3.135 \(\int \coth ^7(c+d x) (a+b \text{sech}^2(c+d x))^3 \, dx\)
Optimal. Leaf size=77 \[ -\frac{3 a^2 (a+b) \text{csch}^2(c+d x)}{2 d}+\frac{a^3 \log (\sinh (c+d x))}{d}-\frac{(a+b)^3 \text{csch}^6(c+d x)}{6 d}-\frac{3 a (a+b)^2 \text{csch}^4(c+d x)}{4 d} \]
[Out]
(-3*a^2*(a + b)*Csch[c + d*x]^2)/(2*d) - (3*a*(a + b)^2*Csch[c + d*x]^4)/(4*d) - ((a + b)^3*Csch[c + d*x]^6)/(
6*d) + (a^3*Log[Sinh[c + d*x]])/d
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Rubi [A] time = 0.121003, 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, 444, 43} \[ -\frac{3 a^2 (a+b) \text{csch}^2(c+d x)}{2 d}+\frac{a^3 \log (\sinh (c+d x))}{d}-\frac{(a+b)^3 \text{csch}^6(c+d x)}{6 d}-\frac{3 a (a+b)^2 \text{csch}^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
[In]
Int[Coth[c + d*x]^7*(a + b*Sech[c + d*x]^2)^3,x]
[Out]
(-3*a^2*(a + b)*Csch[c + d*x]^2)/(2*d) - (3*a*(a + b)^2*Csch[c + d*x]^4)/(4*d) - ((a + b)^3*Csch[c + d*x]^6)/(
6*d) + (a^3*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 444
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]
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 )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x \left (b+a x^2\right )^3}{\left (1-x^2\right )^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^3}{(1-x)^4} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{(a+b)^3}{(-1+x)^4}+\frac{3 a (a+b)^2}{(-1+x)^3}+\frac{3 a^2 (a+b)}{(-1+x)^2}+\frac{a^3}{-1+x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{3 a^2 (a+b) \text{csch}^2(c+d x)}{2 d}-\frac{3 a (a+b)^2 \text{csch}^4(c+d x)}{4 d}-\frac{(a+b)^3 \text{csch}^6(c+d x)}{6 d}+\frac{a^3 \log (\sinh (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.773075, size = 98, normalized size = 1.27 \[ -\frac{2 \left (a \cosh ^2(c+d x)+b\right )^3 \left (18 a^2 (a+b) \text{csch}^2(c+d x)-12 a^3 \log (\sinh (c+d x))+2 (a+b)^3 \text{csch}^6(c+d x)+9 a (a+b)^2 \text{csch}^4(c+d x)\right )}{3 d (a \cosh (2 (c+d x))+a+2 b)^3} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[c + d*x]^7*(a + b*Sech[c + d*x]^2)^3,x]
[Out]
(-2*(b + a*Cosh[c + d*x]^2)^3*(18*a^2*(a + b)*Csch[c + d*x]^2 + 9*a*(a + b)^2*Csch[c + d*x]^4 + 2*(a + b)^3*Cs
ch[c + d*x]^6 - 12*a^3*Log[Sinh[c + d*x]]))/(3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)
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Maple [B] time = 0.05, size = 310, normalized size = 4. \begin{align*}{\frac{{a}^{3}\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3} \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{3} \left ({\rm coth} \left (dx+c\right ) \right ) ^{4}}{4\,d}}-{\frac{{a}^{3} \left ({\rm coth} \left (dx+c\right ) \right ) ^{6}}{6\,d}}-{\frac{3\,{a}^{2}b \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}{2\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{a}^{2}b \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{d \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{a}^{2}b \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{2\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{2}b \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{2\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{a{b}^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{2\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}}-{\frac{a{b}^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{4\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a{b}^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{4\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{6\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{b}^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{6\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{b}^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{6\,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)^3,x)
[Out]
1/d*a^3*ln(sinh(d*x+c))-1/2*a^3*coth(d*x+c)^2/d-1/4*a^3*coth(d*x+c)^4/d-1/6*a^3*coth(d*x+c)^6/d-3/2/d*a^2*b/si
nh(d*x+c)^6*cosh(d*x+c)^4+1/d*a^2*b/sinh(d*x+c)^6*cosh(d*x+c)^2+1/2/d*a^2*b/sinh(d*x+c)^4*cosh(d*x+c)^2-1/2/d*
a^2*b*cosh(d*x+c)^2/sinh(d*x+c)^2-1/2/d*a*b^2/sinh(d*x+c)^6*cosh(d*x+c)^2-1/4/d*a*b^2/sinh(d*x+c)^4*cosh(d*x+c
)^2+1/4/d*a*b^2*cosh(d*x+c)^2/sinh(d*x+c)^2-1/6/d*b^3/sinh(d*x+c)^6*cosh(d*x+c)^2+1/6/d*b^3/sinh(d*x+c)^4*cosh
(d*x+c)^2-1/6/d*b^3*cosh(d*x+c)^2/sinh(d*x+c)^2
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Maxima [B] time = 1.26945, size = 981, normalized size = 12.74 \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)^3,x, algorithm="maxima")
[Out]
1/3*a^3*(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)))
+ 2*a^2*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*a*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))) - 32/3
*b^3/(d*(e^(d*x + c) - e^(-d*x - c))^6)
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Fricas [B] time = 2.47787, size = 6665, normalized size = 86.56 \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)^3,x, algorithm="fricas")
[Out]
-1/3*(3*a^3*d*x*cosh(d*x + c)^12 + 36*a^3*d*x*cosh(d*x + c)*sinh(d*x + c)^11 + 3*a^3*d*x*sinh(d*x + c)^12 - 18
*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c)^10 + 18*(11*a^3*d*x*cosh(d*x + c)^2 - a^3*d*x + a^3 + a^2*b)*sinh(d*x +
c)^10 + 60*(11*a^3*d*x*cosh(d*x + c)^3 - 3*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c))*sinh(d*x + c)^9 + 9*(5*a^3*
d*x - 4*a^3 + 4*a*b^2)*cosh(d*x + c)^8 + 9*(165*a^3*d*x*cosh(d*x + c)^4 + 5*a^3*d*x - 4*a^3 + 4*a*b^2 - 90*(a^
3*d*x - a^3 - a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 72*(33*a^3*d*x*cosh(d*x + c)^5 - 30*(a^3*d*x - a^3 - a
^2*b)*cosh(d*x + c)^3 + (5*a^3*d*x - 4*a^3 + 4*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 4*(15*a^3*d*x - 17*a^3
- 15*a^2*b - 6*a*b^2 - 8*b^3)*cosh(d*x + c)^6 + 4*(693*a^3*d*x*cosh(d*x + c)^6 - 15*a^3*d*x - 945*(a^3*d*x - a
^3 - a^2*b)*cosh(d*x + c)^4 + 17*a^3 + 15*a^2*b + 6*a*b^2 + 8*b^3 + 63*(5*a^3*d*x - 4*a^3 + 4*a*b^2)*cosh(d*x
+ c)^2)*sinh(d*x + c)^6 + 24*(99*a^3*d*x*cosh(d*x + c)^7 - 189*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c)^5 + 21*(5
*a^3*d*x - 4*a^3 + 4*a*b^2)*cosh(d*x + c)^3 - (15*a^3*d*x - 17*a^3 - 15*a^2*b - 6*a*b^2 - 8*b^3)*cosh(d*x + c)
)*sinh(d*x + c)^5 + 3*a^3*d*x + 9*(5*a^3*d*x - 4*a^3 + 4*a*b^2)*cosh(d*x + c)^4 + 3*(495*a^3*d*x*cosh(d*x + c)
^8 - 1260*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c)^6 + 15*a^3*d*x + 210*(5*a^3*d*x - 4*a^3 + 4*a*b^2)*cosh(d*x +
c)^4 - 12*a^3 + 12*a*b^2 - 20*(15*a^3*d*x - 17*a^3 - 15*a^2*b - 6*a*b^2 - 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c
)^4 + 4*(165*a^3*d*x*cosh(d*x + c)^9 - 540*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c)^7 + 126*(5*a^3*d*x - 4*a^3 +
4*a*b^2)*cosh(d*x + c)^5 - 20*(15*a^3*d*x - 17*a^3 - 15*a^2*b - 6*a*b^2 - 8*b^3)*cosh(d*x + c)^3 + 9*(5*a^3*d*
x - 4*a^3 + 4*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 18*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c)^2 + 6*(33*a^3*d
*x*cosh(d*x + c)^10 - 135*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c)^8 + 42*(5*a^3*d*x - 4*a^3 + 4*a*b^2)*cosh(d*x
+ c)^6 - 3*a^3*d*x - 10*(15*a^3*d*x - 17*a^3 - 15*a^2*b - 6*a*b^2 - 8*b^3)*cosh(d*x + c)^4 + 3*a^3 + 3*a^2*b +
9*(5*a^3*d*x - 4*a^3 + 4*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*(a^3*cosh(d*x + c)^12 + 12*a^3*cosh(d*x
+ c)*sinh(d*x + c)^11 + a^3*sinh(d*x + c)^12 - 6*a^3*cosh(d*x + c)^10 + 15*a^3*cosh(d*x + c)^8 + 6*(11*a^3*cos
h(d*x + c)^2 - a^3)*sinh(d*x + c)^10 + 20*(11*a^3*cosh(d*x + c)^3 - 3*a^3*cosh(d*x + c))*sinh(d*x + c)^9 - 20*
a^3*cosh(d*x + c)^6 + 15*(33*a^3*cosh(d*x + c)^4 - 18*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^8 + 24*(33*a^3*
cosh(d*x + c)^5 - 30*a^3*cosh(d*x + c)^3 + 5*a^3*cosh(d*x + c))*sinh(d*x + c)^7 + 15*a^3*cosh(d*x + c)^4 + 4*(
231*a^3*cosh(d*x + c)^6 - 315*a^3*cosh(d*x + c)^4 + 105*a^3*cosh(d*x + c)^2 - 5*a^3)*sinh(d*x + c)^6 + 24*(33*
a^3*cosh(d*x + c)^7 - 63*a^3*cosh(d*x + c)^5 + 35*a^3*cosh(d*x + c)^3 - 5*a^3*cosh(d*x + c))*sinh(d*x + c)^5 -
6*a^3*cosh(d*x + c)^2 + 15*(33*a^3*cosh(d*x + c)^8 - 84*a^3*cosh(d*x + c)^6 + 70*a^3*cosh(d*x + c)^4 - 20*a^3
*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^4 + 20*(11*a^3*cosh(d*x + c)^9 - 36*a^3*cosh(d*x + c)^7 + 42*a^3*cosh(d*
x + c)^5 - 20*a^3*cosh(d*x + c)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 + 6*(11*a^3*cosh(d*x + c)^10 -
45*a^3*cosh(d*x + c)^8 + 70*a^3*cosh(d*x + c)^6 - 50*a^3*cosh(d*x + c)^4 + 15*a^3*cosh(d*x + c)^2 - a^3)*sinh(
d*x + c)^2 + 12*(a^3*cosh(d*x + c)^11 - 5*a^3*cosh(d*x + c)^9 + 10*a^3*cosh(d*x + c)^7 - 10*a^3*cosh(d*x + c)^
5 + 5*a^3*cosh(d*x + c)^3 - a^3*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^3*d*x*cosh(d*x + c)^11 - 15*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c)^9 + 6*(5*a^3*d*x - 4*a^3 + 4*
a*b^2)*cosh(d*x + c)^7 - 2*(15*a^3*d*x - 17*a^3 - 15*a^2*b - 6*a*b^2 - 8*b^3)*cosh(d*x + c)^5 + 3*(5*a^3*d*x -
4*a^3 + 4*a*b^2)*cosh(d*x + c)^3 - 3*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^1
2 + 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*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*cos
h(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 [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)**3,x)
[Out]
Timed out
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Giac [B] time = 1.7339, size = 323, normalized size = 4.19 \begin{align*} -\frac{60 \, a^{3} d x - 60 \, a^{3} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac{147 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} - 522 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 360 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 1485 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 720 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 1580 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1200 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 480 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 640 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1485 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 720 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 522 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 360 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 147 \, a^{3}}{{\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)^3,x, algorithm="giac")
[Out]
-1/60*(60*a^3*d*x - 60*a^3*log(abs(e^(2*d*x + 2*c) - 1)) + (147*a^3*e^(12*d*x + 12*c) - 522*a^3*e^(10*d*x + 10
*c) + 360*a^2*b*e^(10*d*x + 10*c) + 1485*a^3*e^(8*d*x + 8*c) + 720*a*b^2*e^(8*d*x + 8*c) - 1580*a^3*e^(6*d*x +
6*c) + 1200*a^2*b*e^(6*d*x + 6*c) + 480*a*b^2*e^(6*d*x + 6*c) + 640*b^3*e^(6*d*x + 6*c) + 1485*a^3*e^(4*d*x +
4*c) + 720*a*b^2*e^(4*d*x + 4*c) - 522*a^3*e^(2*d*x + 2*c) + 360*a^2*b*e^(2*d*x + 2*c) + 147*a^3)/(e^(2*d*x +
2*c) - 1)^6)/d