Integrand size = 23, antiderivative size = 128 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {\left (8 a^2+12 a b+5 b^2\right ) \arctan (\sinh (c+d x))}{16 d}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{16 d}+\frac {b (8 a+5 b) \text {sech}^3(c+d x) \tanh (c+d x)}{24 d}+\frac {b \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{6 d} \]
1/16*(8*a^2+12*a*b+5*b^2)*arctan(sinh(d*x+c))/d+1/16*(8*a^2+12*a*b+5*b^2)* sech(d*x+c)*tanh(d*x+c)/d+1/24*b*(8*a+5*b)*sech(d*x+c)^3*tanh(d*x+c)/d+1/6 *b*sech(d*x+c)^5*(a+b+a*sinh(d*x+c)^2)*tanh(d*x+c)/d
Time = 0.05 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.46 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {a^2 \arctan (\sinh (c+d x))}{2 d}+\frac {3 a b \arctan (\sinh (c+d x))}{4 d}+\frac {5 b^2 \arctan (\sinh (c+d x))}{16 d}+\frac {a^2 \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {3 a b \text {sech}(c+d x) \tanh (c+d x)}{4 d}+\frac {5 b^2 \text {sech}(c+d x) \tanh (c+d x)}{16 d}+\frac {a b \text {sech}^3(c+d x) \tanh (c+d x)}{2 d}+\frac {5 b^2 \text {sech}^3(c+d x) \tanh (c+d x)}{24 d}+\frac {b^2 \text {sech}^5(c+d x) \tanh (c+d x)}{6 d} \]
(a^2*ArcTan[Sinh[c + d*x]])/(2*d) + (3*a*b*ArcTan[Sinh[c + d*x]])/(4*d) + (5*b^2*ArcTan[Sinh[c + d*x]])/(16*d) + (a^2*Sech[c + d*x]*Tanh[c + d*x])/( 2*d) + (3*a*b*Sech[c + d*x]*Tanh[c + d*x])/(4*d) + (5*b^2*Sech[c + d*x]*Ta nh[c + d*x])/(16*d) + (a*b*Sech[c + d*x]^3*Tanh[c + d*x])/(2*d) + (5*b^2*S ech[c + d*x]^3*Tanh[c + d*x])/(24*d) + (b^2*Sech[c + d*x]^5*Tanh[c + d*x]) /(6*d)
Time = 0.31 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 4635, 315, 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.
| \(\displaystyle \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (i c+i d x)^3 \left (a+b \sec (i c+i d x)^2\right )^2dx\) |
| \(\Big \downarrow \) 4635 |
| \(\displaystyle \frac {\int \frac {\left (a \sinh ^2(c+d x)+a+b\right )^2}{\left (\sinh ^2(c+d x)+1\right )^4}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\frac {1}{6} \int \frac {3 a (2 a+b) \sinh ^2(c+d x)+(a+b) (6 a+5 b)}{\left (\sinh ^2(c+d x)+1\right )^3}d\sinh (c+d x)+\frac {b \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \left (8 a^2+12 a b+5 b^2\right ) \int \frac {1}{\left (\sinh ^2(c+d x)+1\right )^2}d\sinh (c+d x)+\frac {b (8 a+5 b) \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )+\frac {b \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \left (8 a^2+12 a b+5 b^2\right ) \left (\frac {1}{2} \int \frac {1}{\sinh ^2(c+d x)+1}d\sinh (c+d x)+\frac {\sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )+\frac {b (8 a+5 b) \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )+\frac {b \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \left (8 a^2+12 a b+5 b^2\right ) \left (\frac {1}{2} \arctan (\sinh (c+d x))+\frac {\sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )+\frac {b (8 a+5 b) \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )+\frac {b \sinh (c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\) |
((b*Sinh[c + d*x]*(a + b + a*Sinh[c + d*x]^2))/(6*(1 + Sinh[c + d*x]^2)^3) + ((b*(8*a + 5*b)*Sinh[c + d*x])/(4*(1 + Sinh[c + d*x]^2)^2) + (3*(8*a^2 + 12*a*b + 5*b^2)*(ArcTan[Sinh[c + d*x]]/2 + Sinh[c + d*x]/(2*(1 + Sinh[c + d*x]^2))))/4)/6)/d
3.1.63.3.1 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[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 = 1.95 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+2 a b \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+b^{2} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )}{d}\) | \(122\) |
| default | \(\frac {a^{2} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+2 a b \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+b^{2} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )}{d}\) | \(122\) |
| parts | \(\frac {a^{2} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}+\frac {b^{2} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )}{d}+\frac {2 a b \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )}{d}\) | \(127\) |
| parallelrisch | \(\frac {-360 i \left (\frac {2}{3}+\frac {\cosh \left (6 d x +6 c \right )}{15}+\frac {2 \cosh \left (4 d x +4 c \right )}{5}+\cosh \left (2 d x +2 c \right )\right ) \left (a^{2}+\frac {5}{8} b^{2}+\frac {3}{2} a b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )+360 i \left (\frac {2}{3}+\frac {\cosh \left (6 d x +6 c \right )}{15}+\frac {2 \cosh \left (4 d x +4 c \right )}{5}+\cosh \left (2 d x +2 c \right )\right ) \left (a^{2}+\frac {5}{8} b^{2}+\frac {3}{2} a b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )+\left (144 a^{2}+408 a b +170 b^{2}\right ) \sinh \left (3 d x +3 c \right )+\left (48 a^{2}+72 a b +30 b^{2}\right ) \sinh \left (5 d x +5 c \right )+96 \sinh \left (d x +c \right ) \left (a^{2}+\frac {7}{2} a b +\frac {33}{8} b^{2}\right )}{48 d \left (10+\cosh \left (6 d x +6 c \right )+6 \cosh \left (4 d x +4 c \right )+15 \cosh \left (2 d x +2 c \right )\right )}\) | \(237\) |
| risch | \(\frac {{\mathrm e}^{d x +c} \left (24 a^{2} {\mathrm e}^{10 d x +10 c}+36 a b \,{\mathrm e}^{10 d x +10 c}+15 b^{2} {\mathrm e}^{10 d x +10 c}+72 a^{2} {\mathrm e}^{8 d x +8 c}+204 a b \,{\mathrm e}^{8 d x +8 c}+85 b^{2} {\mathrm e}^{8 d x +8 c}+48 a^{2} {\mathrm e}^{6 d x +6 c}+168 a b \,{\mathrm e}^{6 d x +6 c}+198 b^{2} {\mathrm e}^{6 d x +6 c}-48 a^{2} {\mathrm e}^{4 d x +4 c}-168 a b \,{\mathrm e}^{4 d x +4 c}-198 \,{\mathrm e}^{4 d x +4 c} b^{2}-72 a^{2} {\mathrm e}^{2 d x +2 c}-204 a b \,{\mathrm e}^{2 d x +2 c}-85 \,{\mathrm e}^{2 d x +2 c} b^{2}-24 a^{2}-36 a b -15 b^{2}\right )}{24 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{6}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{2 d}+\frac {3 i b a \ln \left ({\mathrm e}^{d x +c}+i\right )}{4 d}+\frac {5 i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right )}{16 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 d}-\frac {3 i b a \ln \left ({\mathrm e}^{d x +c}-i\right )}{4 d}-\frac {5 i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{16 d}\) | \(358\) |
1/d*(a^2*(1/2*sech(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c)))+2*a*b*((1/4*sech (d*x+c)^3+3/8*sech(d*x+c))*tanh(d*x+c)+3/4*arctan(exp(d*x+c)))+b^2*((1/6*s ech(d*x+c)^5+5/24*sech(d*x+c)^3+5/16*sech(d*x+c))*tanh(d*x+c)+5/8*arctan(e xp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 2946 vs. \(2 (120) = 240\).
Time = 0.28 (sec) , antiderivative size = 2946, normalized size of antiderivative = 23.02 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\text {Too large to display} \]
1/24*(3*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^11 + 33*(8*a^2 + 12*a*b + 5 *b^2)*cosh(d*x + c)*sinh(d*x + c)^10 + 3*(8*a^2 + 12*a*b + 5*b^2)*sinh(d*x + c)^11 + (72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^9 + (165*(8*a^2 + 12* a*b + 5*b^2)*cosh(d*x + c)^2 + 72*a^2 + 204*a*b + 85*b^2)*sinh(d*x + c)^9 + 9*(55*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^3 + (72*a^2 + 204*a*b + 85* b^2)*cosh(d*x + c))*sinh(d*x + c)^8 + 6*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^7 + 6*(165*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^4 + 6*(72*a^2 + 20 4*a*b + 85*b^2)*cosh(d*x + c)^2 + 8*a^2 + 28*a*b + 33*b^2)*sinh(d*x + c)^7 + 42*(33*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^5 + 2*(72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^3 + (8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c))*sinh(d *x + c)^6 - 6*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^5 + 6*(231*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^6 + 21*(72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^4 + 21*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^2 - 8*a^2 - 28*a*b - 3 3*b^2)*sinh(d*x + c)^5 + 6*(165*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^7 + 21*(72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^5 + 35*(8*a^2 + 28*a*b + 33* b^2)*cosh(d*x + c)^3 - 5*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - (72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^3 + (495*(8*a^2 + 12*a *b + 5*b^2)*cosh(d*x + c)^8 + 84*(72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c) ^6 + 210*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^4 - 60*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^2 - 72*a^2 - 204*a*b - 85*b^2)*sinh(d*x + c)^3 + ...
\[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \operatorname {sech}^{3}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (120) = 240\).
Time = 0.30 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.72 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {1}{24} \, b^{2} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {15 \, e^{\left (-d x - c\right )} + 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} - 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} - 15 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} - \frac {1}{2} \, a b {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - a^{2} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \]
-1/24*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))) - 1/2*a*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^2*(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. 293 vs. \(2 (120) = 240\).
Time = 0.31 (sec) , antiderivative size = 293, normalized size of antiderivative = 2.29 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} + \frac {4 \, {\left (24 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 36 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 15 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 192 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 384 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 160 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 384 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 960 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 528 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{3}}}{96 \, d} \]
1/96*(3*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(8*a^2 + 1 2*a*b + 5*b^2) + 4*(24*a^2*(e^(d*x + c) - e^(-d*x - c))^5 + 36*a*b*(e^(d*x + c) - e^(-d*x - c))^5 + 15*b^2*(e^(d*x + c) - e^(-d*x - c))^5 + 192*a^2* (e^(d*x + c) - e^(-d*x - c))^3 + 384*a*b*(e^(d*x + c) - e^(-d*x - c))^3 + 160*b^2*(e^(d*x + c) - e^(-d*x - c))^3 + 384*a^2*(e^(d*x + c) - e^(-d*x - c)) + 960*a*b*(e^(d*x + c) - e^(-d*x - c)) + 528*b^2*(e^(d*x + c) - e^(-d* x - c)))/((e^(d*x + c) - e^(-d*x - c))^2 + 4)^3)/d
Time = 2.11 (sec) , antiderivative size = 569, normalized size of antiderivative = 4.45 \[ \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (8\,a^2\,\sqrt {d^2}+5\,b^2\,\sqrt {d^2}+12\,a\,b\,\sqrt {d^2}\right )}{d\,\sqrt {64\,a^4+192\,a^3\,b+224\,a^2\,b^2+120\,a\,b^3+25\,b^4}}\right )\,\sqrt {64\,a^4+192\,a^3\,b+224\,a^2\,b^2+120\,a\,b^3+25\,b^4}}{8\,\sqrt {d^2}}-\frac {\frac {2\,a^2\,{\mathrm {e}}^{c+d\,x}}{3\,d}+\frac {2\,a^2\,{\mathrm {e}}^{9\,c+9\,d\,x}}{3\,d}+\frac {4\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (3\,a^2+8\,a\,b+8\,b^2\right )}{3\,d}+\frac {8\,a\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (a+2\,b\right )}{3\,d}+\frac {8\,a\,{\mathrm {e}}^{7\,c+7\,d\,x}\,\left (a+2\,b\right )}{3\,d}}{6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1}+\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (4\,a\,b-11\,b^2\right )}{3\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {16\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (8\,a^2+12\,a\,b+5\,b^2\right )}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (-16\,a^2+12\,a\,b+5\,b^2\right )}{12\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (20\,a\,b-b^2\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )} \]
(atan((exp(d*x)*exp(c)*(8*a^2*(d^2)^(1/2) + 5*b^2*(d^2)^(1/2) + 12*a*b*(d^ 2)^(1/2)))/(d*(120*a*b^3 + 192*a^3*b + 64*a^4 + 25*b^4 + 224*a^2*b^2)^(1/2 )))*(120*a*b^3 + 192*a^3*b + 64*a^4 + 25*b^4 + 224*a^2*b^2)^(1/2))/(8*(d^2 )^(1/2)) - ((2*a^2*exp(c + d*x))/(3*d) + (2*a^2*exp(9*c + 9*d*x))/(3*d) + (4*exp(5*c + 5*d*x)*(8*a*b + 3*a^2 + 8*b^2))/(3*d) + (8*a*exp(3*c + 3*d*x) *(a + 2*b))/(3*d) + (8*a*exp(7*c + 7*d*x)*(a + 2*b))/(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*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1) + (2*exp(c + d*x)*(4*a*b - 11*b^2))/(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)) + (16*b^2*exp(c + d*x))/(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(1 0*c + 10*d*x) + 1)) + (exp(c + d*x)*(12*a*b + 8*a^2 + 5*b^2))/(8*d*(exp(2* c + 2*d*x) + 1)) + (exp(c + d*x)*(12*a*b - 16*a^2 + 5*b^2))/(12*d*(2*exp(2 *c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (exp(c + d*x)*(20*a*b - b^2))/(3*d* (3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1))