^{4 d x +4 c}+252 b^{3} {\mathrm e}^{4 d x +4 c}+630 a^{3} {\mathrm e}^{2 d x +2 c}+735 a^{2} b \,{\mathrm e}^{2 d x +2 c}+392 a \,b^{2} {\mathrm e}^{2 d x +2 c}+84 b^{3} {\mathrm e}^{2 d x +2 c}+105 a^{3}+105 a^{2} b +56 a \,b^{2}+12 b^{3}\right )}{105 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{7}}\)	\(310\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sech(d*x+c)^2)^4,x,method=_RETURNVERBOSE)

[Out]

x*a^4-8/105*b*(105*a^3*exp(12*d*x+12*c)+630*a^3*exp(10*d*x+10*c)+315*a^2*b*exp(10*d*x+10*c)+1575*a^3*exp(8*d*x
+8*c)+1365*a^2*b*exp(8*d*x+8*c)+560*a*b^2*exp(8*d*x+8*c)+2100*a^3*exp(6*d*x+6*c)+2310*a^2*b*exp(6*d*x+6*c)+140
0*a*b^2*exp(6*d*x+6*c)+420*b^3*exp(6*d*x+6*c)+1575*a^3*exp(4*d*x+4*c)+1890*a^2*b*exp(4*d*x+4*c)+1176*a*b^2*exp
(4*d*x+4*c)+252*b^3*exp(4*d*x+4*c)+630*a^3*exp(2*d*x+2*c)+735*a^2*b*exp(2*d*x+2*c)+392*a*b^2*exp(2*d*x+2*c)+84
*b^3*exp(2*d*x+2*c)+105*a^3+105*a^2*b+56*a*b^2+12*b^3)/d/(1+exp(2*d*x+2*c))^7

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (105) = 210\).
time = 0.29, size = 703, normalized size = 6.33 \begin {gather*} a^{4} x + \frac {32}{35} \, b^{4} {\left (\frac {7 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}} + \frac {21 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}} + \frac {35 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}} + \frac {1}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}}\right )} + \frac {64}{15} \, a b^{3} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + 8 \, a^{2} b^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + \frac {8 \, a^{3} b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 941 vs. \(2 (105) = 210\).
time = 0.39, size = 941, normalized size = 8.48 \begin {gather*} \frac {{\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )^{7} + 7 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + 4 \, {\left (105 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \sinh \left (d x + c\right )^{7} + 7 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )^{5} + 28 \, {\left (75 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4} + 3 \, {\left (105 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} + 35 \, {\left ({\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )^{3} + {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 21 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )^{3} + 28 \, {\left (5 \, {\left (105 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cosh \left (d x + c\right )^{4} + 135 \, a^{3} b + 225 \, a^{2} b^{2} + 168 \, a b^{3} + 36 \, b^{4} + 10 \, {\left (75 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 7 \, {\left (3 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )^{5} + 10 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )^{3} + 9 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right ) + 28 \, {\left ({\left (105 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cosh \left (d x + c\right )^{6} + 5 \, {\left (75 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cosh \left (d x + c\right )^{4} + 75 \, a^{3} b + 135 \, a^{2} b^{2} + 120 \, a b^{3} + 60 \, b^{4} + 9 \, {\left (45 \, a^{3} b + 75 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{105 \, {\left (d \cosh \left (d x + c\right )^{7} + 7 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + 7 \, d \cosh \left (d x + c\right )^{5} + 35 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 21 \, d \cosh \left (d x + c\right )^{3} + 7 \, {\left (3 \, d \cosh \left (d x + c\right )^{5} + 10 \, d \cosh \left (d x + c\right )^{3} + 9 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, d \cosh \left (d x + c\right )\right )}} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{4}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (105) = 210\).
time = 0.42, size = 334, normalized size = 3.01 \begin {gather*} \frac {105 \, {\left (d x + c\right )} a^{4} - \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 )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}}}{105 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(105*(d*x + c)*a^4 - 8*(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 + 10*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)/(e^(2*d*x + 2*c) + 1)^7)/d

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Mupad [B]
time = 0.20, size = 1083, normalized size = 9.76 \begin {gather*} a^4\,x-\frac {\frac {8\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{21\,d}+\frac {16\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{21\,d}+\frac {16\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (5\,a^3\,b+9\,a^2\,b^2+8\,a\,b^3+4\,b^4\right )}{7\,d}+\frac {40\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{10\,c+10\,d\,x}}{7\,d}}{6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1}-\frac {\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{7\,d}+\frac {8\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{7\,d}+\frac {8\,a^3\,b}{7\,d}+\frac {32\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (5\,a^3\,b+9\,a^2\,b^2+8\,a\,b^3+4\,b^4\right )}{7\,d}+\frac {48\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {48\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{12\,c+12\,d\,x}}{7\,d}}{7\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}+35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}+7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}+1}-\frac {\frac {8\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {8\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{105\,d}+\frac {16\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{35\,d}+\frac {32\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (5\,a^3\,b+9\,a^2\,b^2+8\,a\,b^3+4\,b^4\right )}{35\,d}+\frac {32\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}}{7\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1}-\frac {\frac {8\,\left (5\,a^3\,b+9\,a^2\,b^2+8\,a\,b^3+4\,b^4\right )}{35\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{35\,d}+\frac {24\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{7\,d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {\frac {8\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{105\,d}+\frac {16\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{7\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {8\,a^3\,b}{7\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cosh(c + d*x)^2)^4,x)

[Out]

a^4*x - ((8*(a^3*b + a^2*b^2))/(7*d) + (8*exp(2*c + 2*d*x)*(16*a*b^3 + 15*a^3*b + 24*a^2*b^2))/(21*d) + (16*ex
p(6*c + 6*d*x)*(16*a*b^3 + 15*a^3*b + 24*a^2*b^2))/(21*d) + (16*exp(4*c + 4*d*x)*(8*a*b^3 + 5*a^3*b + 4*b^4 +
9*a^2*b^2))/(7*d) + (40*exp(8*c + 8*d*x)*(a^3*b + a^2*b^2))/(7*d) + (8*a^3*b*exp(10*c + 10*d*x))/(7*d))/(6*exp
(2*c + 2*d*x) + 15*exp(4*c + 4*d*x) + 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) + 6*exp(10*c + 10*d*x) + exp(1
2*c + 12*d*x) + 1) - ((8*exp(4*c + 4*d*x)*(16*a*b^3 + 15*a^3*b + 24*a^2*b^2))/(7*d) + (8*exp(8*c + 8*d*x)*(16*
a*b^3 + 15*a^3*b + 24*a^2*b^2))/(7*d) + (8*a^3*b)/(7*d) + (32*exp(6*c + 6*d*x)*(8*a*b^3 + 5*a^3*b + 4*b^4 + 9*
a^2*b^2))/(7*d) + (48*exp(2*c + 2*d*x)*(a^3*b + a^2*b^2))/(7*d) + (48*exp(10*c + 10*d*x)*(a^3*b + a^2*b^2))/(7
*d) + (8*a^3*b*exp(12*c + 12*d*x))/(7*d))/(7*exp(2*c + 2*d*x) + 21*exp(4*c + 4*d*x) + 35*exp(6*c + 6*d*x) + 35
*exp(8*c + 8*d*x) + 21*exp(10*c + 10*d*x) + 7*exp(12*c + 12*d*x) + exp(14*c + 14*d*x) + 1) - ((8*(a^3*b + a^2*
b^2))/(7*d) + (8*a^3*b*exp(2*c + 2*d*x))/(7*d))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1) - ((8*(16*a*b^3 +
15*a^3*b + 24*a^2*b^2))/(105*d) + (16*exp(4*c + 4*d*x)*(16*a*b^3 + 15*a^3*b + 24*a^2*b^2))/(35*d) + (32*exp(2*
c + 2*d*x)*(8*a*b^3 + 5*a^3*b + 4*b^4 + 9*a^2*b^2))/(35*d) + (32*exp(6*c + 6*d*x)*(a^3*b + a^2*b^2))/(7*d) + (
8*a^3*b*exp(8*c + 8*d*x))/(7*d))/(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) + 5*exp(8*c +
8*d*x) + exp(10*c + 10*d*x) + 1) - ((8*(8*a*b^3 + 5*a^3*b + 4*b^4 + 9*a^2*b^2))/(35*d) + (8*exp(2*c + 2*d*x)*
(16*a*b^3 + 15*a^3*b + 24*a^2*b^2))/(35*d) + (24*exp(4*c + 4*d*x)*(a^3*b + a^2*b^2))/(7*d) + (8*a^3*b*exp(6*c
+ 6*d*x))/(7*d))/(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1) - ((8*(
16*a*b^3 + 15*a^3*b + 24*a^2*b^2))/(105*d) + (16*exp(2*c + 2*d*x)*(a^3*b + a^2*b^2))/(7*d) + (8*a^3*b*exp(4*c
+ 4*d*x))/(7*d))/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1) - (8*a^3*b)/(7*d*(exp(2*c +
2*d*x) + 1))

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