3.137 \(\int (a+b \text{sech}^2(c+d x))^5 \, dx\)

Optimal. Leaf size=163 \[ \frac{b^3 \left (10 a^2+15 a b+6 b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac{b^2 \left (20 a^2 b+10 a^3+15 a b^2+4 b^3\right ) \tanh ^3(c+d x)}{3 d}+\frac{b \left (10 a^2 b^2+10 a^3 b+5 a^4+5 a b^3+b^4\right ) \tanh (c+d x)}{d}+a^5 x-\frac{b^4 (5 a+4 b) \tanh ^7(c+d x)}{7 d}+\frac{b^5 \tanh ^9(c+d x)}{9 d} \]

[Out]

a^5*x + (b*(5*a^4 + 10*a^3*b + 10*a^2*b^2 + 5*a*b^3 + b^4)*Tanh[c + d*x])/d - (b^2*(10*a^3 + 20*a^2*b + 15*a*b
^2 + 4*b^3)*Tanh[c + d*x]^3)/(3*d) + (b^3*(10*a^2 + 15*a*b + 6*b^2)*Tanh[c + d*x]^5)/(5*d) - (b^4*(5*a + 4*b)*
Tanh[c + d*x]^7)/(7*d) + (b^5*Tanh[c + d*x]^9)/(9*d)

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Rubi [A]  time = 0.0952181, antiderivative size = 163, 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} \[ \frac{b^3 \left (10 a^2+15 a b+6 b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac{b^2 \left (20 a^2 b+10 a^3+15 a b^2+4 b^3\right ) \tanh ^3(c+d x)}{3 d}+\frac{b \left (10 a^2 b^2+10 a^3 b+5 a^4+5 a b^3+b^4\right ) \tanh (c+d x)}{d}+a^5 x-\frac{b^4 (5 a+4 b) \tanh ^7(c+d x)}{7 d}+\frac{b^5 \tanh ^9(c+d x)}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sech[c + d*x]^2)^5,x]

[Out]

a^5*x + (b*(5*a^4 + 10*a^3*b + 10*a^2*b^2 + 5*a*b^3 + b^4)*Tanh[c + d*x])/d - (b^2*(10*a^3 + 20*a^2*b + 15*a*b
^2 + 4*b^3)*Tanh[c + d*x]^3)/(3*d) + (b^3*(10*a^2 + 15*a*b + 6*b^2)*Tanh[c + d*x]^5)/(5*d) - (b^4*(5*a + 4*b)*
Tanh[c + d*x]^7)/(7*d) + (b^5*Tanh[c + d*x]^9)/(9*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 )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-b x^2\right )^5}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right )-b^2 \left (10 a^3+20 a^2 b+15 a b^2+4 b^3\right ) x^2+b^3 \left (10 a^2+15 a b+6 b^2\right ) x^4-b^4 (5 a+4 b) x^6+b^5 x^8+\frac{a^5}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac{b^2 \left (10 a^3+20 a^2 b+15 a b^2+4 b^3\right ) \tanh ^3(c+d x)}{3 d}+\frac{b^3 \left (10 a^2+15 a b+6 b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac{b^4 (5 a+4 b) \tanh ^7(c+d x)}{7 d}+\frac{b^5 \tanh ^9(c+d x)}{9 d}+\frac{a^5 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^5 x+\frac{b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac{b^2 \left (10 a^3+20 a^2 b+15 a b^2+4 b^3\right ) \tanh ^3(c+d x)}{3 d}+\frac{b^3 \left (10 a^2+15 a b+6 b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac{b^4 (5 a+4 b) \tanh ^7(c+d x)}{7 d}+\frac{b^5 \tanh ^9(c+d x)}{9 d}\\ \end{align*}

Mathematica [B]  time = 6.54179, size = 724, normalized size = 4.44 \[ \frac{32 \text{sech}(c) \cosh ^9(c+d x) \left (1680 a^2 b^3 \sinh (d x)+2100 a^3 b^2 \sinh (d x)+1575 a^4 b \sinh (d x)+720 a b^4 \sinh (d x)+128 b^5 \sinh (d x)\right ) \left (a+b \text{sech}^2(c+d x)\right )^5}{315 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{64 \text{sech}(c) \left (420 a^2 b^3 \sinh (c)+525 a^3 b^2 \sinh (c)+180 a b^4 \sinh (c)+32 b^5 \sinh (c)\right ) \cosh ^8(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^5}{315 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{64 \text{sech}(c) \cosh ^7(c+d x) \left (420 a^2 b^3 \sinh (d x)+525 a^3 b^2 \sinh (d x)+180 a b^4 \sinh (d x)+32 b^5 \sinh (d x)\right ) \left (a+b \text{sech}^2(c+d x)\right )^5}{315 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{64 \text{sech}(c) \left (105 a^2 b^3 \sinh (c)+45 a b^4 \sinh (c)+8 b^5 \sinh (c)\right ) \cosh ^6(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^5}{105 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{64 \text{sech}(c) \cosh ^5(c+d x) \left (105 a^2 b^3 \sinh (d x)+45 a b^4 \sinh (d x)+8 b^5 \sinh (d x)\right ) \left (a+b \text{sech}^2(c+d x)\right )^5}{105 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{32 a^5 x \cosh ^{10}(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^5}{(a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{32 \text{sech}(c) \left (45 a b^4 \sinh (c)+8 b^5 \sinh (c)\right ) \cosh ^4(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^5}{63 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{32 \text{sech}(c) \cosh ^3(c+d x) \left (45 a b^4 \sinh (d x)+8 b^5 \sinh (d x)\right ) \left (a+b \text{sech}^2(c+d x)\right )^5}{63 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{32 b^5 \tanh (c) \cosh ^2(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^5}{9 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{32 b^5 \text{sech}(c) \sinh (d x) \cosh (c+d x) \left (a+b \text{sech}^2(c+d x)\right )^5}{9 d (a \cosh (2 c+2 d x)+a+2 b)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sech[c + d*x]^2)^5,x]

[Out]

(32*a^5*x*Cosh[c + d*x]^10*(a + b*Sech[c + d*x]^2)^5)/(a + 2*b + a*Cosh[2*c + 2*d*x])^5 + (32*Cosh[c + d*x]^4*
Sech[c]*(a + b*Sech[c + d*x]^2)^5*(45*a*b^4*Sinh[c] + 8*b^5*Sinh[c]))/(63*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5)
+ (64*Cosh[c + d*x]^6*Sech[c]*(a + b*Sech[c + d*x]^2)^5*(105*a^2*b^3*Sinh[c] + 45*a*b^4*Sinh[c] + 8*b^5*Sinh[
c]))/(105*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5) + (64*Cosh[c + d*x]^8*Sech[c]*(a + b*Sech[c + d*x]^2)^5*(525*a^
3*b^2*Sinh[c] + 420*a^2*b^3*Sinh[c] + 180*a*b^4*Sinh[c] + 32*b^5*Sinh[c]))/(315*d*(a + 2*b + a*Cosh[2*c + 2*d*
x])^5) + (32*b^5*Cosh[c + d*x]*Sech[c]*(a + b*Sech[c + d*x]^2)^5*Sinh[d*x])/(9*d*(a + 2*b + a*Cosh[2*c + 2*d*x
])^5) + (32*Cosh[c + d*x]^3*Sech[c]*(a + b*Sech[c + d*x]^2)^5*(45*a*b^4*Sinh[d*x] + 8*b^5*Sinh[d*x]))/(63*d*(a
+ 2*b + a*Cosh[2*c + 2*d*x])^5) + (64*Cosh[c + d*x]^5*Sech[c]*(a + b*Sech[c + d*x]^2)^5*(105*a^2*b^3*Sinh[d*x
] + 45*a*b^4*Sinh[d*x] + 8*b^5*Sinh[d*x]))/(105*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5) + (64*Cosh[c + d*x]^7*Sec
h[c]*(a + b*Sech[c + d*x]^2)^5*(525*a^3*b^2*Sinh[d*x] + 420*a^2*b^3*Sinh[d*x] + 180*a*b^4*Sinh[d*x] + 32*b^5*S
inh[d*x]))/(315*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5) + (32*Cosh[c + d*x]^9*Sech[c]*(a + b*Sech[c + d*x]^2)^5*(
1575*a^4*b*Sinh[d*x] + 2100*a^3*b^2*Sinh[d*x] + 1680*a^2*b^3*Sinh[d*x] + 720*a*b^4*Sinh[d*x] + 128*b^5*Sinh[d*
x]))/(315*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5) + (32*b^5*Cosh[c + d*x]^2*(a + b*Sech[c + d*x]^2)^5*Tanh[c])/(9
*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5)

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Maple [A]  time = 0.039, size = 185, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({a}^{5} \left ( dx+c \right ) +5\,{a}^{4}b\tanh \left ( dx+c \right ) +10\,{a}^{3}{b}^{2} \left ( 2/3+1/3\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) \tanh \left ( dx+c \right ) +10\,{a}^{2}{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 ) +5\,a{b}^{4} \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}^{5} \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((a+b*sech(d*x+c)^2)^5,x)

[Out]

1/d*(a^5*(d*x+c)+5*a^4*b*tanh(d*x+c)+10*a^3*b^2*(2/3+1/3*sech(d*x+c)^2)*tanh(d*x+c)+10*a^2*b^3*(8/15+1/5*sech(
d*x+c)^4+4/15*sech(d*x+c)^2)*tanh(d*x+c)+5*a*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)+b^5*(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)*ta
nh(d*x+c))

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Maxima [B]  time = 1.21151, size = 1724, normalized size = 10.58 \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)^5,x, algorithm="maxima")

[Out]

a^5*x + 256/315*b^5*(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) + 1
26*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) + 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)) + 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 - 1
2*c) + 36*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))) + 32/7*a*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) + 3
5*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))) + 32/3*a^2*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))) + 40/3*a^3*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))) + 10*a^4*b/(d*(e^(-2*d*x - 2*c) +
1))

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Fricas [B]  time = 2.15477, size = 4396, normalized size = 26.97 \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)^5,x, algorithm="fricas")

[Out]

1/315*((315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x + c)^9 + 9*(315
*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x + c)*sinh(d*x + c)^8 + (15
75*a^4*b + 2100*a^3*b^2 + 1680*a^2*b^3 + 720*a*b^4 + 128*b^5)*sinh(d*x + c)^9 + 9*(315*a^5*d*x - 1575*a^4*b -
2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x + c)^7 + 9*(1225*a^4*b + 2100*a^3*b^2 + 1680*a^2*b
^3 + 720*a*b^4 + 128*b^5 + 4*(1575*a^4*b + 2100*a^3*b^2 + 1680*a^2*b^3 + 720*a*b^4 + 128*b^5)*cosh(d*x + c)^2)
*sinh(d*x + c)^7 + 21*(4*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d
*x + c)^3 + 3*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x + c))*si
nh(d*x + c)^6 + 36*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x + c
)^5 + 9*(3500*a^4*b + 7000*a^3*b^2 + 6720*a^2*b^3 + 2880*a*b^4 + 512*b^5 + 14*(1575*a^4*b + 2100*a^3*b^2 + 168
0*a^2*b^3 + 720*a*b^4 + 128*b^5)*cosh(d*x + c)^4 + 21*(1225*a^4*b + 2100*a^3*b^2 + 1680*a^2*b^3 + 720*a*b^4 +
128*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 9*(14*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 72
0*a*b^4 - 128*b^5)*cosh(d*x + c)^5 + 35*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 -
128*b^5)*cosh(d*x + c)^3 + 20*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*c
osh(d*x + c))*sinh(d*x + c)^4 + 84*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b
^5)*cosh(d*x + c)^3 + 3*(28*(1575*a^4*b + 2100*a^3*b^2 + 1680*a^2*b^3 + 720*a*b^4 + 128*b^5)*cosh(d*x + c)^6 +
14700*a^4*b + 32200*a^3*b^2 + 35840*a^2*b^3 + 20160*a*b^4 + 3584*b^5 + 105*(1225*a^4*b + 2100*a^3*b^2 + 1680*
a^2*b^3 + 720*a*b^4 + 128*b^5)*cosh(d*x + c)^4 + 120*(875*a^4*b + 1750*a^3*b^2 + 1680*a^2*b^3 + 720*a*b^4 + 12
8*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 9*(4*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a
*b^4 - 128*b^5)*cosh(d*x + c)^7 + 21*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128
*b^5)*cosh(d*x + c)^5 + 40*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh
(d*x + c)^3 + 28*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x + c))
*sinh(d*x + c)^2 + 126*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x
+ c) + 9*((1575*a^4*b + 2100*a^3*b^2 + 1680*a^2*b^3 + 720*a*b^4 + 128*b^5)*cosh(d*x + c)^8 + 7*(1225*a^4*b +
2100*a^3*b^2 + 1680*a^2*b^3 + 720*a*b^4 + 128*b^5)*cosh(d*x + c)^6 + 2450*a^4*b + 5600*a^3*b^2 + 6720*a^2*b^3
+ 4480*a*b^4 + 1792*b^5 + 20*(875*a^4*b + 1750*a^3*b^2 + 1680*a^2*b^3 + 720*a*b^4 + 128*b^5)*cosh(d*x + c)^4 +
28*(525*a^4*b + 1150*a^3*b^2 + 1280*a^2*b^3 + 720*a*b^4 + 128*b^5)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*
x + c)^9 + 9*d*cosh(d*x + c)*sinh(d*x + c)^8 + 9*d*cosh(d*x + c)^7 + 21*(4*d*cosh(d*x + c)^3 + 3*d*cosh(d*x +
c))*sinh(d*x + c)^6 + 36*d*cosh(d*x + c)^5 + 9*(14*d*cosh(d*x + c)^5 + 35*d*cosh(d*x + c)^3 + 20*d*cosh(d*x +
c))*sinh(d*x + c)^4 + 84*d*cosh(d*x + c)^3 + 9*(4*d*cosh(d*x + c)^7 + 21*d*cosh(d*x + c)^5 + 40*d*cosh(d*x + c
)^3 + 28*d*cosh(d*x + c))*sinh(d*x + c)^2 + 126*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 )^{5}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)**2)**5,x)

[Out]

Integral((a + b*sech(c + d*x)**2)**5, x)

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Giac [B]  time = 1.18922, size = 725, normalized size = 4.45 \begin{align*} \frac{{\left (d x + c\right )} a^{5}}{d} - \frac{2 \,{\left (1575 \, a^{4} b e^{\left (16 \, d x + 16 \, c\right )} + 12600 \, a^{4} b e^{\left (14 \, d x + 14 \, c\right )} + 6300 \, a^{3} b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 44100 \, a^{4} b e^{\left (12 \, d x + 12 \, c\right )} + 39900 \, a^{3} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 16800 \, a^{2} b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 88200 \, a^{4} b e^{\left (10 \, d x + 10 \, c\right )} + 107100 \, a^{3} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 75600 \, a^{2} b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 25200 \, a b^{4} e^{\left (10 \, d x + 10 \, c\right )} + 110250 \, a^{4} b e^{\left (8 \, d x + 8 \, c\right )} + 157500 \, a^{3} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 136080 \, a^{2} b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 65520 \, a b^{4} e^{\left (8 \, d x + 8 \, c\right )} + 16128 \, b^{5} e^{\left (8 \, d x + 8 \, c\right )} + 88200 \, a^{4} b e^{\left (6 \, d x + 6 \, c\right )} + 136500 \, a^{3} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 124320 \, a^{2} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 60480 \, a b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 10752 \, b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 44100 \, a^{4} b e^{\left (4 \, d x + 4 \, c\right )} + 69300 \, a^{3} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 60480 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 25920 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 4608 \, b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 12600 \, a^{4} b e^{\left (2 \, d x + 2 \, c\right )} + 18900 \, a^{3} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 15120 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 6480 \, a b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 1152 \, b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 1575 \, a^{4} b + 2100 \, a^{3} b^{2} + 1680 \, a^{2} b^{3} + 720 \, a b^{4} + 128 \, b^{5}\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((a+b*sech(d*x+c)^2)^5,x, algorithm="giac")

[Out]

(d*x + c)*a^5/d - 2/315*(1575*a^4*b*e^(16*d*x + 16*c) + 12600*a^4*b*e^(14*d*x + 14*c) + 6300*a^3*b^2*e^(14*d*x
+ 14*c) + 44100*a^4*b*e^(12*d*x + 12*c) + 39900*a^3*b^2*e^(12*d*x + 12*c) + 16800*a^2*b^3*e^(12*d*x + 12*c) +
88200*a^4*b*e^(10*d*x + 10*c) + 107100*a^3*b^2*e^(10*d*x + 10*c) + 75600*a^2*b^3*e^(10*d*x + 10*c) + 25200*a*
b^4*e^(10*d*x + 10*c) + 110250*a^4*b*e^(8*d*x + 8*c) + 157500*a^3*b^2*e^(8*d*x + 8*c) + 136080*a^2*b^3*e^(8*d*
x + 8*c) + 65520*a*b^4*e^(8*d*x + 8*c) + 16128*b^5*e^(8*d*x + 8*c) + 88200*a^4*b*e^(6*d*x + 6*c) + 136500*a^3*
b^2*e^(6*d*x + 6*c) + 124320*a^2*b^3*e^(6*d*x + 6*c) + 60480*a*b^4*e^(6*d*x + 6*c) + 10752*b^5*e^(6*d*x + 6*c)
+ 44100*a^4*b*e^(4*d*x + 4*c) + 69300*a^3*b^2*e^(4*d*x + 4*c) + 60480*a^2*b^3*e^(4*d*x + 4*c) + 25920*a*b^4*e
^(4*d*x + 4*c) + 4608*b^5*e^(4*d*x + 4*c) + 12600*a^4*b*e^(2*d*x + 2*c) + 18900*a^3*b^2*e^(2*d*x + 2*c) + 1512
0*a^2*b^3*e^(2*d*x + 2*c) + 6480*a*b^4*e^(2*d*x + 2*c) + 1152*b^5*e^(2*d*x + 2*c) + 1575*a^4*b + 2100*a^3*b^2
+ 1680*a^2*b^3 + 720*a*b^4 + 128*b^5)/(d*(e^(2*d*x + 2*c) + 1)^9)