3.124 \(\int (a+b \text{sech}^2(c+d x))^3 \tanh ^4(c+d x) \, dx\)
Optimal. Leaf size=110 \[ \frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac{a^3 \tanh ^3(c+d x)}{3 d}-\frac{a^3 \tanh (c+d x)}{d}+a^3 x-\frac{b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac{b^3 \tanh ^9(c+d x)}{9 d} \]
[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)
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Rubi [A] time = 0.118163, antiderivative size = 110, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, 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 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])
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 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 )^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*}
Mathematica [B] time = 6.14437, size = 301, normalized size = 2.74 \[ \frac{8 \text{sech}^9(c+d x) \left (a \cosh ^2(c+d x)+b\right )^3 \left (\left (-378 a^2 b+105 a^3+27 a b^2+4 b^3\right ) \tanh (c) \cosh ^7(c+d x)+3 b \left (63 a^2-72 a b+b^2\right ) \tanh (c) \cosh ^5(c+d x)-\left (-189 a^2 b+420 a^3-54 a b^2-8 b^3\right ) \text{sech}(c) \sinh (d x) \cosh ^8(c+d x)+\left (-378 a^2 b+105 a^3+27 a b^2+4 b^3\right ) \text{sech}(c) \sinh (d x) \cosh ^6(c+d x)+3 b \left (63 a^2-72 a b+b^2\right ) \text{sech}(c) \sinh (d x) \cosh ^4(c+d x)+315 a^3 d x \cosh ^9(c+d x)+5 b^2 (27 a-10 b) \tanh (c) \cosh ^3(c+d x)+5 b^2 (27 a-10 b) \text{sech}(c) \sinh (d x) \cosh ^2(c+d x)+35 b^3 \tanh (c) \cosh (c+d x)+35 b^3 \text{sech}(c) \sinh (d x)\right )}{315 d (a \cosh (2 (c+d x))+a+2 b)^3} \]
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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Maple [B] time = 0.049, size = 274, normalized size = 2.5 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( dx+c-\tanh \left ( dx+c \right ) -{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{3}} \right ) +3\,{a}^{2}b \left ( -1/2\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}-3/8\,{\frac{\sinh \left ( dx+c \right ) }{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+3/8\, \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 ) \right ) +3\,a{b}^{2} \left ( -1/4\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{7}}}-1/8\,{\frac{\sinh \left ( dx+c \right ) }{ \left ( \cosh \left ( dx+c \right ) \right ) ^{7}}}+1/8\, \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 ) \right ) +{b}^{3} \left ( -{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{6\, \left ( \cosh \left ( dx+c \right ) \right ) ^{9}}}-{\frac{\sinh \left ( dx+c \right ) }{16\, \left ( \cosh \left ( dx+c \right ) \right ) ^{9}}}+{\frac{\tanh \left ( dx+c \right ) }{16} \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 ) } \right ) \right ) } \end{align*}
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 = 1.18216, size = 1962, normalized size = 17.84 \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)^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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Fricas [B] time = 2.21913, size = 3414, normalized size = 31.04 \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)^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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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} \tanh ^{4}{\left (c + d x \right )}\, dx \end{align*}
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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Giac [B] time = 1.58587, size = 637, normalized size = 5.79 \begin{align*} \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} \end{align*}
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