\(\int (a+b \text {sech}^2(c+d x))^3 \tanh ^3(c+d x) \, dx\) [125]

Optimal result

Integrand size = 23, antiderivative size = 103 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=\frac {a^3 \log (\cosh (c+d x))}{d}-\frac {3 a^2 b \text {sech}^2(c+d x)}{2 d}-\frac {3 a b^2 \text {sech}^4(c+d x)}{4 d}-\frac {b^3 \text {sech}^6(c+d x)}{6 d}+\frac {\left (b+a \cosh ^2(c+d x)\right )^4 \text {sech}^8(c+d x)}{8 b d} \] Output:

a^3*ln(cosh(d*x+c))/d-3/2*a^2*b*sech(d*x+c)^2/d-3/4*a*b^2*sech(d*x+c)^4/d- 
1/6*b^3*sech(d*x+c)^6/d+1/8*(b+a*cosh(d*x+c)^2)^4*sech(d*x+c)^8/b/d
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.24 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=\frac {\cosh ^6(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \left (24 a^3 \log (\cosh (c+d x))+12 a^2 (a-3 b) \text {sech}^2(c+d x)+18 a (a-b) b \text {sech}^4(c+d x)+4 (3 a-b) b^2 \text {sech}^6(c+d x)+3 b^3 \text {sech}^8(c+d x)\right )}{3 d (a+2 b+a \cosh (2 c+2 d x))^3} \] Input:

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

Output:

(Cosh[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3*(24*a^3*Log[Cosh[c + d*x]] + 12 
*a^2*(a - 3*b)*Sech[c + d*x]^2 + 18*a*(a - b)*b*Sech[c + d*x]^4 + 4*(3*a - 
 b)*b^2*Sech[c + d*x]^6 + 3*b^3*Sech[c + d*x]^8))/(3*d*(a + 2*b + a*Cosh[2 
*c + 2*d*x])^3)
 

Rubi [A] (warning: unable to verify)

Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.91, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 26, 4626, 354, 87, 49, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int i \tan (i c+i d x)^3 \left (a+b \sec (i c+i d x)^2\right )^3dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \left (b \sec (i c+i d x)^2+a\right )^3 \tan (i c+i d x)^3dx\)

\(\Big \downarrow \) 4626

\(\displaystyle -\frac {\int \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^3 \text {sech}^9(c+d x)d\cosh (c+d x)}{d}\)

\(\Big \downarrow \) 354

\(\displaystyle -\frac {\int \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^3 \text {sech}^5(c+d x)d\cosh ^2(c+d x)}{2 d}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\int \left (a \cosh ^2(c+d x)+b\right )^3 \text {sech}^4(c+d x)d\cosh ^2(c+d x)-\frac {\text {sech}^4(c+d x) \left (a \cosh ^2(c+d x)+b\right )^4}{4 b}}{2 d}\)

\(\Big \downarrow \) 49

\(\displaystyle -\frac {-\int \left (b^3 \text {sech}^4(c+d x)+3 a b^2 \text {sech}^3(c+d x)+3 a^2 b \text {sech}^2(c+d x)+a^3 \text {sech}(c+d x)\right )d\cosh ^2(c+d x)-\frac {\text {sech}^4(c+d x) \left (a \cosh ^2(c+d x)+b\right )^4}{4 b}}{2 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-a^3 \log \left (\cosh ^2(c+d x)\right )+3 a^2 b \text {sech}(c+d x)+\frac {3}{2} a b^2 \text {sech}^2(c+d x)-\frac {\text {sech}^4(c+d x) \left (a \cosh ^2(c+d x)+b\right )^4}{4 b}+\frac {1}{3} b^3 \text {sech}^3(c+d x)}{2 d}\)

Input:

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

Output:

-1/2*(-(a^3*Log[Cosh[c + d*x]^2]) + 3*a^2*b*Sech[c + d*x] + (3*a*b^2*Sech[ 
c + d*x]^2)/2 + (b^3*Sech[c + d*x]^3)/3 - ((b + a*Cosh[c + d*x]^2)^4*Sech[ 
c + d*x]^4)/(4*b))/d
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 49

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}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(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] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4626

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_ 
)]^(m_.), x_Symbol] :> Module[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-(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]
 

Maple [A] (verified)

Time = 106.65 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\frac {\frac {b^{3} \operatorname {sech}\left (d x +c \right )^{8}}{8}+\frac {a \,b^{2} \operatorname {sech}\left (d x +c \right )^{6}}{2}-\frac {\operatorname {sech}\left (d x +c \right )^{6} b^{3}}{6}+\frac {3 a^{2} b \operatorname {sech}\left (d x +c \right )^{4}}{4}-\frac {3 \operatorname {sech}\left (d x +c \right )^{4} a \,b^{2}}{4}+\frac {a^{3} \operatorname {sech}\left (d x +c \right )^{2}}{2}-\frac {3 \operatorname {sech}\left (d x +c \right )^{2} a^{2} b}{2}-a^{3} \ln \left (\operatorname {sech}\left (d x +c \right )\right )}{d}\)	\(113\)
default	\(\frac {\frac {b^{3} \operatorname {sech}\left (d x +c \right )^{8}}{8}+\frac {a \,b^{2} \operatorname {sech}\left (d x +c \right )^{6}}{2}-\frac {\operatorname {sech}\left (d x +c \right )^{6} b^{3}}{6}+\frac {3 a^{2} b \operatorname {sech}\left (d x +c \right )^{4}}{4}-\frac {3 \operatorname {sech}\left (d x +c \right )^{4} a \,b^{2}}{4}+\frac {a^{3} \operatorname {sech}\left (d x +c \right )^{2}}{2}-\frac {3 \operatorname {sech}\left (d x +c \right )^{2} a^{2} b}{2}-a^{3} \ln \left (\operatorname {sech}\left (d x +c \right )\right )}{d}\)	\(113\)
parts	\(\frac {a^{3} \left (-\frac {\tanh \left (d x +c \right )^{2}}{2}-\frac {\ln \left (-1+\tanh \left (d x +c \right )\right )}{2}-\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {b^{3} \left (\frac {\operatorname {sech}\left (d x +c \right )^{8}}{8}-\frac {\operatorname {sech}\left (d x +c \right )^{6}}{6}\right )}{d}+\frac {3 a^{2} b \tanh \left (d x +c \right )^{4}}{4 d}+\frac {3 a \,b^{2} \left (-\frac {\tanh \left (d x +c \right )^{6}}{6}+\frac {\tanh \left (d x +c \right )^{4}}{4}\right )}{d}\)	\(117\)
risch	\(-a^{3} x -\frac {2 a^{3} c}{d}+\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (3 a^{3} {\mathrm e}^{12 d x +12 c}-9 a^{2} b \,{\mathrm e}^{12 d x +12 c}+18 a^{3} {\mathrm e}^{10 d x +10 c}-36 a^{2} b \,{\mathrm e}^{10 d x +10 c}-18 a \,b^{2} {\mathrm e}^{10 d x +10 c}+45 a^{3} {\mathrm e}^{8 d x +8 c}-63 a^{2} b \,{\mathrm e}^{8 d x +8 c}-24 a \,b^{2} {\mathrm e}^{8 d x +8 c}-16 b^{3} {\mathrm e}^{8 d x +8 c}+60 a^{3} {\mathrm e}^{6 d x +6 c}-72 a^{2} b \,{\mathrm e}^{6 d x +6 c}-12 a \,b^{2} {\mathrm e}^{6 d x +6 c}+16 b^{3} {\mathrm e}^{6 d x +6 c}+45 a^{3} {\mathrm e}^{4 d x +4 c}-63 a^{2} b \,{\mathrm e}^{4 d x +4 c}-24 a \,b^{2} {\mathrm e}^{4 d x +4 c}-16 b^{3} {\mathrm e}^{4 d x +4 c}+18 a^{3} {\mathrm e}^{2 d x +2 c}-36 a^{2} b \,{\mathrm e}^{2 d x +2 c}-18 a \,b^{2} {\mathrm e}^{2 d x +2 c}+3 a^{3}-9 a^{2} b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d}\)	\(366\)

Input:

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

Output:

1/d*(1/8*b^3*sech(d*x+c)^8+1/2*a*b^2*sech(d*x+c)^6-1/6*sech(d*x+c)^6*b^3+3 
/4*a^2*b*sech(d*x+c)^4-3/4*sech(d*x+c)^4*a*b^2+1/2*a^3*sech(d*x+c)^2-3/2*s 
ech(d*x+c)^2*a^2*b-a^3*ln(sech(d*x+c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4658 vs. \(2 (95) = 190\).

Time = 0.48 (sec) , antiderivative size = 4658, normalized size of antiderivative = 45.22 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [A] (verification not implemented)

Time = 1.25 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.73 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=\begin {cases} a^{3} x - \frac {a^{3} \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} - \frac {a^{3} \tanh ^{2}{\left (c + d x \right )}}{2 d} - \frac {3 a^{2} b \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}^{2}{\left (c + d x \right )}}{4 d} - \frac {3 a^{2} b \operatorname {sech}^{2}{\left (c + d x \right )}}{4 d} - \frac {a b^{2} \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}^{4}{\left (c + d x \right )}}{2 d} - \frac {a b^{2} \operatorname {sech}^{4}{\left (c + d x \right )}}{4 d} - \frac {b^{3} \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}^{6}{\left (c + d x \right )}}{8 d} - \frac {b^{3} \operatorname {sech}^{6}{\left (c + d x \right )}}{24 d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {sech}^{2}{\left (c \right )}\right )^{3} \tanh ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a**3*x - a**3*log(tanh(c + d*x) + 1)/d - a**3*tanh(c + d*x)**2/ 
(2*d) - 3*a**2*b*tanh(c + d*x)**2*sech(c + d*x)**2/(4*d) - 3*a**2*b*sech(c 
 + d*x)**2/(4*d) - a*b**2*tanh(c + d*x)**2*sech(c + d*x)**4/(2*d) - a*b**2 
*sech(c + d*x)**4/(4*d) - b**3*tanh(c + d*x)**2*sech(c + d*x)**6/(8*d) - b 
**3*sech(c + d*x)**6/(24*d), Ne(d, 0)), (x*(a + b*sech(c)**2)**3*tanh(c)** 
3, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (95) = 190\).

Time = 0.15 (sec) , antiderivative size = 652, normalized size of antiderivative = 6.33 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx =\text {Too large to display} \] Input:

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

Output:

3/4*a^2*b*tanh(d*x + c)^4/d + a^3*(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*a 
*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))) - 32/3*b^3*(e^(-6*d*x - 6*c)/(d*( 
8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8* 
d*x - 8*c) + 56*e^(-10*d*x - 10*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x 
- 14*c) + e^(-16*d*x - 16*c) + 1)) - e^(-8*d*x - 8*c)/(d*(8*e^(-2*d*x - 2* 
c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56* 
e^(-10*d*x - 10*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16 
*d*x - 16*c) + 1)) + e^(-10*d*x - 10*c)/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4* 
d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 1 
0*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 
 1)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (95) = 190\).

Time = 0.19 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.76 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=-\frac {840 \, {\left (d x + c\right )} a^{3} - 840 \, a^{3} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac {2283 \, a^{3} e^{\left (16 \, d x + 16 \, c\right )} + 16584 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} + 5040 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 53844 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 20160 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 10080 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 102648 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 35280 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 13440 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 8960 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 126210 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 40320 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 6720 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 8960 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 102648 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 35280 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 13440 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 8960 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 53844 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 20160 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 10080 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 16584 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5040 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2283 \, a^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{840 \, d} \] Input:

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

Output:

-1/840*(840*(d*x + c)*a^3 - 840*a^3*log(e^(2*d*x + 2*c) + 1) + (2283*a^3*e 
^(16*d*x + 16*c) + 16584*a^3*e^(14*d*x + 14*c) + 5040*a^2*b*e^(14*d*x + 14 
*c) + 53844*a^3*e^(12*d*x + 12*c) + 20160*a^2*b*e^(12*d*x + 12*c) + 10080* 
a*b^2*e^(12*d*x + 12*c) + 102648*a^3*e^(10*d*x + 10*c) + 35280*a^2*b*e^(10 
*d*x + 10*c) + 13440*a*b^2*e^(10*d*x + 10*c) + 8960*b^3*e^(10*d*x + 10*c) 
+ 126210*a^3*e^(8*d*x + 8*c) + 40320*a^2*b*e^(8*d*x + 8*c) + 6720*a*b^2*e^ 
(8*d*x + 8*c) - 8960*b^3*e^(8*d*x + 8*c) + 102648*a^3*e^(6*d*x + 6*c) + 35 
280*a^2*b*e^(6*d*x + 6*c) + 13440*a*b^2*e^(6*d*x + 6*c) + 8960*b^3*e^(6*d* 
x + 6*c) + 53844*a^3*e^(4*d*x + 4*c) + 20160*a^2*b*e^(4*d*x + 4*c) + 10080 
*a*b^2*e^(4*d*x + 4*c) + 16584*a^3*e^(2*d*x + 2*c) + 5040*a^2*b*e^(2*d*x + 
 2*c) + 2283*a^3)/(e^(2*d*x + 2*c) + 1)^8)/d
 

Mupad [B] (verification not implemented)

Time = 2.51 (sec) , antiderivative size = 573, normalized size of antiderivative = 5.56 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=\frac {32\,\left (3\,a\,b^2-5\,b^3\right )}{d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-a^3\,x-\frac {128\,b^3}{d\,\left (7\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}+35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}+7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}+1\right )}-\frac {32\,\left (3\,a\,b^2-19\,b^3\right )}{3\,d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}+\frac {32\,b^3}{d\,\left (8\,{\mathrm {e}}^{2\,c+2\,d\,x}+28\,{\mathrm {e}}^{4\,c+4\,d\,x}+56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}+56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}+8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )}-\frac {8\,\left (9\,a^2\,b-21\,a\,b^2+4\,b^3\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {4\,\left (3\,a^2\,b-27\,a\,b^2+16\,b^3\right )}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {2\,\left (3\,a^2\,b-a^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {a^3\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1\right )}{d}-\frac {2\,\left (a^3-9\,a^2\,b+6\,a\,b^2\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \] Input:

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

Output:

(32*(3*a*b^2 - 5*b^3))/(d*(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*e 
xp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1)) - a^3*x - 
(128*b^3)/(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)) - (32*(3*a*b^2 - 19*b^3))/(3*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*ex 
p(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1)) + (32*b^3)/(d*(8*exp(2*c + 2*d 
*x) + 28*exp(4*c + 4*d*x) + 56*exp(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(1 
6*c + 16*d*x) + 1)) - (8*(9*a^2*b - 21*a*b^2 + 4*b^3))/(3*d*(3*exp(2*c + 2 
*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (4*(3*a^2*b - 27*a*b 
^2 + 16*b^3))/(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)) - (2*(3*a^2*b - a^3))/(d*(exp(2*c + 2*d*x) + 
 1)) + (a^3*log(exp(2*c)*exp(2*d*x) + 1))/d - (2*(6*a*b^2 - 9*a^2*b + a^3) 
)/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 605, normalized size of antiderivative = 5.87 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=\frac {-e^{4 d x +4 c} \mathrm {sech}\left (d x +c \right )^{6} b^{3}-3 e^{4 d x +4 c} \mathrm {sech}\left (d x +c \right )^{6} \tanh \left (d x +c \right )^{2} b^{3}-6 e^{4 d x +4 c} \mathrm {sech}\left (d x +c \right )^{4} a \,b^{2}-18 e^{4 d x +4 c} \mathrm {sech}\left (d x +c \right )^{2} a^{2} b -6 e^{2 d x +2 c} \mathrm {sech}\left (d x +c \right )^{6} \tanh \left (d x +c \right )^{2} b^{3}-12 e^{2 d x +2 c} \mathrm {sech}\left (d x +c \right )^{4} a \,b^{2}-36 e^{2 d x +2 c} \mathrm {sech}\left (d x +c \right )^{2} a^{2} b -12 \mathrm {sech}\left (d x +c \right )^{4} \tanh \left (d x +c \right )^{2} a \,b^{2}-18 \mathrm {sech}\left (d x +c \right )^{2} \tanh \left (d x +c \right )^{2} a^{2} b -24 a^{3}+24 e^{4 d x +4 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) a^{3}+48 e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) a^{3}-2 e^{2 d x +2 c} \mathrm {sech}\left (d x +c \right )^{6} b^{3}-3 \mathrm {sech}\left (d x +c \right )^{6} \tanh \left (d x +c \right )^{2} b^{3}-6 \mathrm {sech}\left (d x +c \right )^{4} a \,b^{2}-18 \mathrm {sech}\left (d x +c \right )^{2} a^{2} b -12 e^{4 d x +4 c} \mathrm {sech}\left (d x +c \right )^{4} \tanh \left (d x +c \right )^{2} a \,b^{2}-18 e^{4 d x +4 c} \mathrm {sech}\left (d x +c \right )^{2} \tanh \left (d x +c \right )^{2} a^{2} b -24 e^{2 d x +2 c} \mathrm {sech}\left (d x +c \right )^{4} \tanh \left (d x +c \right )^{2} a \,b^{2}-36 e^{2 d x +2 c} \mathrm {sech}\left (d x +c \right )^{2} \tanh \left (d x +c \right )^{2} a^{2} b -24 e^{4 d x +4 c} a^{3}-24 e^{4 d x +4 c} a^{3} d x -24 a^{3} d x -48 e^{2 d x +2 c} a^{3} d x -\mathrm {sech}\left (d x +c \right )^{6} b^{3}+24 \,\mathrm {log}\left (e^{2 d x +2 c}+1\right ) a^{3}}{24 d \left (e^{4 d x +4 c}+2 e^{2 d x +2 c}+1\right )} \] Input:

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

Output:

(24*e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x) + 1)*a**3 - 3*e**(4*c + 4*d*x)*s 
ech(c + d*x)**6*tanh(c + d*x)**2*b**3 - e**(4*c + 4*d*x)*sech(c + d*x)**6* 
b**3 - 12*e**(4*c + 4*d*x)*sech(c + d*x)**4*tanh(c + d*x)**2*a*b**2 - 6*e* 
*(4*c + 4*d*x)*sech(c + d*x)**4*a*b**2 - 18*e**(4*c + 4*d*x)*sech(c + d*x) 
**2*tanh(c + d*x)**2*a**2*b - 18*e**(4*c + 4*d*x)*sech(c + d*x)**2*a**2*b 
- 24*e**(4*c + 4*d*x)*a**3*d*x - 24*e**(4*c + 4*d*x)*a**3 + 48*e**(2*c + 2 
*d*x)*log(e**(2*c + 2*d*x) + 1)*a**3 - 6*e**(2*c + 2*d*x)*sech(c + d*x)**6 
*tanh(c + d*x)**2*b**3 - 2*e**(2*c + 2*d*x)*sech(c + d*x)**6*b**3 - 24*e** 
(2*c + 2*d*x)*sech(c + d*x)**4*tanh(c + d*x)**2*a*b**2 - 12*e**(2*c + 2*d* 
x)*sech(c + d*x)**4*a*b**2 - 36*e**(2*c + 2*d*x)*sech(c + d*x)**2*tanh(c + 
 d*x)**2*a**2*b - 36*e**(2*c + 2*d*x)*sech(c + d*x)**2*a**2*b - 48*e**(2*c 
 + 2*d*x)*a**3*d*x + 24*log(e**(2*c + 2*d*x) + 1)*a**3 - 3*sech(c + d*x)** 
6*tanh(c + d*x)**2*b**3 - sech(c + d*x)**6*b**3 - 12*sech(c + d*x)**4*tanh 
(c + d*x)**2*a*b**2 - 6*sech(c + d*x)**4*a*b**2 - 18*sech(c + d*x)**2*tanh 
(c + d*x)**2*a**2*b - 18*sech(c + d*x)**2*a**2*b - 24*a**3*d*x - 24*a**3)/ 
(24*d*(e**(4*c + 4*d*x) + 2*e**(2*c + 2*d*x) + 1))