3.123 \(\int \coth ^7(c+d x) (a+b \text{sech}^2(c+d x))^2 \, dx\)
Optimal. Leaf size=86 \[ \frac{a^2 \log (\sinh (c+d x))}{d}-\frac{(a+b)^2 \text{csch}^4(c+d x)}{4 d}-\frac{a (a+b) \text{csch}^2(c+d x)}{d}-\frac{\text{csch}^6(c+d x) \left (a \cosh ^2(c+d x)+b\right )^3}{6 d (a+b)} \]
[Out]
-((a*(a + b)*Csch[c + d*x]^2)/d) - ((a + b)^2*Csch[c + d*x]^4)/(4*d) - ((b + a*Cosh[c + d*x]^2)^3*Csch[c + d*x
]^6)/(6*(a + b)*d) + (a^2*Log[Sinh[c + d*x]])/d
________________________________________________________________________________________
Rubi [A] time = 0.124845, antiderivative size = 86, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used =
{4138, 446, 78, 43} \[ \frac{a^2 \log (\sinh (c+d x))}{d}-\frac{(a+b)^2 \text{csch}^4(c+d x)}{4 d}-\frac{a (a+b) \text{csch}^2(c+d x)}{d}-\frac{\text{csch}^6(c+d x) \left (a \cosh ^2(c+d x)+b\right )^3}{6 d (a+b)} \]
Antiderivative was successfully verified.
[In]
Int[Coth[c + d*x]^7*(a + b*Sech[c + d*x]^2)^2,x]
[Out]
-((a*(a + b)*Csch[c + d*x]^2)/d) - ((a + b)^2*Csch[c + d*x]^4)/(4*d) - ((b + a*Cosh[c + d*x]^2)^3*Csch[c + d*x
]^6)/(6*(a + b)*d) + (a^2*Log[Sinh[c + d*x]])/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 78
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ
[p, n]))))
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int \coth ^7(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3 \left (b+a x^2\right )^2}{\left (1-x^2\right )^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x (b+a x)^2}{(1-x)^4} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{\left (b+a \cosh ^2(c+d x)\right )^3 \text{csch}^6(c+d x)}{6 (a+b) d}-\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^2}{(1-x)^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{\left (b+a \cosh ^2(c+d x)\right )^3 \text{csch}^6(c+d x)}{6 (a+b) d}-\frac{\operatorname{Subst}\left (\int \left (-\frac{(a+b)^2}{(-1+x)^3}-\frac{2 a (a+b)}{(-1+x)^2}-\frac{a^2}{-1+x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{a (a+b) \text{csch}^2(c+d x)}{d}-\frac{(a+b)^2 \text{csch}^4(c+d x)}{4 d}-\frac{\left (b+a \cosh ^2(c+d x)\right )^3 \text{csch}^6(c+d x)}{6 (a+b) d}+\frac{a^2 \log (\sinh (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.50803, size = 107, normalized size = 1.24 \[ -\frac{\left (a \cosh ^2(c+d x)+b\right )^2 \left (3 \left (3 a^2+4 a b+b^2\right ) \text{csch}^4(c+d x)-12 a^2 \log (\sinh (c+d x))+2 (a+b)^2 \text{csch}^6(c+d x)+6 a (3 a+2 b) \text{csch}^2(c+d x)\right )}{3 d (a \cosh (2 (c+d x))+a+2 b)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[c + d*x]^7*(a + b*Sech[c + d*x]^2)^2,x]
[Out]
-((b + a*Cosh[c + d*x]^2)^2*(6*a*(3*a + 2*b)*Csch[c + d*x]^2 + 3*(3*a^2 + 4*a*b + b^2)*Csch[c + d*x]^4 + 2*(a
+ b)^2*Csch[c + d*x]^6 - 12*a^2*Log[Sinh[c + d*x]]))/(3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)
________________________________________________________________________________________
Maple [B] time = 0.053, size = 228, normalized size = 2.7 \begin{align*}{\frac{{a}^{2}\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{2} \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{2} \left ({\rm coth} \left (dx+c\right ) \right ) ^{4}}{4\,d}}-{\frac{{a}^{2} \left ({\rm coth} \left (dx+c\right ) \right ) ^{6}}{6\,d}}-{\frac{ab \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}{d \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}}+{\frac{2\,ab \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{3\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}}+{\frac{ab \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{3\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}}-{\frac{ab \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{3\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{6\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}}-{\frac{{b}^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{12\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{b}^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{12\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(d*x+c)^7*(a+b*sech(d*x+c)^2)^2,x)
[Out]
1/d*a^2*ln(sinh(d*x+c))-1/2*a^2*coth(d*x+c)^2/d-1/4*a^2*coth(d*x+c)^4/d-1/6/d*a^2*coth(d*x+c)^6-1/d*a*b/sinh(d
*x+c)^6*cosh(d*x+c)^4+2/3/d*a*b/sinh(d*x+c)^6*cosh(d*x+c)^2+1/3/d*a*b/sinh(d*x+c)^4*cosh(d*x+c)^2-1/3/d*a*b*co
sh(d*x+c)^2/sinh(d*x+c)^2-1/6/d*b^2/sinh(d*x+c)^6*cosh(d*x+c)^2-1/12/d*b^2/sinh(d*x+c)^4*cosh(d*x+c)^2+1/12/d*
b^2*cosh(d*x+c)^2/sinh(d*x+c)^2
________________________________________________________________________________________
Maxima [B] time = 1.21822, size = 940, normalized size = 10.93 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)^7*(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
1/3*a^2*(3*x + 3*c/d + 3*log(e^(-d*x - c) + 1)/d + 3*log(e^(-d*x - c) - 1)/d + 2*(9*e^(-2*d*x - 2*c) - 18*e^(-
4*d*x - 4*c) + 34*e^(-6*d*x - 6*c) - 18*e^(-8*d*x - 8*c) + 9*e^(-10*d*x - 10*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)))
+ 4/3*a*b*(3*e^(-2*d*x - 2*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)) + 10*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^(-10*d*x - 10*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))) + 4/3*b^2*(3*e^(-4*d*x - 4*c)/(d*(6*e^(-2*d*x - 2*c) - 1
5*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)))
________________________________________________________________________________________
Fricas [B] time = 2.46665, size = 6593, normalized size = 76.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)^7*(a+b*sech(d*x+c)^2)^2,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 - 3*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 + 3*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 - 3*a^2 - 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^9 + 3
*(15*a^2*d*x - 12*a^2 + 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 -
3*a^2 - 2*a*b)*cosh(d*x + c)^2 - 12*a^2 + 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 - 3*a^2 - 2*a*b)*cosh(d*x + c)^3 + (15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 4*(15*a
^2*d*x - 17*a^2 - 10*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 - 3*a^2 -
2*a*b)*cosh(d*x + c)^4 - 15*a^2*d*x + 21*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^2 + 17*a^2 + 10*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 - 3*a^2 - 2*a*b)*cosh(d*x + c)^5 + 7*(15*
a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^3 - (15*a^2*d*x - 17*a^2 - 10*a*b - 2*b^2)*cosh(d*x + c))*sinh(d*x + c
)^5 + 3*(15*a^2*d*x - 12*a^2 + 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 - 3*a^
2 - 2*a*b)*cosh(d*x + c)^6 + 70*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^4 + 15*a^2*d*x - 20*(15*a^2*d*x -
17*a^2 - 10*a*b - 2*b^2)*cosh(d*x + c)^2 - 12*a^2 + 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 - 3*a^2 - 2*a*b)*cosh(d*x + c)^7 + 42*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^5
- 20*(15*a^2*d*x - 17*a^2 - 10*a*b - 2*b^2)*cosh(d*x + c)^3 + 3*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c))*
sinh(d*x + c)^3 - 6*(3*a^2*d*x - 3*a^2 - 2*a*b)*cosh(d*x + c)^2 + 6*(33*a^2*d*x*cosh(d*x + c)^10 - 45*(3*a^2*d
*x - 3*a^2 - 2*a*b)*cosh(d*x + c)^8 + 14*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^6 - 10*(15*a^2*d*x - 17*a
^2 - 10*a*b - 2*b^2)*cosh(d*x + c)^4 - 3*a^2*d*x + 3*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^2 + 3*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)*sinh(d*x + c)^10 + 15*a^2*cosh(d*x + c)^8 + 20*(1
1*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*cosh(d*x + c)^5 - 20*a^2*cosh(d*x + c)^3 + 3*a^2*co
sh(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*sinh(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 - 3*a^2 - 2*a*b)*cosh(d*x + c)^9 + 2*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^7 - 2*(15*a^2*d*x -
17*a^2 - 10*a*b - 2*b^2)*cosh(d*x + c)^5 + (15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^3 - (3*a^2*d*x - 3*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*cosh(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*cosh(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*co
sh(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 - 4
5*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)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)**7*(a+b*sech(d*x+c)**2)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.65332, size = 292, normalized size = 3.4 \begin{align*} -\frac{60 \, a^{2} d x - 60 \, a^{2} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac{147 \, a^{2} e^{\left (12 \, d x + 12 \, c\right )} - 522 \, a^{2} e^{\left (10 \, d x + 10 \, c\right )} + 240 \, a b e^{\left (10 \, d x + 10 \, c\right )} + 1485 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 240 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 1580 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 800 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 160 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1485 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 240 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 522 \, 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(coth(d*x+c)^7*(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")
[Out]
-1/60*(60*a^2*d*x - 60*a^2*log(abs(e^(2*d*x + 2*c) - 1)) + (147*a^2*e^(12*d*x + 12*c) - 522*a^2*e^(10*d*x + 10
*c) + 240*a*b*e^(10*d*x + 10*c) + 1485*a^2*e^(8*d*x + 8*c) + 240*b^2*e^(8*d*x + 8*c) - 1580*a^2*e^(6*d*x + 6*c
) + 800*a*b*e^(6*d*x + 6*c) + 160*b^2*e^(6*d*x + 6*c) + 1485*a^2*e^(4*d*x + 4*c) + 240*b^2*e^(4*d*x + 4*c) - 5
22*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