3.124 \(\int (a+b \text {sech}^2(c+d x))^3 \tanh ^4(c+d x) \, dx\)

Optimal. Leaf size=110 \[ -\frac {a^3 \tanh ^3(c+d x)}{3 d}-\frac {a^3 \tanh (c+d x)}{d}+a^3 x+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d} \]

[Out]

a^3*x-a^3*tanh(d*x+c)/d-1/3*a^3*tanh(d*x+c)^3/d+1/5*b*(3*a^2+3*a*b+b^2)*tanh(d*x+c)^5/d-1/7*b^2*(3*a+2*b)*tanh
(d*x+c)^7/d+1/9*b^3*tanh(d*x+c)^9/d

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Rubi [A]  time = 0.12, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4141, 1802, 206} \[ \frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {a^3 \tanh ^3(c+d x)}{3 d}-\frac {a^3 \tanh (c+d x)}{d}+a^3 x-\frac {b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^4,x]

[Out]

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

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])

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 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])

Rubi steps

\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^4(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (a+b \left (1-x^2\right )\right )^3}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a^3-a^3 x^2+b \left (3 a^2+3 a b+b^2\right ) x^4-b^2 (3 a+2 b) x^6+b^3 x^8+\frac {a^3}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a^3 \tanh (c+d x)}{d}-\frac {a^3 \tanh ^3(c+d x)}{3 d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^3 x-\frac {a^3 \tanh (c+d x)}{d}-\frac {a^3 \tanh ^3(c+d x)}{3 d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d}\\ \end {align*}

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Mathematica [B]  time = 5.94, size = 301, normalized size = 2.74 \[ \frac {8 \text {sech}^9(c+d x) \left (a \cosh ^2(c+d x)+b\right )^3 \left (315 a^3 d x \cosh ^9(c+d x)+3 b \left (63 a^2-72 a b+b^2\right ) \tanh (c) \cosh ^5(c+d x)+3 b \left (63 a^2-72 a b+b^2\right ) \text {sech}(c) \sinh (d x) \cosh ^4(c+d x)+\left (105 a^3-378 a^2 b+27 a b^2+4 b^3\right ) \tanh (c) \cosh ^7(c+d x)-\left (420 a^3-189 a^2 b-54 a b^2-8 b^3\right ) \text {sech}(c) \sinh (d x) \cosh ^8(c+d x)+\left (105 a^3-378 a^2 b+27 a b^2+4 b^3\right ) \text {sech}(c) \sinh (d x) \cosh ^6(c+d x)+5 b^2 (27 a-10 b) \tanh (c) \cosh ^3(c+d x)+5 b^2 (27 a-10 b) \text {sech}(c) \sinh (d x) \cosh ^2(c+d x)+35 b^3 \tanh (c) \cosh (c+d x)+35 b^3 \text {sech}(c) \sinh (d x)\right )}{315 d (a \cosh (2 (c+d x))+a+2 b)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^4,x]

[Out]

(8*(b + a*Cosh[c + d*x]^2)^3*Sech[c + d*x]^9*(315*a^3*d*x*Cosh[c + d*x]^9 + 35*b^3*Sech[c]*Sinh[d*x] + 5*(27*a
- 10*b)*b^2*Cosh[c + d*x]^2*Sech[c]*Sinh[d*x] + 3*b*(63*a^2 - 72*a*b + b^2)*Cosh[c + d*x]^4*Sech[c]*Sinh[d*x]
+ (105*a^3 - 378*a^2*b + 27*a*b^2 + 4*b^3)*Cosh[c + d*x]^6*Sech[c]*Sinh[d*x] - (420*a^3 - 189*a^2*b - 54*a*b^
2 - 8*b^3)*Cosh[c + d*x]^8*Sech[c]*Sinh[d*x] + 35*b^3*Cosh[c + d*x]*Tanh[c] + 5*(27*a - 10*b)*b^2*Cosh[c + d*x
]^3*Tanh[c] + 3*b*(63*a^2 - 72*a*b + b^2)*Cosh[c + d*x]^5*Tanh[c] + (105*a^3 - 378*a^2*b + 27*a*b^2 + 4*b^3)*C
osh[c + d*x]^7*Tanh[c]))/(315*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)

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fricas [B]  time = 0.43, size = 1323, normalized size = 12.03 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)^2)^3*tanh(d*x+c)^4,x, algorithm="fricas")

[Out]

1/315*((315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^9 + 9*(315*a^3*d*x + 420*a^3 - 189
*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)*sinh(d*x + c)^8 - (420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*sinh(d*x +
c)^9 + 9*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^7 - 9*(280*a^3 + 21*a^2*b - 54*
a*b^2 - 8*b^3 + 4*(420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 21*(4*(315*a^3*d
*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^3 + 3*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2
- 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 36*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x
+ c)^5 - 9*(14*(420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^4 + 700*a^3 + 84*a^2*b + 204*a*b^2 - 32*
b^3 + 21*(280*a^3 + 21*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 9*(14*(315*a^3*d*x + 420*a
^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^5 + 35*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)
*cosh(d*x + c)^3 + 20*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 +
84*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^3 - 3*(28*(420*a^3 - 189*a^2*b - 54*a*
b^2 - 8*b^3)*cosh(d*x + c)^6 + 105*(280*a^3 + 21*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^4 + 2660*a^3 - 252*a^
2*b - 252*a*b^2 + 896*b^3 + 120*(175*a^3 + 21*a^2*b + 51*a*b^2 - 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 9*(
4*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^7 + 21*(315*a^3*d*x + 420*a^3 - 189*a^2
*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^5 + 40*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x +
c)^3 + 28*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 126*(315*a^3
*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c) - 9*((420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cos
h(d*x + c)^8 + 7*(280*a^3 + 21*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^6 + 20*(175*a^3 + 21*a^2*b + 51*a*b^2 -
8*b^3)*cosh(d*x + c)^4 + 420*a^3 - 126*a^2*b - 336*a*b^2 - 672*b^3 + 28*(95*a^3 - 9*a^2*b - 9*a*b^2 + 32*b^3)
*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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giac [B]  time = 0.30, size = 472, normalized size = 4.29 \[ \frac {315 \, a^{3} d x + \frac {2 \, {\left (630 \, a^{3} e^{\left (16 \, d x + 16 \, c\right )} - 945 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} + 4410 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} - 3780 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} - 1890 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 13650 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} - 7560 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} - 1890 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 1680 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 24570 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} - 11340 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 1890 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 2520 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 28350 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 12474 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 4914 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 3528 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 21630 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 8316 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 2646 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1008 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 10710 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 3024 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 54 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 288 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 3150 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 756 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 486 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 72 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{9}}}{315 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)^2)^3*tanh(d*x+c)^4,x, algorithm="giac")

[Out]

1/315*(315*a^3*d*x + 2*(630*a^3*e^(16*d*x + 16*c) - 945*a^2*b*e^(16*d*x + 16*c) + 4410*a^3*e^(14*d*x + 14*c) -
3780*a^2*b*e^(14*d*x + 14*c) - 1890*a*b^2*e^(14*d*x + 14*c) + 13650*a^3*e^(12*d*x + 12*c) - 7560*a^2*b*e^(12*
d*x + 12*c) - 1890*a*b^2*e^(12*d*x + 12*c) - 1680*b^3*e^(12*d*x + 12*c) + 24570*a^3*e^(10*d*x + 10*c) - 11340*
a^2*b*e^(10*d*x + 10*c) - 1890*a*b^2*e^(10*d*x + 10*c) + 2520*b^3*e^(10*d*x + 10*c) + 28350*a^3*e^(8*d*x + 8*c
) - 12474*a^2*b*e^(8*d*x + 8*c) - 4914*a*b^2*e^(8*d*x + 8*c) - 3528*b^3*e^(8*d*x + 8*c) + 21630*a^3*e^(6*d*x +
6*c) - 8316*a^2*b*e^(6*d*x + 6*c) - 2646*a*b^2*e^(6*d*x + 6*c) + 1008*b^3*e^(6*d*x + 6*c) + 10710*a^3*e^(4*d*
x + 4*c) - 3024*a^2*b*e^(4*d*x + 4*c) - 54*a*b^2*e^(4*d*x + 4*c) - 288*b^3*e^(4*d*x + 4*c) + 3150*a^3*e^(2*d*x
+ 2*c) - 756*a^2*b*e^(2*d*x + 2*c) - 486*a*b^2*e^(2*d*x + 2*c) - 72*b^3*e^(2*d*x + 2*c) + 420*a^3 - 189*a^2*b
- 54*a*b^2 - 8*b^3)/(e^(2*d*x + 2*c) + 1)^9)/d

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maple [B]  time = 0.55, size = 274, normalized size = 2.49 \[ \frac {a^{3} \left (d x +c -\tanh \left (d x +c \right )-\frac {\left (\tanh ^{3}\left (d x +c \right )\right )}{3}\right )+3 a^{2} b \left (-\frac {\sinh ^{3}\left (d x +c \right )}{2 \cosh \left (d x +c \right )^{5}}-\frac {3 \sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{5}}+\frac {3 \left (\frac {8}{15}+\frac {\mathrm {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{8}\right )+3 a \,b^{2} \left (-\frac {\sinh ^{3}\left (d x +c \right )}{4 \cosh \left (d x +c \right )^{7}}-\frac {\sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{7}}+\frac {\left (\frac {16}{35}+\frac {\mathrm {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \mathrm {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \mathrm {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )}{8}\right )+b^{3} \left (-\frac {\sinh ^{3}\left (d x +c \right )}{6 \cosh \left (d x +c \right )^{9}}-\frac {\sinh \left (d x +c \right )}{16 \cosh \left (d x +c \right )^{9}}+\frac {\left (\frac {128}{315}+\frac {\mathrm {sech}\left (d x +c \right )^{8}}{9}+\frac {8 \mathrm {sech}\left (d x +c \right )^{6}}{63}+\frac {16 \mathrm {sech}\left (d x +c \right )^{4}}{105}+\frac {64 \mathrm {sech}\left (d x +c \right )^{2}}{315}\right ) \tanh \left (d x +c \right )}{16}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sech(d*x+c)^2)^3*tanh(d*x+c)^4,x)

[Out]

1/d*(a^3*(d*x+c-tanh(d*x+c)-1/3*tanh(d*x+c)^3)+3*a^2*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))+3*a*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
))+b^3*(-1/6*sinh(d*x+c)^3/cosh(d*x+c)^9-1/16*sinh(d*x+c)/cosh(d*x+c)^9+1/16*(128/315+1/9*sech(d*x+c)^8+8/63*s
ech(d*x+c)^6+16/105*sech(d*x+c)^4+64/315*sech(d*x+c)^2)*tanh(d*x+c)))

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maxima [B]  time = 0.39, size = 1453, normalized size = 13.21 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)^2)^3*tanh(d*x+c)^4,x, algorithm="maxima")

[Out]

3/5*a^2*b*tanh(d*x + c)^5/d + 1/3*a^3*(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))) + 16/315*b^3*(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) + 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)) + 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)) - 126*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)) + 441*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 - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e
^(-18*d*x - 18*c) + 1)) - 315*e^(-10*d*x - 10*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)) + 210*e^(-12*d*x - 12*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)) + 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))) + 12/35*a*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) +
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^(-1
0*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)))

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mupad [B]  time = 1.62, size = 1834, normalized size = 16.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(c + d*x)^4*(a + b/cosh(c + d*x)^2)^3,x)

[Out]

((3*a*b^2 + 13*a^3 + 16*b^3)/(63*d) + (10*exp(4*c + 4*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(63*d) + (20*exp(6*c +
6*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^3))/(63*d) - (2*exp(2*c + 2*d*x)*(8*a*b^2 + 3*a^2*b - 10*a^3 + 16*b^3
))/(21*d) - (5*exp(8*c + 8*d*x)*(a*b^2 - a^3))/(3*d) - (2*exp(10*c + 10*d*x)*(3*a^2*b - 2*a^3))/(9*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(12
*c + 12*d*x) + 1) - ((2*exp(2*c + 2*d*x)*(a*b^2 - a^3))/(3*d) - (2*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^3))/(63*d)
+ (2*exp(4*c + 4*d*x)*(3*a^2*b - 2*a^3))/(9*d))/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) +
1) + ((3*a*b^2 + 13*a^3 + 16*b^3)/(63*d) + (2*exp(2*c + 2*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^3))/(21*d) -
(exp(4*c + 4*d*x)*(a*b^2 - a^3))/d - (2*exp(6*c + 6*d*x)*(3*a^2*b - 2*a^3))/(9*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) - ((a*b^2 - a^3)/(3*d) + (2*exp(2*c + 2*d*x)*(3*a^2
*b - 2*a^3))/(9*d))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1) + a^3*x + ((2*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b
^3))/(63*d) + (2*exp(2*c + 2*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(21*d) + (20*exp(6*c + 6*d*x)*(3*a*b^2 + 13*a^3
+ 16*b^3))/(63*d) + (10*exp(8*c + 8*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^3))/(21*d) - (2*exp(4*c + 4*d*x)*(8
*a*b^2 + 3*a^2*b - 10*a^3 + 16*b^3))/(7*d) - (2*exp(10*c + 10*d*x)*(a*b^2 - a^3))/d - (2*exp(12*c + 12*d*x)*(3
*a^2*b - 2*a^3))/(9*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) - ((a*b^2 - a^3)/(3*d) - (exp(4*c + 4
*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(3*d) - (5*exp(8*c + 8*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(9*d) - (2*exp(2*c
+ 2*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^3))/(9*d) - (2*exp(10*c + 10*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^
3))/(3*d) + (2*exp(6*c + 6*d*x)*(8*a*b^2 + 3*a^2*b - 10*a^3 + 16*b^3))/(3*d) + (7*exp(12*c + 12*d*x)*(a*b^2 -
a^3))/(3*d) + (2*exp(14*c + 14*d*x)*(3*a^2*b - 2*a^3))/(9*d))/(8*exp(2*c + 2*d*x) + 28*exp(4*c + 4*d*x) + 56*e
xp(6*c + 6*d*x) + 70*exp(8*c + 8*d*x) + 56*exp(10*c + 10*d*x) + 28*exp(12*c + 12*d*x) + 8*exp(14*c + 14*d*x) +
exp(16*c + 16*d*x) + 1) - ((2*(3*a^2*b - 2*a^3))/(9*d) - (8*exp(6*c + 6*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(9*
d) - (8*exp(10*c + 10*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(9*d) - (8*exp(4*c + 4*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3
- 4*b^3))/(9*d) - (8*exp(12*c + 12*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^3))/(9*d) + (4*exp(8*c + 8*d*x)*(8*a
*b^2 + 3*a^2*b - 10*a^3 + 16*b^3))/(3*d) + (8*exp(2*c + 2*d*x)*(a*b^2 - a^3))/(3*d) + (8*exp(14*c + 14*d*x)*(a
*b^2 - a^3))/(3*d) + (2*exp(16*c + 16*d*x)*(3*a^2*b - 2*a^3))/(9*d))/(9*exp(2*c + 2*d*x) + 36*exp(4*c + 4*d*x)
+ 84*exp(6*c + 6*d*x) + 126*exp(8*c + 8*d*x) + 126*exp(10*c + 10*d*x) + 84*exp(12*c + 12*d*x) + 36*exp(14*c +
14*d*x) + 9*exp(16*c + 16*d*x) + exp(18*c + 18*d*x) + 1) - ((2*(8*a*b^2 + 3*a^2*b - 10*a^3 + 16*b^3))/(105*d)
- (4*exp(2*c + 2*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(63*d) - (4*exp(4*c + 4*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 -
4*b^3))/(21*d) + (4*exp(6*c + 6*d*x)*(a*b^2 - a^3))/(3*d) + (2*exp(8*c + 8*d*x)*(3*a^2*b - 2*a^3))/(9*d))/(5*e
xp(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) - (
2*(3*a^2*b - 2*a^3))/(9*d*(exp(2*c + 2*d*x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3} \tanh ^{4}{\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)**2)**3*tanh(d*x+c)**4,x)

[Out]

Integral((a + b*sech(c + d*x)**2)**3*tanh(c + d*x)**4, x)

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