Integrand size = 23, antiderivative size = 196 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {\left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \arctan (\sinh (c+d x))}{128 d}+\frac {\left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \text {sech}(c+d x) \tanh (c+d x)}{128 d}+\frac {b \left (72 a^2+92 a b+35 b^2\right ) \text {sech}^3(c+d x) \tanh (c+d x)}{192 d}+\frac {b (12 a+7 b) \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{48 d}+\frac {b \text {sech}^7(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{8 d} \]
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*(72*a^2+ 92*a*b+35*b^2)*sech(d*x+c)^3*tanh(d*x+c)/d+1/48*b*(12*a+7*b)*sech(d*x+c)^5 *(a+b+a*sinh(d*x+c)^2)*tanh(d*x+c)/d+1/8*b*sech(d*x+c)^7*(a+b+a*sinh(d*x+c )^2)^2*tanh(d*x+c)/d
Time = 10.73 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.52 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {\left (b+a \cosh ^2(c+d x)\right )^3 \text {sech}^8(c+d x) \left (6 \left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right ) \cosh ^8(c+d x)+48 b^3 \text {sech}(c) \sinh (d x)+8 b^2 (24 a+7 b) \cosh ^2(c+d x) \text {sech}(c) \sinh (d x)+2 b \left (144 a^2+120 a b+35 b^2\right ) \cosh ^4(c+d x) \text {sech}(c) \sinh (d x)+3 \left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \cosh ^6(c+d x) \text {sech}(c) \sinh (d x)+48 b^3 \cosh (c+d x) \tanh (c)+8 b^2 (24 a+7 b) \cosh ^3(c+d x) \tanh (c)+2 b \left (144 a^2+120 a b+35 b^2\right ) \cosh ^5(c+d x) \tanh (c)+3 \left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \cosh ^7(c+d x) \tanh (c)\right )}{48 d (a+2 b+a \cosh (2 (c+d x)))^3} \]
((b + a*Cosh[c + d*x]^2)^3*Sech[c + d*x]^8*(6*(64*a^3 + 144*a^2*b + 120*a* b^2 + 35*b^3)*ArcTan[Tanh[(c + d*x)/2]]*Cosh[c + d*x]^8 + 48*b^3*Sech[c]*S inh[d*x] + 8*b^2*(24*a + 7*b)*Cosh[c + d*x]^2*Sech[c]*Sinh[d*x] + 2*b*(144 *a^2 + 120*a*b + 35*b^2)*Cosh[c + d*x]^4*Sech[c]*Sinh[d*x] + 3*(64*a^3 + 1 44*a^2*b + 120*a*b^2 + 35*b^3)*Cosh[c + d*x]^6*Sech[c]*Sinh[d*x] + 48*b^3* Cosh[c + d*x]*Tanh[c] + 8*b^2*(24*a + 7*b)*Cosh[c + d*x]^3*Tanh[c] + 2*b*( 144*a^2 + 120*a*b + 35*b^2)*Cosh[c + d*x]^5*Tanh[c] + 3*(64*a^3 + 144*a^2* b + 120*a*b^2 + 35*b^3)*Cosh[c + d*x]^7*Tanh[c]))/(48*d*(a + 2*b + a*Cosh[ 2*(c + d*x)])^3)
Time = 0.41 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.99, 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 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {b \left (72 a^2+92 a b+35 b^2\right ) \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}+\frac {3}{4} \left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \left (\frac {1}{2} \arctan (\sinh (c+d x))+\frac {\sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )\right )+\frac {b (12 a+7 b) \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}\right )+\frac {b \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )^2}{8 \left (\sinh ^2(c+d x)+1\right )^4}}{d}\) |
((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
3.1.71.3.1 Defintions of rubi rules used
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])
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])
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])
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]
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]
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]
Time = 3.17 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.95
| 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 | \(\frac {-10752 i \left (a^{3}+\frac {9}{4} a^{2} b +\frac {15}{8} a \,b^{2}+\frac {35}{64} b^{3}\right ) \left (\frac {5}{8}+\frac {\cosh \left (8 d x +8 c \right )}{56}+\frac {\cosh \left (6 d x +6 c \right )}{7}+\frac {\cosh \left (4 d x +4 c \right )}{2}+\cosh \left (2 d x +2 c \right )\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )+10752 i \left (a^{3}+\frac {9}{4} a^{2} b +\frac {15}{8} a \,b^{2}+\frac {35}{64} b^{3}\right ) \left (\frac {5}{8}+\frac {\cosh \left (8 d x +8 c \right )}{56}+\frac {\cosh \left (6 d x +6 c \right )}{7}+\frac {\cosh \left (4 d x +4 c \right )}{2}+\cosh \left (2 d x +2 c \right )\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )+\left (3456 a^{3}+14688 a^{2} b +18384 a \,b^{2}+5362 b^{3}\right ) \sinh \left (3 d x +3 c \right )+\left (1920 a^{3}+6624 a^{2} b +5520 a \,b^{2}+1610 b^{3}\right ) \sinh \left (5 d x +5 c \right )+\left (384 a^{3}+864 a^{2} b +720 a \,b^{2}+210 b^{3}\right ) \sinh \left (7 d x +7 c \right )+1920 \left (a^{3}+\frac {93}{20} a^{2} b +\frac {283}{40} a \,b^{2}+\frac {5053}{960} b^{3}\right ) \sinh \left (d x +c \right )}{384 d \left (35+\cosh \left (8 d x +8 c \right )+8 \cosh \left (6 d x +6 c \right )+28 \cosh \left (4 d x +4 c \right )+56 \cosh \left (2 d x +2 c \right )\right )}\) | \(343\) |
| risch | \(\frac {{\mathrm e}^{d x +c} \left (-360 a \,b^{2}+960 a^{3} {\mathrm e}^{12 d x +12 c}+2681 b^{3} {\mathrm e}^{10 d x +10 c}+2760 a \,b^{2} {\mathrm e}^{12 d x +12 c}-192 a^{3}-2681 \,{\mathrm e}^{4 d x +4 c} b^{3}+960 a^{3} {\mathrm e}^{8 d x +8 c}-432 a^{2} b -2760 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}-3312 a^{2} b \,{\mathrm e}^{2 d x +2 c}-9192 a \,b^{2} {\mathrm e}^{4 d x +4 c}-7344 a^{2} b \,{\mathrm e}^{4 d x +4 c}-4464 a^{2} b \,{\mathrm e}^{6 d x +6 c}-6792 a \,b^{2} {\mathrm e}^{6 d x +6 c}+192 a^{3} {\mathrm e}^{14 d x +14 c}+105 b^{3} {\mathrm e}^{14 d x +14 c}-105 b^{3}+805 b^{3} {\mathrm e}^{12 d x +12 c}+3312 a^{2} b \,{\mathrm e}^{12 d x +12 c}+360 a \,b^{2} {\mathrm e}^{14 d x +14 c}-960 a^{3} {\mathrm e}^{6 d x +6 c}-5053 \,{\mathrm e}^{6 d x +6 c} b^{3}-1728 a^{3} {\mathrm e}^{4 d x +4 c}+1728 a^{3} {\mathrm e}^{10 d x +10 c}+5053 b^{3} {\mathrm e}^{8 d x +8 c}-960 a^{3} {\mathrm e}^{2 d x +2 c}-805 \,{\mathrm e}^{2 d x +2 c} b^{3}+432 a^{2} b \,{\mathrm e}^{14 d x +14 c}+7344 a^{2} b \,{\mathrm e}^{10 d x +10 c}+4464 a^{2} b \,{\mathrm e}^{8 d x +8 c}+6792 a \,b^{2} {\mathrm e}^{8 d x +8 c}+9192 a \,b^{2} {\mathrm e}^{10 d x +10 c}\right )}{192 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{2 d}+\frac {9 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{8 d}+\frac {15 i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{16 d}+\frac {35 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{128 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{2 d}-\frac {9 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2} b}{8 d}-\frac {15 i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{16 d}-\frac {35 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{128 d}\) | \(582\) |
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))))
Leaf count of result is larger than twice the leaf count of optimal. 6114 vs. \(2 (186) = 372\).
Time = 0.31 (sec) , antiderivative size = 6114, normalized size of antiderivative = 31.19 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]
\[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3} \operatorname {sech}^{3}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (186) = 372\).
Time = 0.28 (sec) , antiderivative size = 556, normalized size of antiderivative = 2.84 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {1}{192} \, b^{3} {\left (\frac {105 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {105 \, e^{\left (-d x - c\right )} + 805 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2681 \, e^{\left (-5 \, d x - 5 \, c\right )} + 5053 \, e^{\left (-7 \, d x - 7 \, c\right )} - 5053 \, e^{\left (-9 \, d x - 9 \, c\right )} - 2681 \, e^{\left (-11 \, d x - 11 \, c\right )} - 805 \, e^{\left (-13 \, d x - 13 \, c\right )} - 105 \, e^{\left (-15 \, d x - 15 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}}\right )} - \frac {1}{8} \, a b^{2} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {15 \, e^{\left (-d x - c\right )} + 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} - 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} - 15 \, e^{\left (-11 \, d x - 11 \, 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 )} - \frac {3}{4} \, a^{2} b {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - a^{3} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \]
-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)))
Leaf count of result is larger than twice the leaf count of optimal. 485 vs. \(2 (186) = 372\).
Time = 0.31 (sec) , antiderivative size = 485, normalized size of antiderivative = 2.47 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (64 \, a^{3} + 144 \, a^{2} b + 120 \, a b^{2} + 35 \, b^{3}\right )} + \frac {4 \, {\left (192 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 432 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 360 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 105 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 2304 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 6336 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 5280 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 1540 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 9216 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 29952 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 28032 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 8176 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 12288 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 46080 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 50688 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 17856 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{4}}}{768 \, d} \]
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
Time = 2.24 (sec) , antiderivative size = 931, normalized size of antiderivative = 4.75 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (64\,a^3\,\sqrt {d^2}+35\,b^3\,\sqrt {d^2}+120\,a\,b^2\,\sqrt {d^2}+144\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {4096\,a^6+18432\,a^5\,b+36096\,a^4\,b^2+39040\,a^3\,b^3+24480\,a^2\,b^4+8400\,a\,b^5+1225\,b^6}}\right )\,\sqrt {4096\,a^6+18432\,a^5\,b+36096\,a^4\,b^2+39040\,a^3\,b^3+24480\,a^2\,b^4+8400\,a\,b^5+1225\,b^6}}{64\,\sqrt {d^2}}-\frac {\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{2\,d}+\frac {2\,{\mathrm {e}}^{7\,c+7\,d\,x}\,\left (5\,a^3+18\,a^2\,b+24\,a\,b^2+16\,b^3\right )}{d}+\frac {a^3\,{\mathrm {e}}^{13\,c+13\,d\,x}}{2\,d}+\frac {3\,a\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{2\,d}+\frac {3\,a\,{\mathrm {e}}^{9\,c+9\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{2\,d}+\frac {3\,a^2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (a+2\,b\right )}{d}+\frac {3\,a^2\,{\mathrm {e}}^{11\,c+11\,d\,x}\,\left (a+2\,b\right )}{d}}{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}+\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (48\,a\,b^2-37\,b^3\right )}{3\,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 )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (24\,a^2\,b-120\,a\,b^2+b^3\right )}{4\,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 {16\,b^3\,{\mathrm {e}}^{c+d\,x}}{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 {{\mathrm {e}}^{c+d\,x}\,\left (64\,a^3+144\,a^2\,b+120\,a\,b^2+35\,b^3\right )}{64\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {4\,{\mathrm {e}}^{c+d\,x}\,\left (6\,a\,b^2-29\,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 {{\mathrm {e}}^{c+d\,x}\,\left (-144\,a^3+144\,a^2\,b+120\,a\,b^2+35\,b^3\right )}{96\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (-288\,a^2\,b+24\,a\,b^2+7\,b^3\right )}{24\,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 )} \]
(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...