Integrand size = 23, antiderivative size = 74 \[ \int \text {sech}^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {b (a+b)^2 \tanh ^3(c+d x)}{d}+\frac {3 b^2 (a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^7(c+d x)}{7 d} \]
(a+b)^3*tanh(d*x+c)/d-b*(a+b)^2*tanh(d*x+c)^3/d+3/5*b^2*(a+b)*tanh(d*x+c)^ 5/d-1/7*b^3*tanh(d*x+c)^7/d
Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(74)=148\).
Time = 3.62 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.19 \[ \int \text {sech}^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {a^3 \tanh (c+d x)}{d}+\frac {3 a^2 b \tanh (c+d x)}{d}+\frac {3 a b^2 \tanh (c+d x)}{d}+\frac {b^3 \tanh (c+d x)}{d}-\frac {a^2 b \tanh ^3(c+d x)}{d}-\frac {2 a b^2 \tanh ^3(c+d x)}{d}-\frac {b^3 \tanh ^3(c+d x)}{d}+\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {3 b^3 \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^7(c+d x)}{7 d} \]
(a^3*Tanh[c + d*x])/d + (3*a^2*b*Tanh[c + d*x])/d + (3*a*b^2*Tanh[c + d*x] )/d + (b^3*Tanh[c + d*x])/d - (a^2*b*Tanh[c + d*x]^3)/d - (2*a*b^2*Tanh[c + d*x]^3)/d - (b^3*Tanh[c + d*x]^3)/d + (3*a*b^2*Tanh[c + d*x]^5)/(5*d) + (3*b^3*Tanh[c + d*x]^5)/(5*d) - (b^3*Tanh[c + d*x]^7)/(7*d)
Time = 0.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4634, 210, 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.
| \(\displaystyle \int \text {sech}^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (i c+i d x)^2 \left (a+b \sec (i c+i d x)^2\right )^3dx\) |
| \(\Big \downarrow \) 4634 |
| \(\displaystyle \frac {\int \left (-b \tanh ^2(c+d x)+a+b\right )^3d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle \frac {\int \left (-b^3 \tanh ^6(c+d x)+3 b^2 (a+b) \tanh ^4(c+d x)-3 b (a+b)^2 \tanh ^2(c+d x)+a^3 \left (\frac {b \left (3 a^2+3 b a+b^2\right )}{a^3}+1\right )\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3}{5} b^2 (a+b) \tanh ^5(c+d x)-b (a+b)^2 \tanh ^3(c+d x)+(a+b)^3 \tanh (c+d x)-\frac {1}{7} b^3 \tanh ^7(c+d x)}{d}\) |
((a + b)^3*Tanh[c + d*x] - b*(a + b)^2*Tanh[c + d*x]^3 + (3*b^2*(a + b)*Ta nh[c + d*x]^5)/5 - (b^3*Tanh[c + d*x]^7)/7)/d
3.1.70.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_) )^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ [m/2] && IntegerQ[n/2]
Time = 1.78 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.57
| method | result | size |
| derivativedivides | \(\frac {a^{3} \tanh \left (d x +c \right )+3 a^{2} b \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+3 a \,b^{2} \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )+b^{3} \left (\frac {16}{35}+\frac {\operatorname {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \operatorname {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )}{d}\) | \(116\) |
| default | \(\frac {a^{3} \tanh \left (d x +c \right )+3 a^{2} b \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+3 a \,b^{2} \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )+b^{3} \left (\frac {16}{35}+\frac {\operatorname {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \operatorname {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )}{d}\) | \(116\) |
| parts | \(\frac {a^{3} \tanh \left (d x +c \right )}{d}+\frac {b^{3} \left (\frac {16}{35}+\frac {\operatorname {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \operatorname {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )}{d}+\frac {3 a \,b^{2} \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{d}+\frac {3 a^{2} b \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )}{d}\) | \(124\) |
| parallelrisch | \(\frac {\left (315 a^{3}+1050 a^{2} b +1176 a \,b^{2}+336 b^{3}\right ) \sinh \left (3 d x +3 c \right )+\left (175 a^{3}+490 a^{2} b +392 a \,b^{2}+112 b^{3}\right ) \sinh \left (5 d x +5 c \right )+\left (35 a^{3}+70 a^{2} b +56 a \,b^{2}+16 b^{3}\right ) \sinh \left (7 d x +7 c \right )+175 \left (a +2 b \right ) \left (a^{2}+\frac {8}{5} a b +\frac {8}{5} b^{2}\right ) \sinh \left (d x +c \right )}{35 d \left (\cosh \left (7 d x +7 c \right )+7 \cosh \left (5 d x +5 c \right )+21 \cosh \left (3 d x +3 c \right )+35 \cosh \left (d x +c \right )\right )}\) | \(174\) |
| risch | \(-\frac {2 \left (35 a^{3} {\mathrm e}^{12 d x +12 c}+210 a^{3} {\mathrm e}^{10 d x +10 c}+210 a^{2} b \,{\mathrm e}^{10 d x +10 c}+525 a^{3} {\mathrm e}^{8 d x +8 c}+910 a^{2} b \,{\mathrm e}^{8 d x +8 c}+560 a \,b^{2} {\mathrm e}^{8 d x +8 c}+700 a^{3} {\mathrm e}^{6 d x +6 c}+1540 a^{2} b \,{\mathrm e}^{6 d x +6 c}+1400 a \,b^{2} {\mathrm e}^{6 d x +6 c}+560 \,{\mathrm e}^{6 d x +6 c} b^{3}+525 a^{3} {\mathrm e}^{4 d x +4 c}+1260 a^{2} b \,{\mathrm e}^{4 d x +4 c}+1176 a \,b^{2} {\mathrm e}^{4 d x +4 c}+336 \,{\mathrm e}^{4 d x +4 c} b^{3}+210 a^{3} {\mathrm e}^{2 d x +2 c}+490 a^{2} b \,{\mathrm e}^{2 d x +2 c}+392 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+112 \,{\mathrm e}^{2 d x +2 c} b^{3}+35 a^{3}+70 a^{2} b +56 a \,b^{2}+16 b^{3}\right )}{35 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{7}}\) | \(303\) |
1/d*(a^3*tanh(d*x+c)+3*a^2*b*(2/3+1/3*sech(d*x+c)^2)*tanh(d*x+c)+3*a*b^2*( 8/15+1/5*sech(d*x+c)^4+4/15*sech(d*x+c)^2)*tanh(d*x+c)+b^3*(16/35+1/7*sech (d*x+c)^6+6/35*sech(d*x+c)^4+8/35*sech(d*x+c)^2)*tanh(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 816 vs. \(2 (70) = 140\).
Time = 0.27 (sec) , antiderivative size = 816, normalized size of antiderivative = 11.03 \[ \int \text {sech}^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {4 \, {\left ({\left (35 \, a^{3} + 35 \, a^{2} b + 28 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} - 6 \, {\left (35 \, a^{2} b + 28 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + {\left (35 \, a^{3} + 35 \, a^{2} b + 28 \, a b^{2} + 8 \, b^{3}\right )} \sinh \left (d x + c\right )^{6} + 14 \, {\left (15 \, a^{3} + 25 \, a^{2} b + 14 \, a b^{2} + 4 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + {\left (210 \, a^{3} + 350 \, a^{2} b + 196 \, a b^{2} + 56 \, b^{3} + 15 \, {\left (35 \, a^{3} + 35 \, a^{2} b + 28 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} - 4 \, {\left (5 \, {\left (35 \, a^{2} b + 28 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 28 \, {\left (5 \, a^{2} b + 7 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 350 \, a^{3} + 770 \, a^{2} b + 700 \, a b^{2} + 280 \, b^{3} + 7 \, {\left (75 \, a^{3} + 155 \, a^{2} b + 124 \, a b^{2} + 24 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + {\left (15 \, {\left (35 \, a^{3} + 35 \, a^{2} b + 28 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 525 \, a^{3} + 1085 \, a^{2} b + 868 \, a b^{2} + 168 \, b^{3} + 84 \, {\left (15 \, a^{3} + 25 \, a^{2} b + 14 \, a b^{2} + 4 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 2 \, {\left (3 \, {\left (35 \, a^{2} b + 28 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 56 \, {\left (5 \, a^{2} b + 7 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 7 \, {\left (25 \, a^{2} b + 44 \, a b^{2} + 24 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}{35 \, {\left (d \cosh \left (d x + c\right )^{8} + 8 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + d \sinh \left (d x + c\right )^{8} + 8 \, d \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )^{6} + 4 \, {\left (14 \, d \cosh \left (d x + c\right )^{3} + 9 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 28 \, d \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, d \cosh \left (d x + c\right )^{4} + 60 \, d \cosh \left (d x + c\right )^{2} + 14 \, d\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{5} + 15 \, d \cosh \left (d x + c\right )^{3} + 7 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 56 \, d \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{6} + 30 \, d \cosh \left (d x + c\right )^{4} + 42 \, d \cosh \left (d x + c\right )^{2} + 14 \, d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (2 \, d \cosh \left (d x + c\right )^{7} + 9 \, d \cosh \left (d x + c\right )^{5} + 14 \, d \cosh \left (d x + c\right )^{3} + 7 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 35 \, d\right )}} \]
-4/35*((35*a^3 + 35*a^2*b + 28*a*b^2 + 8*b^3)*cosh(d*x + c)^6 - 6*(35*a^2* b + 28*a*b^2 + 8*b^3)*cosh(d*x + c)*sinh(d*x + c)^5 + (35*a^3 + 35*a^2*b + 28*a*b^2 + 8*b^3)*sinh(d*x + c)^6 + 14*(15*a^3 + 25*a^2*b + 14*a*b^2 + 4* b^3)*cosh(d*x + c)^4 + (210*a^3 + 350*a^2*b + 196*a*b^2 + 56*b^3 + 15*(35* a^3 + 35*a^2*b + 28*a*b^2 + 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 4*(5 *(35*a^2*b + 28*a*b^2 + 8*b^3)*cosh(d*x + c)^3 + 28*(5*a^2*b + 7*a*b^2 + 2 *b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 350*a^3 + 770*a^2*b + 700*a*b^2 + 2 80*b^3 + 7*(75*a^3 + 155*a^2*b + 124*a*b^2 + 24*b^3)*cosh(d*x + c)^2 + (15 *(35*a^3 + 35*a^2*b + 28*a*b^2 + 8*b^3)*cosh(d*x + c)^4 + 525*a^3 + 1085*a ^2*b + 868*a*b^2 + 168*b^3 + 84*(15*a^3 + 25*a^2*b + 14*a*b^2 + 4*b^3)*cos h(d*x + c)^2)*sinh(d*x + c)^2 - 2*(3*(35*a^2*b + 28*a*b^2 + 8*b^3)*cosh(d* x + c)^5 + 56*(5*a^2*b + 7*a*b^2 + 2*b^3)*cosh(d*x + c)^3 + 7*(25*a^2*b + 44*a*b^2 + 24*b^3)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^8 + 8*d* cosh(d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 + 8*d*cosh(d*x + c)^6 + 4*(7*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^6 + 4*(14*d*cosh(d*x + c)^3 + 9*d*cosh(d*x + c))*sinh(d*x + c)^5 + 28*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d *x + c)^4 + 60*d*cosh(d*x + c)^2 + 14*d)*sinh(d*x + c)^4 + 8*(7*d*cosh(d*x + c)^5 + 15*d*cosh(d*x + c)^3 + 7*d*cosh(d*x + c))*sinh(d*x + c)^3 + 56*d *cosh(d*x + c)^2 + 4*(7*d*cosh(d*x + c)^6 + 30*d*cosh(d*x + c)^4 + 42*d*co sh(d*x + c)^2 + 14*d)*sinh(d*x + c)^2 + 4*(2*d*cosh(d*x + c)^7 + 9*d*co...
\[ \int \text {sech}^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3} \operatorname {sech}^{2}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (70) = 140\).
Time = 0.21 (sec) , antiderivative size = 695, normalized size of antiderivative = 9.39 \[ \int \text {sech}^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {32}{35} \, b^{3} {\left (\frac {7 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}} + \frac {21 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}} + \frac {35 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}} + \frac {1}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}}\right )} + \frac {16}{5} \, a b^{2} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + 4 \, a^{2} b {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + \frac {2 \, a^{3}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \]
32/35*b^3*(7*e^(-2*d*x - 2*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e ^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 21*e^(-4*d*x - 4*c)/(d*(7*e ^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c ) + 1)) + 35*e^(-6*d*x - 6*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e ^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 1/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-1 0*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1))) + 16/5*a* b^2*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10* e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) + 10*e^(- 4*d*x - 4*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) + 1/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1))) + 4*a^2*b*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1)) + 1/(d*(3*e^(-2*d*x - 2 *c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1))) + 2*a^3/(d*(e^(-2*d*x - 2*c) + 1))
Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (70) = 140\).
Time = 0.31 (sec) , antiderivative size = 302, normalized size of antiderivative = 4.08 \[ \int \text {sech}^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {2 \, {\left (35 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 210 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 210 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 525 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 910 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 560 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 700 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1540 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 1400 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 560 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 525 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 1260 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 1176 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 336 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 210 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 490 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 392 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 112 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 35 \, a^{3} + 70 \, a^{2} b + 56 \, a b^{2} + 16 \, b^{3}\right )}}{35 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}} \]
-2/35*(35*a^3*e^(12*d*x + 12*c) + 210*a^3*e^(10*d*x + 10*c) + 210*a^2*b*e^ (10*d*x + 10*c) + 525*a^3*e^(8*d*x + 8*c) + 910*a^2*b*e^(8*d*x + 8*c) + 56 0*a*b^2*e^(8*d*x + 8*c) + 700*a^3*e^(6*d*x + 6*c) + 1540*a^2*b*e^(6*d*x + 6*c) + 1400*a*b^2*e^(6*d*x + 6*c) + 560*b^3*e^(6*d*x + 6*c) + 525*a^3*e^(4 *d*x + 4*c) + 1260*a^2*b*e^(4*d*x + 4*c) + 1176*a*b^2*e^(4*d*x + 4*c) + 33 6*b^3*e^(4*d*x + 4*c) + 210*a^3*e^(2*d*x + 2*c) + 490*a^2*b*e^(2*d*x + 2*c ) + 392*a*b^2*e^(2*d*x + 2*c) + 112*b^3*e^(2*d*x + 2*c) + 35*a^3 + 70*a^2* b + 56*a*b^2 + 16*b^3)/(d*(e^(2*d*x + 2*c) + 1)^7)
Time = 2.12 (sec) , antiderivative size = 978, normalized size of antiderivative = 13.22 \[ \int \text {sech}^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {\frac {2\,\left (5\,a^3+18\,a^2\,b+24\,a\,b^2+16\,b^3\right )}{35\,d}+\frac {2\,a^3\,{\mathrm {e}}^{6\,c+6\,d\,x}}{7\,d}+\frac {6\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{35\,d}+\frac {6\,a^2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a+2\,b\right )}{7\,d}}{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}-\frac {\frac {2\,a^2\,\left (a+2\,b\right )}{7\,d}+\frac {2\,a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {2\,a\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{35\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (5\,a^3+18\,a^2\,b+24\,a\,b^2+16\,b^3\right )}{35\,d}+\frac {2\,a^3\,{\mathrm {e}}^{8\,c+8\,d\,x}}{7\,d}+\frac {12\,a\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{35\,d}+\frac {8\,a^2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a+2\,b\right )}{7\,d}}{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}-\frac {\frac {2\,a^3}{7\,d}+\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (5\,a^3+18\,a^2\,b+24\,a\,b^2+16\,b^3\right )}{7\,d}+\frac {2\,a^3\,{\mathrm {e}}^{12\,c+12\,d\,x}}{7\,d}+\frac {6\,a\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{7\,d}+\frac {6\,a\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{7\,d}+\frac {12\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+2\,b\right )}{7\,d}+\frac {12\,a^2\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (a+2\,b\right )}{7\,d}}{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}-\frac {\frac {2\,a\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{35\,d}+\frac {2\,a^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{7\,d}+\frac {4\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+2\,b\right )}{7\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {\frac {2\,a^2\,\left (a+2\,b\right )}{7\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (5\,a^3+18\,a^2\,b+24\,a\,b^2+16\,b^3\right )}{7\,d}+\frac {2\,a^3\,{\mathrm {e}}^{10\,c+10\,d\,x}}{7\,d}+\frac {2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{7\,d}+\frac {4\,a\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{7\,d}+\frac {10\,a^2\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a+2\,b\right )}{7\,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\,a^3}{7\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
- ((2*(24*a*b^2 + 18*a^2*b + 5*a^3 + 16*b^3))/(35*d) + (2*a^3*exp(6*c + 6* d*x))/(7*d) + (6*a*exp(2*c + 2*d*x)*(16*a*b + 5*a^2 + 16*b^2))/(35*d) + (6 *a^2*exp(4*c + 4*d*x)*(a + 2*b))/(7*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*a^2*(a + 2*b))/( 7*d) + (2*a^3*exp(2*c + 2*d*x))/(7*d))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d *x) + 1) - ((2*a*(16*a*b + 5*a^2 + 16*b^2))/(35*d) + (8*exp(2*c + 2*d*x)*( 24*a*b^2 + 18*a^2*b + 5*a^3 + 16*b^3))/(35*d) + (2*a^3*exp(8*c + 8*d*x))/( 7*d) + (12*a*exp(4*c + 4*d*x)*(16*a*b + 5*a^2 + 16*b^2))/(35*d) + (8*a^2*e xp(6*c + 6*d*x)*(a + 2*b))/(7*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) - ( (2*a^3)/(7*d) + (8*exp(6*c + 6*d*x)*(24*a*b^2 + 18*a^2*b + 5*a^3 + 16*b^3) )/(7*d) + (2*a^3*exp(12*c + 12*d*x))/(7*d) + (6*a*exp(4*c + 4*d*x)*(16*a*b + 5*a^2 + 16*b^2))/(7*d) + (6*a*exp(8*c + 8*d*x)*(16*a*b + 5*a^2 + 16*b^2 ))/(7*d) + (12*a^2*exp(2*c + 2*d*x)*(a + 2*b))/(7*d) + (12*a^2*exp(10*c + 10*d*x)*(a + 2*b))/(7*d))/(7*exp(2*c + 2*d*x) + 21*exp(4*c + 4*d*x) + 35*e xp(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) - ((2*a*(16*a*b + 5*a^2 + 16*b^2))/(3 5*d) + (2*a^3*exp(4*c + 4*d*x))/(7*d) + (4*a^2*exp(2*c + 2*d*x)*(a + 2*b)) /(7*d))/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1) - ((2*a^2*(a + 2*b))/(7*d) + (4*exp(4*c + 4*d*x)*(24*a*b^2 + 18*a^2*b + ...