\(\int \text {sech}^3(c+d x) (a+b \text {sech}^2(c+d x))^3 \, dx\) [71]

Optimal result

Integrand size = 23, antiderivative size = 172 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {\left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \arctan (\sinh (c+d x))}{128 d}+\frac {\left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \text {sech}(c+d x) \tanh (c+d x)}{128 d}+\frac {b \left (144 a^2+120 a b+35 b^2\right ) \text {sech}^3(c+d x) \tanh (c+d x)}{192 d}+\frac {b^2 (24 a+7 b) \text {sech}^5(c+d x) \tanh (c+d x)}{48 d}+\frac {b^3 \text {sech}^7(c+d x) \tanh (c+d x)}{8 d} \] Output:

1/128*(64*a^3+144*a^2*b+120*a*b^2+35*b^3)*arctan(sinh(d*x+c))/d+1/128*(64* 
a^3+144*a^2*b+120*a*b^2+35*b^3)*sech(d*x+c)*tanh(d*x+c)/d+1/192*b*(144*a^2 
+120*a*b+35*b^2)*sech(d*x+c)^3*tanh(d*x+c)/d+1/48*b^2*(24*a+7*b)*sech(d*x+ 
c)^5*tanh(d*x+c)/d+1/8*b^3*sech(d*x+c)^7*tanh(d*x+c)/d
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.49 (sec) , antiderivative size = 1618, normalized size of antiderivative = 9.41 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(Coth[c + d*x]^6*Csch[c + d*x]*(a + b*Sech[c + d*x]^2)^3*(344123325*(a + b 
)^3*Sinh[c + d*x]^2 + 760096575*a*(a + b)^2*Sinh[c + d*x]^4 + 213089100*(a 
 + b)^3*Sinh[c + d*x]^4 + 578580975*a^2*(a + b)*Sinh[c + d*x]^6 + 48196260 
0*a*(a + b)^2*Sinh[c + d*x]^6 + 12757815*(a + b)^3*Sinh[c + d*x]^6 + 15347 
5245*a^3*Sinh[c + d*x]^8 + 372265740*a^2*(a + b)*Sinh[c + d*x]^8 + 2867602 
5*a*(a + b)^2*Sinh[c + d*x]^8 + 99450960*a^3*Sinh[c + d*x]^10 + 22639365*a 
^2*(a + b)*Sinh[c + d*x]^10 - 257600*(a + b)^3*HypergeometricPFQ[{3/2, 2, 
2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^10 - 65408*(a + 
 b)^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c 
+ d*x]^2]*Sinh[c + d*x]^10 - 8960*(a + b)^3*HypergeometricPFQ[{3/2, 2, 2, 
2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^10 - 5 
12*(a + b)^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 
 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^10 + 6134499*a^3*Sinh[c + d*x]^ 
12 - 613440*a*(a + b)^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/ 
2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^12 - 171648*a*(a + b)^2*Hypergeometric 
PFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d 
*x]^12 - 25344*a*(a + b)^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 
1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^12 - 1536*a*(a + b)^2*H 
ypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 1, 11/2}, -Si 
nh[c + d*x]^2]*Sinh[c + d*x]^12 - 495552*a^2*(a + b)*HypergeometricPFQ[...
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 4635, 315, 401, 25, 298, 215, 216}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4635

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

\(\Big \downarrow \) 315

\(\displaystyle \frac {\frac {1}{8} \int \frac {\left (a \sinh ^2(c+d x)+a+b\right ) \left (a (8 a+3 b) \sinh ^2(c+d x)+(a+b) (8 a+7 b)\right )}{\left (\sinh ^2(c+d x)+1\right )^4}d\sinh (c+d x)+\frac {b \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )^2}{8 \left (\sinh ^2(c+d x)+1\right )^4}}{d}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {\frac {1}{8} \left (\frac {b (12 a+7 b) \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}-\frac {1}{6} \int -\frac {3 a \left (16 a^2+18 b a+7 b^2\right ) \sinh ^2(c+d x)+(a+b) \left (48 a^2+78 b a+35 b^2\right )}{\left (\sinh ^2(c+d x)+1\right )^3}d\sinh (c+d x)\right )+\frac {b \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )^2}{8 \left (\sinh ^2(c+d x)+1\right )^4}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \int \frac {3 a \left (16 a^2+18 b a+7 b^2\right ) \sinh ^2(c+d x)+(a+b) \left (48 a^2+78 b a+35 b^2\right )}{\left (\sinh ^2(c+d x)+1\right )^3}d\sinh (c+d x)+\frac {b (12 a+7 b) \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}\right )+\frac {b \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )^2}{8 \left (\sinh ^2(c+d x)+1\right )^4}}{d}\)

\(\Big \downarrow \) 298

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {3}{4} \left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \int \frac {1}{\left (\sinh ^2(c+d x)+1\right )^2}d\sinh (c+d x)+\frac {b \left (72 a^2+92 a b+35 b^2\right ) \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )+\frac {b (12 a+7 b) \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}\right )+\frac {b \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )^2}{8 \left (\sinh ^2(c+d x)+1\right )^4}}{d}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {3}{4} \left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \left (\frac {1}{2} \int \frac {1}{\sinh ^2(c+d x)+1}d\sinh (c+d x)+\frac {\sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )+\frac {b \left (72 a^2+92 a b+35 b^2\right ) \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )+\frac {b (12 a+7 b) \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}\right )+\frac {b \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )^2}{8 \left (\sinh ^2(c+d x)+1\right )^4}}{d}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {b \left (72 a^2+92 a b+35 b^2\right ) \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}+\frac {3}{4} \left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \left (\frac {1}{2} \arctan (\sinh (c+d x))+\frac {\sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )\right )+\frac {b (12 a+7 b) \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}\right )+\frac {b \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )^2}{8 \left (\sinh ^2(c+d x)+1\right )^4}}{d}\)

Input:

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

Output:

((b*Sinh[c + d*x]*(a + b + a*Sinh[c + d*x]^2)^2)/(8*(1 + Sinh[c + d*x]^2)^ 
4) + ((b*(12*a + 7*b)*Sinh[c + d*x]*(a + b + a*Sinh[c + d*x]^2))/(6*(1 + S 
inh[c + d*x]^2)^3) + ((b*(72*a^2 + 92*a*b + 35*b^2)*Sinh[c + d*x])/(4*(1 + 
 Sinh[c + d*x]^2)^2) + (3*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*(ArcTa 
n[Sinh[c + d*x]]/2 + Sinh[c + d*x]/(2*(1 + Sinh[c + d*x]^2))))/4)/6)/8)/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
 

rule 401

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

rule 3042

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

rule 4635

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ 
))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f 
 Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^((m 
+ n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && In 
tegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
 

Maple [A] (verified)

Time = 3.64 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+3 a \,b^{2} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )+b^{3} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )+\frac {35 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\)	\(186\)
default	\(\frac {a^{3} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+3 a \,b^{2} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )+b^{3} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )+\frac {35 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\)	\(186\)
parts	\(\frac {a^{3} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}+\frac {b^{3} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )+\frac {35 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}+\frac {3 a \,b^{2} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )}{d}+\frac {3 a^{2} b \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )}{d}\)	\(194\)
parallelrisch	\(\frac {-10752 i \left (a^{3}+\frac {9}{4} a^{2} b +\frac {15}{8} a \,b^{2}+\frac {35}{64} b^{3}\right ) \left (\frac {5}{8}+\frac {\cosh \left (8 d x +8 c \right )}{56}+\frac {\cosh \left (6 d x +6 c \right )}{7}+\frac {\cosh \left (4 d x +4 c \right )}{2}+\cosh \left (2 d x +2 c \right )\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )+10752 i \left (a^{3}+\frac {9}{4} a^{2} b +\frac {15}{8} a \,b^{2}+\frac {35}{64} b^{3}\right ) \left (\frac {5}{8}+\frac {\cosh \left (8 d x +8 c \right )}{56}+\frac {\cosh \left (6 d x +6 c \right )}{7}+\frac {\cosh \left (4 d x +4 c \right )}{2}+\cosh \left (2 d x +2 c \right )\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )+\left (3456 a^{3}+14688 a^{2} b +18384 a \,b^{2}+5362 b^{3}\right ) \sinh \left (3 d x +3 c \right )+\left (1920 a^{3}+6624 a^{2} b +5520 a \,b^{2}+1610 b^{3}\right ) \sinh \left (5 d x +5 c \right )+\left (384 a^{3}+864 a^{2} b +720 a \,b^{2}+210 b^{3}\right ) \sinh \left (7 d x +7 c \right )+1920 \sinh \left (d x +c \right ) \left (a^{3}+\frac {93}{20} a^{2} b +\frac {283}{40} a \,b^{2}+\frac {5053}{960} b^{3}\right )}{384 d \left (35+\cosh \left (8 d x +8 c \right )+8 \cosh \left (6 d x +6 c \right )+28 \cosh \left (4 d x +4 c \right )+56 \cosh \left (2 d x +2 c \right )\right )}\)	\(343\)
risch	\(\frac {{\mathrm e}^{d x +c} \left (-432 a^{2} b -360 a \,b^{2}+192 \,{\mathrm e}^{14 d x +14 c} a^{3}-192 a^{3}-805 b^{3} {\mathrm e}^{2 d x +2 c}-960 a^{3} {\mathrm e}^{2 d x +2 c}-2681 b^{3} {\mathrm e}^{4 d x +4 c}-5053 b^{3} {\mathrm e}^{6 d x +6 c}-1728 a^{3} {\mathrm e}^{4 d x +4 c}-960 a^{3} {\mathrm e}^{6 d x +6 c}-105 b^{3}-4464 a^{2} b \,{\mathrm e}^{6 d x +6 c}-6792 a \,b^{2} {\mathrm e}^{6 d x +6 c}-7344 a^{2} b \,{\mathrm e}^{4 d x +4 c}-9192 a \,b^{2} {\mathrm e}^{4 d x +4 c}+6792 a \,b^{2} {\mathrm e}^{8 d x +8 c}+432 a^{2} b \,{\mathrm e}^{14 d x +14 c}+805 b^{3} {\mathrm e}^{12 d x +12 c}+1728 a^{3} {\mathrm e}^{10 d x +10 c}+960 a^{3} {\mathrm e}^{12 d x +12 c}+2681 b^{3} {\mathrm e}^{10 d x +10 c}+960 a^{3} {\mathrm e}^{8 d x +8 c}+2760 a \,b^{2} {\mathrm e}^{12 d x +12 c}+7344 a^{2} b \,{\mathrm e}^{10 d x +10 c}+9192 a \,b^{2} {\mathrm e}^{10 d x +10 c}+5053 b^{3} {\mathrm e}^{8 d x +8 c}+4464 a^{2} b \,{\mathrm e}^{8 d x +8 c}+3312 a^{2} b \,{\mathrm e}^{12 d x +12 c}-2760 a \,b^{2} {\mathrm e}^{2 d x +2 c}-3312 a^{2} b \,{\mathrm e}^{2 d x +2 c}+360 \,{\mathrm e}^{14 d x +14 c} a \,b^{2}+105 b^{3} {\mathrm e}^{14 d x +14 c}\right )}{192 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{2 d}+\frac {9 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{8 d}+\frac {15 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a \,b^{2}}{16 d}+\frac {35 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{128 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{2 d}-\frac {9 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{8 d}-\frac {15 i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{16 d}-\frac {35 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{128 d}\)	\(582\)

Input:

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

Output:

1/d*(a^3*(1/2*sech(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c)))+3*a^2*b*((1/4*se 
ch(d*x+c)^3+3/8*sech(d*x+c))*tanh(d*x+c)+3/4*arctan(exp(d*x+c)))+3*a*b^2*( 
(1/6*sech(d*x+c)^5+5/24*sech(d*x+c)^3+5/16*sech(d*x+c))*tanh(d*x+c)+5/8*ar 
ctan(exp(d*x+c)))+b^3*((1/8*sech(d*x+c)^7+7/48*sech(d*x+c)^5+35/192*sech(d 
*x+c)^3+35/128*sech(d*x+c))*tanh(d*x+c)+35/64*arctan(exp(d*x+c))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6114 vs. \(2 (162) = 324\).

Time = 0.32 (sec) , antiderivative size = 6114, normalized size of antiderivative = 35.55 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

\[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3} \operatorname {sech}^{3}{\left (c + d x \right )}\, dx \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (162) = 324\).

Time = 0.15 (sec) , antiderivative size = 556, normalized size of antiderivative = 3.23 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

-1/192*b^3*(105*arctan(e^(-d*x - c))/d - (105*e^(-d*x - c) + 805*e^(-3*d*x 
 - 3*c) + 2681*e^(-5*d*x - 5*c) + 5053*e^(-7*d*x - 7*c) - 5053*e^(-9*d*x - 
 9*c) - 2681*e^(-11*d*x - 11*c) - 805*e^(-13*d*x - 13*c) - 105*e^(-15*d*x 
- 15*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))) - 1/8*a*b^2*(15*arctan(e^ 
(-d*x - c))/d - (15*e^(-d*x - c) + 85*e^(-3*d*x - 3*c) + 198*e^(-5*d*x - 5 
*c) - 198*e^(-7*d*x - 7*c) - 85*e^(-9*d*x - 9*c) - 15*e^(-11*d*x - 11*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))) - 3/4*a^ 
2*b*(3*arctan(e^(-d*x - c))/d - (3*e^(-d*x - c) + 11*e^(-3*d*x - 3*c) - 11 
*e^(-5*d*x - 5*c) - 3*e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) + 6*e^(-4*d 
*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1))) - a^3*(arctan(e^( 
-d*x - c))/d - (e^(-d*x - c) - e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) + 
e^(-4*d*x - 4*c) + 1)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 485 vs. \(2 (162) = 324\).

Time = 0.14 (sec) , antiderivative size = 485, normalized size of antiderivative = 2.82 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (64 \, a^{3} + 144 \, a^{2} b + 120 \, a b^{2} + 35 \, b^{3}\right )} + \frac {4 \, {\left (192 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 432 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 360 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 105 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 2304 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 6336 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 5280 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 1540 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 9216 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 29952 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 28032 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 8176 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 12288 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 46080 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 50688 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 17856 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{4}}}{768 \, d} \] Input:

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

Output:

1/768*(3*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(64*a^3 + 
 144*a^2*b + 120*a*b^2 + 35*b^3) + 4*(192*a^3*(e^(d*x + c) - e^(-d*x - c)) 
^7 + 432*a^2*b*(e^(d*x + c) - e^(-d*x - c))^7 + 360*a*b^2*(e^(d*x + c) - e 
^(-d*x - c))^7 + 105*b^3*(e^(d*x + c) - e^(-d*x - c))^7 + 2304*a^3*(e^(d*x 
 + c) - e^(-d*x - c))^5 + 6336*a^2*b*(e^(d*x + c) - e^(-d*x - c))^5 + 5280 
*a*b^2*(e^(d*x + c) - e^(-d*x - c))^5 + 1540*b^3*(e^(d*x + c) - e^(-d*x - 
c))^5 + 9216*a^3*(e^(d*x + c) - e^(-d*x - c))^3 + 29952*a^2*b*(e^(d*x + c) 
 - e^(-d*x - c))^3 + 28032*a*b^2*(e^(d*x + c) - e^(-d*x - c))^3 + 8176*b^3 
*(e^(d*x + c) - e^(-d*x - c))^3 + 12288*a^3*(e^(d*x + c) - e^(-d*x - c)) + 
 46080*a^2*b*(e^(d*x + c) - e^(-d*x - c)) + 50688*a*b^2*(e^(d*x + c) - e^( 
-d*x - c)) + 17856*b^3*(e^(d*x + c) - e^(-d*x - c)))/((e^(d*x + c) - e^(-d 
*x - c))^2 + 4)^4)/d
 

Mupad [B] (verification not implemented)

Time = 2.40 (sec) , antiderivative size = 931, normalized size of antiderivative = 5.41 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(atan((exp(d*x)*exp(c)*(64*a^3*(d^2)^(1/2) + 35*b^3*(d^2)^(1/2) + 120*a*b^ 
2*(d^2)^(1/2) + 144*a^2*b*(d^2)^(1/2)))/(d*(8400*a*b^5 + 18432*a^5*b + 409 
6*a^6 + 1225*b^6 + 24480*a^2*b^4 + 39040*a^3*b^3 + 36096*a^4*b^2)^(1/2)))* 
(8400*a*b^5 + 18432*a^5*b + 4096*a^6 + 1225*b^6 + 24480*a^2*b^4 + 39040*a^ 
3*b^3 + 36096*a^4*b^2)^(1/2))/(64*(d^2)^(1/2)) - ((a^3*exp(c + d*x))/(2*d) 
 + (2*exp(7*c + 7*d*x)*(24*a*b^2 + 18*a^2*b + 5*a^3 + 16*b^3))/d + (a^3*ex 
p(13*c + 13*d*x))/(2*d) + (3*a*exp(5*c + 5*d*x)*(16*a*b + 5*a^2 + 16*b^2)) 
/(2*d) + (3*a*exp(9*c + 9*d*x)*(16*a*b + 5*a^2 + 16*b^2))/(2*d) + (3*a^2*e 
xp(3*c + 3*d*x)*(a + 2*b))/d + (3*a^2*exp(11*c + 11*d*x)*(a + 2*b))/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(16*c + 16*d*x) + 1) + (2*exp(c + d*x)*(48*a*b^2 - 37*b^3))/(3* 
d*(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) + 5*exp( 
8*c + 8*d*x) + exp(10*c + 10*d*x) + 1)) + (exp(c + d*x)*(24*a^2*b - 120*a* 
b^2 + b^3))/(4*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)) - (16*b^3*exp(c + d*x))/(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)) + (ex 
p(c + d*x)*(120*a*b^2 + 144*a^2*b + 64*a^3 + 35*b^3))/(64*d*(exp(2*c + 2*d 
*x) + 1)) - (4*exp(c + d*x)*(6*a*b^2 - 29*b^3))/(3*d*(6*exp(2*c + 2*d*x...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1394, normalized size of antiderivative = 8.10 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(192*e**(16*c + 16*d*x)*atan(e**(c + d*x))*a**3 + 432*e**(16*c + 16*d*x)*a 
tan(e**(c + d*x))*a**2*b + 360*e**(16*c + 16*d*x)*atan(e**(c + d*x))*a*b** 
2 + 105*e**(16*c + 16*d*x)*atan(e**(c + d*x))*b**3 + 1536*e**(14*c + 14*d* 
x)*atan(e**(c + d*x))*a**3 + 3456*e**(14*c + 14*d*x)*atan(e**(c + d*x))*a* 
*2*b + 2880*e**(14*c + 14*d*x)*atan(e**(c + d*x))*a*b**2 + 840*e**(14*c + 
14*d*x)*atan(e**(c + d*x))*b**3 + 5376*e**(12*c + 12*d*x)*atan(e**(c + d*x 
))*a**3 + 12096*e**(12*c + 12*d*x)*atan(e**(c + d*x))*a**2*b + 10080*e**(1 
2*c + 12*d*x)*atan(e**(c + d*x))*a*b**2 + 2940*e**(12*c + 12*d*x)*atan(e** 
(c + d*x))*b**3 + 10752*e**(10*c + 10*d*x)*atan(e**(c + d*x))*a**3 + 24192 
*e**(10*c + 10*d*x)*atan(e**(c + d*x))*a**2*b + 20160*e**(10*c + 10*d*x)*a 
tan(e**(c + d*x))*a*b**2 + 5880*e**(10*c + 10*d*x)*atan(e**(c + d*x))*b**3 
 + 13440*e**(8*c + 8*d*x)*atan(e**(c + d*x))*a**3 + 30240*e**(8*c + 8*d*x) 
*atan(e**(c + d*x))*a**2*b + 25200*e**(8*c + 8*d*x)*atan(e**(c + d*x))*a*b 
**2 + 7350*e**(8*c + 8*d*x)*atan(e**(c + d*x))*b**3 + 10752*e**(6*c + 6*d* 
x)*atan(e**(c + d*x))*a**3 + 24192*e**(6*c + 6*d*x)*atan(e**(c + d*x))*a** 
2*b + 20160*e**(6*c + 6*d*x)*atan(e**(c + d*x))*a*b**2 + 5880*e**(6*c + 6* 
d*x)*atan(e**(c + d*x))*b**3 + 5376*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a* 
*3 + 12096*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**2*b + 10080*e**(4*c + 4* 
d*x)*atan(e**(c + d*x))*a*b**2 + 2940*e**(4*c + 4*d*x)*atan(e**(c + d*x))* 
b**3 + 1536*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**3 + 3456*e**(2*c + 2...