3.113 \(\int (a+b \text{sech}^2(c+d x))^2 \tanh ^3(c+d x) \, dx\)
Optimal. Leaf size=77 \[ \frac{a^2 \log (\cosh (c+d x))}{d}+\frac{b (2 a-b) \text{sech}^4(c+d x)}{4 d}+\frac{a (a-2 b) \text{sech}^2(c+d x)}{2 d}+\frac{b^2 \text{sech}^6(c+d x)}{6 d} \]
[Out]
(a^2*Log[Cosh[c + d*x]])/d + (a*(a - 2*b)*Sech[c + d*x]^2)/(2*d) + ((2*a - b)*b*Sech[c + d*x]^4)/(4*d) + (b^2*
Sech[c + d*x]^6)/(6*d)
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Rubi [A] time = 0.094726, antiderivative size = 77, 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 =
{4138, 446, 76} \[ \frac{a^2 \log (\cosh (c+d x))}{d}+\frac{b (2 a-b) \text{sech}^4(c+d x)}{4 d}+\frac{a (a-2 b) \text{sech}^2(c+d x)}{2 d}+\frac{b^2 \text{sech}^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Sech[c + d*x]^2)^2*Tanh[c + d*x]^3,x]
[Out]
(a^2*Log[Cosh[c + d*x]])/d + (a*(a - 2*b)*Sech[c + d*x]^2)/(2*d) + ((2*a - b)*b*Sech[c + d*x]^4)/(4*d) + (b^2*
Sech[c + d*x]^6)/(6*d)
Rule 4138
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Module[{ff =
FreeFactors[Cos[e + f*x], x]}, -Dist[(f*ff^(m + n*p - 1))^(-1), Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*
(ff*x)^n)^p)/x^(m + n*p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n] && IntegerQ[p]
Rule 446
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rule 76
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Rubi steps
\begin{align*} \int \left (a+b \text{sech}^2(c+d x)\right )^2 \tanh ^3(c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right ) \left (b+a x^2\right )^2}{x^7} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(1-x) (b+a x)^2}{x^4} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{b^2}{x^4}+\frac{(2 a-b) b}{x^3}+\frac{a (a-2 b)}{x^2}-\frac{a^2}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac{a^2 \log (\cosh (c+d x))}{d}+\frac{a (a-2 b) \text{sech}^2(c+d x)}{2 d}+\frac{(2 a-b) b \text{sech}^4(c+d x)}{4 d}+\frac{b^2 \text{sech}^6(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.306637, size = 107, normalized size = 1.39 \[ \frac{\cosh ^4(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^2 \left (12 a^2 \log (\cosh (c+d x))+3 b (2 a-b) \text{sech}^4(c+d x)+6 a (a-2 b) \text{sech}^2(c+d x)+2 b^2 \text{sech}^6(c+d x)\right )}{3 d (a \cosh (2 c+2 d x)+a+2 b)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Sech[c + d*x]^2)^2*Tanh[c + d*x]^3,x]
[Out]
(Cosh[c + d*x]^4*(a + b*Sech[c + d*x]^2)^2*(12*a^2*Log[Cosh[c + d*x]] + 6*a*(a - 2*b)*Sech[c + d*x]^2 + 3*(2*a
- b)*b*Sech[c + d*x]^4 + 2*b^2*Sech[c + d*x]^6))/(3*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^2)
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Maple [B] time = 0.043, size = 150, normalized size = 2. \begin{align*}{\frac{{a}^{2}\ln \left ( \cosh \left ( dx+c \right ) \right ) }{d}}-{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{2}{a}^{2}}{2\,d}}-{\frac{ab \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{2\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{ab \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{2\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{6\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{12\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{12\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*sech(d*x+c)^2)^2*tanh(d*x+c)^3,x)
[Out]
a^2*ln(cosh(d*x+c))/d-1/2/d*tanh(d*x+c)^2*a^2-1/2/d*a*b*sinh(d*x+c)^2/cosh(d*x+c)^4+1/2/d*a*b*sinh(d*x+c)^2/co
sh(d*x+c)^2-1/6/d*b^2*sinh(d*x+c)^2/cosh(d*x+c)^6+1/12/d*b^2*sinh(d*x+c)^2/cosh(d*x+c)^4+1/12/d*b^2*sinh(d*x+c
)^2/cosh(d*x+c)^2
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Maxima [B] time = 2.35945, size = 450, normalized size = 5.84 \begin{align*} \frac{a b \tanh \left (d x + c\right )^{4}}{2 \, d} + a^{2}{\left (x + \frac{c}{d} + \frac{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac{2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} - \frac{4}{3} \, b^{2}{\left (\frac{3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d{\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}} - \frac{2 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d{\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}} + \frac{3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d{\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c)^2)^2*tanh(d*x+c)^3,x, algorithm="maxima")
[Out]
1/2*a*b*tanh(d*x + c)^4/d + a^2*(x + c/d + log(e^(-2*d*x - 2*c) + 1)/d + 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x -
2*c) + e^(-4*d*x - 4*c) + 1))) - 4/3*b^2*(3*e^(-4*d*x - 4*c)/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20
*e^(-6*d*x - 6*c) + 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c) + 1)) - 2*e^(-6*d*x - 6*c)
/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) + 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*
c) + e^(-12*d*x - 12*c) + 1)) + 3*e^(-8*d*x - 8*c)/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x
- 6*c) + 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c) + 1)))
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Fricas [B] time = 2.43469, size = 6669, normalized size = 86.61 \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)^2*tanh(d*x+c)^3,x, algorithm="fricas")
[Out]
-1/3*(3*a^2*d*x*cosh(d*x + c)^12 + 36*a^2*d*x*cosh(d*x + c)*sinh(d*x + c)^11 + 3*a^2*d*x*sinh(d*x + c)^12 + 6*
(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^10 + 6*(33*a^2*d*x*cosh(d*x + c)^2 + 3*a^2*d*x - a^2 + 2*a*b)*sinh(d*x
+ c)^10 + 60*(11*a^2*d*x*cosh(d*x + c)^3 + (3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^9 + 3*(15*a
^2*d*x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^8 + 3*(495*a^2*d*x*cosh(d*x + c)^4 + 15*a^2*d*x + 90*(3*a^2*d*x
- a^2 + 2*a*b)*cosh(d*x + c)^2 - 8*a^2 + 8*a*b + 4*b^2)*sinh(d*x + c)^8 + 24*(99*a^2*d*x*cosh(d*x + c)^5 + 30*
(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^3 + (15*a^2*d*x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^
7 + 4*(15*a^2*d*x - 9*a^2 + 6*a*b - 2*b^2)*cosh(d*x + c)^6 + 4*(693*a^2*d*x*cosh(d*x + c)^6 + 315*(3*a^2*d*x -
a^2 + 2*a*b)*cosh(d*x + c)^4 + 15*a^2*d*x + 21*(15*a^2*d*x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^2 - 9*a^2 +
6*a*b - 2*b^2)*sinh(d*x + c)^6 + 24*(99*a^2*d*x*cosh(d*x + c)^7 + 63*(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^
5 + 7*(15*a^2*d*x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^3 + (15*a^2*d*x - 9*a^2 + 6*a*b - 2*b^2)*cosh(d*x + c
))*sinh(d*x + c)^5 + 3*(15*a^2*d*x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^4 + 3*(495*a^2*d*x*cosh(d*x + c)^8 +
420*(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^6 + 70*(15*a^2*d*x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^4 + 15*
a^2*d*x + 20*(15*a^2*d*x - 9*a^2 + 6*a*b - 2*b^2)*cosh(d*x + c)^2 - 8*a^2 + 8*a*b + 4*b^2)*sinh(d*x + c)^4 + 3
*a^2*d*x + 4*(165*a^2*d*x*cosh(d*x + c)^9 + 180*(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^7 + 42*(15*a^2*d*x - 8
*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^5 + 20*(15*a^2*d*x - 9*a^2 + 6*a*b - 2*b^2)*cosh(d*x + c)^3 + 3*(15*a^2*d*
x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^2 + 6*(3
3*a^2*d*x*cosh(d*x + c)^10 + 45*(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^8 + 14*(15*a^2*d*x - 8*a^2 + 8*a*b + 4
*b^2)*cosh(d*x + c)^6 + 10*(15*a^2*d*x - 9*a^2 + 6*a*b - 2*b^2)*cosh(d*x + c)^4 + 3*a^2*d*x + 3*(15*a^2*d*x -
8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^2 - a^2 + 2*a*b)*sinh(d*x + c)^2 - 3*(a^2*cosh(d*x + c)^12 + 12*a^2*cosh(
d*x + c)*sinh(d*x + c)^11 + a^2*sinh(d*x + c)^12 + 6*a^2*cosh(d*x + c)^10 + 6*(11*a^2*cosh(d*x + c)^2 + a^2)*s
inh(d*x + c)^10 + 15*a^2*cosh(d*x + c)^8 + 20*(11*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x + c)^9 +
15*(33*a^2*cosh(d*x + c)^4 + 18*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^8 + 20*a^2*cosh(d*x + c)^6 + 24*(33*
a^2*cosh(d*x + c)^5 + 30*a^2*cosh(d*x + c)^3 + 5*a^2*cosh(d*x + c))*sinh(d*x + c)^7 + 4*(231*a^2*cosh(d*x + c)
^6 + 315*a^2*cosh(d*x + c)^4 + 105*a^2*cosh(d*x + c)^2 + 5*a^2)*sinh(d*x + c)^6 + 15*a^2*cosh(d*x + c)^4 + 24*
(33*a^2*cosh(d*x + c)^7 + 63*a^2*cosh(d*x + c)^5 + 35*a^2*cosh(d*x + c)^3 + 5*a^2*cosh(d*x + c))*sinh(d*x + c)
^5 + 15*(33*a^2*cosh(d*x + c)^8 + 84*a^2*cosh(d*x + c)^6 + 70*a^2*cosh(d*x + c)^4 + 20*a^2*cosh(d*x + c)^2 + a
^2)*sinh(d*x + c)^4 + 6*a^2*cosh(d*x + c)^2 + 20*(11*a^2*cosh(d*x + c)^9 + 36*a^2*cosh(d*x + c)^7 + 42*a^2*cos
h(d*x + c)^5 + 20*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(11*a^2*cosh(d*x + c)^10 + 45
*a^2*cosh(d*x + c)^8 + 70*a^2*cosh(d*x + c)^6 + 50*a^2*cosh(d*x + c)^4 + 15*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*
x + c)^2 + a^2 + 12*(a^2*cosh(d*x + c)^11 + 5*a^2*cosh(d*x + c)^9 + 10*a^2*cosh(d*x + c)^7 + 10*a^2*cosh(d*x +
c)^5 + 5*a^2*cosh(d*x + c)^3 + a^2*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*
x + c))) + 12*(3*a^2*d*x*cosh(d*x + c)^11 + 5*(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^9 + 2*(15*a^2*d*x - 8*a^
2 + 8*a*b + 4*b^2)*cosh(d*x + c)^7 + 2*(15*a^2*d*x - 9*a^2 + 6*a*b - 2*b^2)*cosh(d*x + c)^5 + (15*a^2*d*x - 8*
a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^3 + (3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c
)^12 + 12*d*cosh(d*x + c)*sinh(d*x + c)^11 + d*sinh(d*x + c)^12 + 6*d*cosh(d*x + c)^10 + 6*(11*d*cosh(d*x + c)
^2 + d)*sinh(d*x + c)^10 + 20*(11*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^9 + 15*d*cosh(d*x + c)^
8 + 15*(33*d*cosh(d*x + c)^4 + 18*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^8 + 24*(33*d*cosh(d*x + c)^5 + 30*d*cos
h(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)^7 + 20*d*cosh(d*x + c)^6 + 4*(231*d*cosh(d*x + c)^6 + 315*d*co
sh(d*x + c)^4 + 105*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^6 + 24*(33*d*cosh(d*x + c)^7 + 63*d*cosh(d*x + c)^5
+ 35*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)^5 + 15*d*cosh(d*x + c)^4 + 15*(33*d*cosh(d*x + c)^8
+ 84*d*cosh(d*x + c)^6 + 70*d*cosh(d*x + c)^4 + 20*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^4 + 20*(11*d*cosh(d*x
+ c)^9 + 36*d*cosh(d*x + c)^7 + 42*d*cosh(d*x + c)^5 + 20*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c
)^3 + 6*d*cosh(d*x + c)^2 + 6*(11*d*cosh(d*x + c)^10 + 45*d*cosh(d*x + c)^8 + 70*d*cosh(d*x + c)^6 + 50*d*cosh
(d*x + c)^4 + 15*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 12*(d*cosh(d*x + c)^11 + 5*d*cosh(d*x + c)^9 + 10*d*
cosh(d*x + c)^7 + 10*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) + d)
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Sympy [A] time = 8.80859, size = 129, normalized size = 1.68 \begin{align*} \begin{cases} a^{2} x - \frac{a^{2} \log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{d} - \frac{a^{2} \tanh ^{2}{\left (c + d x \right )}}{2 d} - \frac{a b \tanh ^{2}{\left (c + d x \right )} \operatorname{sech}^{2}{\left (c + d x \right )}}{2 d} - \frac{a b \operatorname{sech}^{2}{\left (c + d x \right )}}{2 d} - \frac{b^{2} \tanh ^{2}{\left (c + d x \right )} \operatorname{sech}^{4}{\left (c + d x \right )}}{6 d} - \frac{b^{2} \operatorname{sech}^{4}{\left (c + d x \right )}}{12 d} & \text{for}\: d \neq 0 \\x \left (a + b \operatorname{sech}^{2}{\left (c \right )}\right )^{2} \tanh ^{3}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c)**2)**2*tanh(d*x+c)**3,x)
[Out]
Piecewise((a**2*x - a**2*log(tanh(c + d*x) + 1)/d - a**2*tanh(c + d*x)**2/(2*d) - a*b*tanh(c + d*x)**2*sech(c
+ d*x)**2/(2*d) - a*b*sech(c + d*x)**2/(2*d) - b**2*tanh(c + d*x)**2*sech(c + d*x)**4/(6*d) - b**2*sech(c + d*
x)**4/(12*d), Ne(d, 0)), (x*(a + b*sech(c)**2)**2*tanh(c)**3, True))
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Giac [B] time = 1.28324, size = 325, normalized size = 4.22 \begin{align*} -\frac{60 \, a^{2} d x - 60 \, a^{2} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac{147 \, a^{2} e^{\left (12 \, d x + 12 \, c\right )} + 762 \, a^{2} e^{\left (10 \, d x + 10 \, c\right )} + 240 \, a b e^{\left (10 \, d x + 10 \, c\right )} + 1725 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 480 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 240 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 2220 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 480 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 160 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1725 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 480 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 240 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 762 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 240 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 147 \, a^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{6}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c)^2)^2*tanh(d*x+c)^3,x, algorithm="giac")
[Out]
-1/60*(60*a^2*d*x - 60*a^2*log(e^(2*d*x + 2*c) + 1) + (147*a^2*e^(12*d*x + 12*c) + 762*a^2*e^(10*d*x + 10*c) +
240*a*b*e^(10*d*x + 10*c) + 1725*a^2*e^(8*d*x + 8*c) + 480*a*b*e^(8*d*x + 8*c) + 240*b^2*e^(8*d*x + 8*c) + 22
20*a^2*e^(6*d*x + 6*c) + 480*a*b*e^(6*d*x + 6*c) - 160*b^2*e^(6*d*x + 6*c) + 1725*a^2*e^(4*d*x + 4*c) + 480*a*
b*e^(4*d*x + 4*c) + 240*b^2*e^(4*d*x + 4*c) + 762*a^2*e^(2*d*x + 2*c) + 240*a*b*e^(2*d*x + 2*c) + 147*a^2)/(e^
(2*d*x + 2*c) + 1)^6)/d