Integrand size = 14, antiderivative size = 163 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^5 \, dx=a^5 x+\frac {b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac {b^2 \left (10 a^3+20 a^2 b+15 a b^2+4 b^3\right ) \tanh ^3(c+d x)}{3 d}+\frac {b^3 \left (10 a^2+15 a b+6 b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^4 (5 a+4 b) \tanh ^7(c+d x)}{7 d}+\frac {b^5 \tanh ^9(c+d x)}{9 d} \]
a^5*x+b*(5*a^4+10*a^3*b+10*a^2*b^2+5*a*b^3+b^4)*tanh(d*x+c)/d-1/3*b^2*(10* a^3+20*a^2*b+15*a*b^2+4*b^3)*tanh(d*x+c)^3/d+1/5*b^3*(10*a^2+15*a*b+6*b^2) *tanh(d*x+c)^5/d-1/7*b^4*(5*a+4*b)*tanh(d*x+c)^7/d+1/9*b^5*tanh(d*x+c)^9/d
Time = 5.11 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.65 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^5 \, dx=a^5 x+\frac {5 a^4 b \tanh (c+d x)}{d}+\frac {10 a^3 b^2 \tanh (c+d x)}{d}+\frac {10 a^2 b^3 \tanh (c+d x)}{d}+\frac {5 a b^4 \tanh (c+d x)}{d}+\frac {b^5 \tanh (c+d x)}{d}-\frac {10 a^3 b^2 \tanh ^3(c+d x)}{3 d}-\frac {20 a^2 b^3 \tanh ^3(c+d x)}{3 d}-\frac {5 a b^4 \tanh ^3(c+d x)}{d}-\frac {4 b^5 \tanh ^3(c+d x)}{3 d}+\frac {2 a^2 b^3 \tanh ^5(c+d x)}{d}+\frac {3 a b^4 \tanh ^5(c+d x)}{d}+\frac {6 b^5 \tanh ^5(c+d x)}{5 d}-\frac {5 a b^4 \tanh ^7(c+d x)}{7 d}-\frac {4 b^5 \tanh ^7(c+d x)}{7 d}+\frac {b^5 \tanh ^9(c+d x)}{9 d} \]
a^5*x + (5*a^4*b*Tanh[c + d*x])/d + (10*a^3*b^2*Tanh[c + d*x])/d + (10*a^2 *b^3*Tanh[c + d*x])/d + (5*a*b^4*Tanh[c + d*x])/d + (b^5*Tanh[c + d*x])/d - (10*a^3*b^2*Tanh[c + d*x]^3)/(3*d) - (20*a^2*b^3*Tanh[c + d*x]^3)/(3*d) - (5*a*b^4*Tanh[c + d*x]^3)/d - (4*b^5*Tanh[c + d*x]^3)/(3*d) + (2*a^2*b^3 *Tanh[c + d*x]^5)/d + (3*a*b^4*Tanh[c + d*x]^5)/d + (6*b^5*Tanh[c + d*x]^5 )/(5*d) - (5*a*b^4*Tanh[c + d*x]^7)/(7*d) - (4*b^5*Tanh[c + d*x]^7)/(7*d) + (b^5*Tanh[c + d*x]^9)/(9*d)
Time = 0.33 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4616, 300, 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.
| \(\displaystyle \int \left (a+b \text {sech}^2(c+d x)\right )^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \sec (i c+i d x)^2\right )^5dx\) |
| \(\Big \downarrow \) 4616 |
| \(\displaystyle \frac {\int \frac {\left (-b \tanh ^2(c+d x)+a+b\right )^5}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left (b^5 \tanh ^8(c+d x)-b^4 (5 a+4 b) \tanh ^6(c+d x)+b^3 \left (10 a^2+15 b a+6 b^2\right ) \tanh ^4(c+d x)-b^2 \left (10 a^3+20 b a^2+15 b^2 a+4 b^3\right ) \tanh ^2(c+d x)+b \left (5 a^4+10 b a^3+10 b^2 a^2+5 b^3 a+b^4\right )+\frac {a^5}{1-\tanh ^2(c+d x)}\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^5 \text {arctanh}(\tanh (c+d x))+\frac {1}{5} b^3 \left (10 a^2+15 a b+6 b^2\right ) \tanh ^5(c+d x)-\frac {1}{3} b^2 \left (10 a^3+20 a^2 b+15 a b^2+4 b^3\right ) \tanh ^3(c+d x)+b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)-\frac {1}{7} b^4 (5 a+4 b) \tanh ^7(c+d x)+\frac {1}{9} b^5 \tanh ^9(c+d x)}{d}\) |
(a^5*ArcTanh[Tanh[c + d*x]] + b*(5*a^4 + 10*a^3*b + 10*a^2*b^2 + 5*a*b^3 + b^4)*Tanh[c + d*x] - (b^2*(10*a^3 + 20*a^2*b + 15*a*b^2 + 4*b^3)*Tanh[c + d*x]^3)/3 + (b^3*(10*a^2 + 15*a*b + 6*b^2)*Tanh[c + d*x]^5)/5 - (b^4*(5*a + 4*b)*Tanh[c + d*x]^7)/7 + (b^5*Tanh[c + d*x]^9)/9)/d
3.2.37.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(a + b + b*ff^2*x^2)^p /(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && NeQ[a + b, 0] && NeQ[p, -1]
Time = 2.41 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(\frac {a^{5} \left (d x +c \right )+5 a^{4} b \tanh \left (d x +c \right )+10 a^{3} b^{2} \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+10 a^{2} b^{3} \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )+5 a \,b^{4} \left (\frac {16}{35}+\frac {\operatorname {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \operatorname {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )+b^{5} \left (\frac {128}{315}+\frac {\operatorname {sech}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {sech}\left (d x +c \right )^{6}}{63}+\frac {16 \operatorname {sech}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {sech}\left (d x +c \right )^{2}}{315}\right ) \tanh \left (d x +c \right )}{d}\) | \(185\) |
| default | \(\frac {a^{5} \left (d x +c \right )+5 a^{4} b \tanh \left (d x +c \right )+10 a^{3} b^{2} \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+10 a^{2} b^{3} \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )+5 a \,b^{4} \left (\frac {16}{35}+\frac {\operatorname {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \operatorname {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )+b^{5} \left (\frac {128}{315}+\frac {\operatorname {sech}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {sech}\left (d x +c \right )^{6}}{63}+\frac {16 \operatorname {sech}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {sech}\left (d x +c \right )^{2}}{315}\right ) \tanh \left (d x +c \right )}{d}\) | \(185\) |
| parts | \(a^{5} x +\frac {b^{5} \left (\frac {128}{315}+\frac {\operatorname {sech}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {sech}\left (d x +c \right )^{6}}{63}+\frac {16 \operatorname {sech}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {sech}\left (d x +c \right )^{2}}{315}\right ) \tanh \left (d x +c \right )}{d}+\frac {5 a^{4} b \tanh \left (d x +c \right )}{d}+\frac {10 a^{3} b^{2} \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )}{d}+\frac {10 a^{2} b^{3} \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{d}+\frac {5 a \,b^{4} \left (\frac {16}{35}+\frac {\operatorname {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \operatorname {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )}{d}\) | \(192\) |
| parallelrisch | \(\frac {\left (44100 a^{4} b +96600 a^{3} b^{2}+107520 a^{2} b^{3}+60480 a \,b^{4}+10752 b^{5}\right ) \sinh \left (3 d x +3 c \right )+\left (31500 a^{4} b +63000 a^{3} b^{2}+60480 a^{2} b^{3}+25920 a \,b^{4}+4608 b^{5}\right ) \sinh \left (5 d x +5 c \right )+\left (11025 a^{4} b +18900 a^{3} b^{2}+15120 a^{2} b^{3}+6480 a \,b^{4}+1152 b^{5}\right ) \sinh \left (7 d x +7 c \right )+\left (1575 a^{4} b +2100 a^{3} b^{2}+1680 a^{2} b^{3}+720 a \,b^{4}+128 b^{5}\right ) \sinh \left (9 d x +9 c \right )+26460 a^{5} d x \cosh \left (3 d x +3 c \right )+11340 a^{5} d x \cosh \left (5 d x +5 c \right )+2835 a^{5} d x \cosh \left (7 d x +7 c \right )+315 a^{5} d x \cosh \left (9 d x +9 c \right )+\left (22050 a^{4} b +50400 a^{3} b^{2}+60480 a^{2} b^{3}+40320 a \,b^{4}+16128 b^{5}\right ) \sinh \left (d x +c \right )+39690 a^{5} d x \cosh \left (d x +c \right )}{315 d \left (\cosh \left (9 d x +9 c \right )+9 \cosh \left (7 d x +7 c \right )+36 \cosh \left (5 d x +5 c \right )+84 \cosh \left (3 d x +3 c \right )+126 \cosh \left (d x +c \right )\right )}\) | \(354\) |
| risch | \(a^{5} x -\frac {2 b \left (88200 a^{4} {\mathrm e}^{6 d x +6 c}+10752 b^{4} {\mathrm e}^{6 d x +6 c}+88200 a^{4} {\mathrm e}^{10 d x +10 c}+18900 a^{3} b \,{\mathrm e}^{2 d x +2 c}+15120 a^{2} b^{2} {\mathrm e}^{2 d x +2 c}+6480 a \,b^{3} {\mathrm e}^{2 d x +2 c}+69300 a^{3} b \,{\mathrm e}^{4 d x +4 c}+1575 a^{4} {\mathrm e}^{16 d x +16 c}+60480 a \,b^{3} {\mathrm e}^{6 d x +6 c}+60480 a^{2} b^{2} {\mathrm e}^{4 d x +4 c}+25920 a \,b^{3} {\mathrm e}^{4 d x +4 c}+25200 a \,b^{3} {\mathrm e}^{10 d x +10 c}+16800 a^{2} b^{2} {\mathrm e}^{12 d x +12 c}+1575 a^{4}+128 b^{4}+157500 a^{3} b \,{\mathrm e}^{8 d x +8 c}+136080 a^{2} b^{2} {\mathrm e}^{8 d x +8 c}+65520 a \,b^{3} {\mathrm e}^{8 d x +8 c}+136500 a^{3} b \,{\mathrm e}^{6 d x +6 c}+1680 a^{2} b^{2}+2100 a^{3} b +720 a \,b^{3}+44100 a^{4} {\mathrm e}^{12 d x +12 c}+6300 a^{3} b \,{\mathrm e}^{14 d x +14 c}+107100 a^{3} b \,{\mathrm e}^{10 d x +10 c}+39900 a^{3} b \,{\mathrm e}^{12 d x +12 c}+124320 a^{2} b^{2} {\mathrm e}^{6 d x +6 c}+75600 a^{2} b^{2} {\mathrm e}^{10 d x +10 c}+110250 a^{4} {\mathrm e}^{8 d x +8 c}+12600 a^{4} {\mathrm e}^{2 d x +2 c}+1152 b^{4} {\mathrm e}^{2 d x +2 c}+16128 b^{4} {\mathrm e}^{8 d x +8 c}+12600 a^{4} {\mathrm e}^{14 d x +14 c}+44100 a^{4} {\mathrm e}^{4 d x +4 c}+4608 b^{4} {\mathrm e}^{4 d x +4 c}\right )}{315 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{9}}\) | \(507\) |
1/d*(a^5*(d*x+c)+5*a^4*b*tanh(d*x+c)+10*a^3*b^2*(2/3+1/3*sech(d*x+c)^2)*ta nh(d*x+c)+10*a^2*b^3*(8/15+1/5*sech(d*x+c)^4+4/15*sech(d*x+c)^2)*tanh(d*x+ c)+5*a*b^4*(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)+b^5*(128/315+1/9*sech(d*x+c)^8+8/63*sech(d*x+c)^6+16/105*sech (d*x+c)^4+64/315*sech(d*x+c)^2)*tanh(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 1652 vs. \(2 (155) = 310\).
Time = 0.27 (sec) , antiderivative size = 1652, normalized size of antiderivative = 10.13 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^5 \, dx=\text {Too large to display} \]
1/315*((315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x + c)^9 + 9*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x + c)*sinh(d*x + c)^8 + (1575 *a^4*b + 2100*a^3*b^2 + 1680*a^2*b^3 + 720*a*b^4 + 128*b^5)*sinh(d*x + c)^ 9 + 9*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x + c)^7 + 9*(1225*a^4*b + 2100*a^3*b^2 + 1680*a^2*b^3 + 720*a*b^4 + 128*b^5 + 4*(1575*a^4*b + 2100*a^3*b^2 + 1680*a^2*b^3 + 720*a *b^4 + 128*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 21*(4*(315*a^5*d*x - 15 75*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x + c )^3 + 3*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^ 4 - 128*b^5)*cosh(d*x + c))*sinh(d*x + c)^6 + 36*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x + c)^5 + 9* (3500*a^4*b + 7000*a^3*b^2 + 6720*a^2*b^3 + 2880*a*b^4 + 512*b^5 + 14*(157 5*a^4*b + 2100*a^3*b^2 + 1680*a^2*b^3 + 720*a*b^4 + 128*b^5)*cosh(d*x + c) ^4 + 21*(1225*a^4*b + 2100*a^3*b^2 + 1680*a^2*b^3 + 720*a*b^4 + 128*b^5)*c osh(d*x + c)^2)*sinh(d*x + c)^5 + 9*(14*(315*a^5*d*x - 1575*a^4*b - 2100*a ^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x + c)^5 + 35*(315*a^5 *d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cos h(d*x + c)^3 + 20*(315*a^5*d*x - 1575*a^4*b - 2100*a^3*b^2 - 1680*a^2*b^3 - 720*a*b^4 - 128*b^5)*cosh(d*x + c))*sinh(d*x + c)^4 + 84*(315*a^5*d*x...
\[ \int \left (a+b \text {sech}^2(c+d x)\right )^5 \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{5}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 1277 vs. \(2 (155) = 310\).
Time = 0.20 (sec) , antiderivative size = 1277, normalized size of antiderivative = 7.83 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^5 \, dx=\text {Too large to display} \]
a^5*x + 256/315*b^5*(9*e^(-2*d*x - 2*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4* d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16 *c) + e^(-18*d*x - 18*c) + 1)) + 36*e^(-4*d*x - 4*c)/(d*(9*e^(-2*d*x - 2*c ) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126 *e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^ (-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) + 84*e^(-6*d*x - 6*c)/(d*(9*e^ (-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) + 126*e^(-8*d*x - 8*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 3 6*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) + 1 /(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126* e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^( -14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1))) + 32/7* a*b^4*(7*e^(-2*d*x - 2*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 3 5*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-1 2*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 ...
Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (155) = 310\).
Time = 0.32 (sec) , antiderivative size = 537, normalized size of antiderivative = 3.29 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^5 \, dx=\frac {315 \, {\left (d x + c\right )} a^{5} - \frac {2 \, {\left (1575 \, a^{4} b e^{\left (16 \, d x + 16 \, c\right )} + 12600 \, a^{4} b e^{\left (14 \, d x + 14 \, c\right )} + 6300 \, a^{3} b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 44100 \, a^{4} b e^{\left (12 \, d x + 12 \, c\right )} + 39900 \, a^{3} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 16800 \, a^{2} b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 88200 \, a^{4} b e^{\left (10 \, d x + 10 \, c\right )} + 107100 \, a^{3} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 75600 \, a^{2} b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 25200 \, a b^{4} e^{\left (10 \, d x + 10 \, c\right )} + 110250 \, a^{4} b e^{\left (8 \, d x + 8 \, c\right )} + 157500 \, a^{3} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 136080 \, a^{2} b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 65520 \, a b^{4} e^{\left (8 \, d x + 8 \, c\right )} + 16128 \, b^{5} e^{\left (8 \, d x + 8 \, c\right )} + 88200 \, a^{4} b e^{\left (6 \, d x + 6 \, c\right )} + 136500 \, a^{3} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 124320 \, a^{2} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 60480 \, a b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 10752 \, b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 44100 \, a^{4} b e^{\left (4 \, d x + 4 \, c\right )} + 69300 \, a^{3} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 60480 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 25920 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 4608 \, b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 12600 \, a^{4} b e^{\left (2 \, d x + 2 \, c\right )} + 18900 \, a^{3} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 15120 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 6480 \, a b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 1152 \, b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 1575 \, a^{4} b + 2100 \, a^{3} b^{2} + 1680 \, a^{2} b^{3} + 720 \, a b^{4} + 128 \, b^{5}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{9}}}{315 \, d} \]
1/315*(315*(d*x + c)*a^5 - 2*(1575*a^4*b*e^(16*d*x + 16*c) + 12600*a^4*b*e ^(14*d*x + 14*c) + 6300*a^3*b^2*e^(14*d*x + 14*c) + 44100*a^4*b*e^(12*d*x + 12*c) + 39900*a^3*b^2*e^(12*d*x + 12*c) + 16800*a^2*b^3*e^(12*d*x + 12*c ) + 88200*a^4*b*e^(10*d*x + 10*c) + 107100*a^3*b^2*e^(10*d*x + 10*c) + 756 00*a^2*b^3*e^(10*d*x + 10*c) + 25200*a*b^4*e^(10*d*x + 10*c) + 110250*a^4* b*e^(8*d*x + 8*c) + 157500*a^3*b^2*e^(8*d*x + 8*c) + 136080*a^2*b^3*e^(8*d *x + 8*c) + 65520*a*b^4*e^(8*d*x + 8*c) + 16128*b^5*e^(8*d*x + 8*c) + 8820 0*a^4*b*e^(6*d*x + 6*c) + 136500*a^3*b^2*e^(6*d*x + 6*c) + 124320*a^2*b^3* e^(6*d*x + 6*c) + 60480*a*b^4*e^(6*d*x + 6*c) + 10752*b^5*e^(6*d*x + 6*c) + 44100*a^4*b*e^(4*d*x + 4*c) + 69300*a^3*b^2*e^(4*d*x + 4*c) + 60480*a^2* b^3*e^(4*d*x + 4*c) + 25920*a*b^4*e^(4*d*x + 4*c) + 4608*b^5*e^(4*d*x + 4* c) + 12600*a^4*b*e^(2*d*x + 2*c) + 18900*a^3*b^2*e^(2*d*x + 2*c) + 15120*a ^2*b^3*e^(2*d*x + 2*c) + 6480*a*b^4*e^(2*d*x + 2*c) + 1152*b^5*e^(2*d*x + 2*c) + 1575*a^4*b + 2100*a^3*b^2 + 1680*a^2*b^3 + 720*a*b^4 + 128*b^5)/(e^ (2*d*x + 2*c) + 1)^9)/d
Time = 0.30 (sec) , antiderivative size = 1952, normalized size of antiderivative = 11.98 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^5 \, dx=\text {Too large to display} \]
a^5*x - ((10*(8*a*b^4 + 7*a^4*b + 16*a^2*b^3 + 15*a^3*b^2))/(63*d) + (10*e xp(2*c + 2*d*x)*(7*a^4*b + 8*a^2*b^3 + 12*a^3*b^2))/(21*d) + (10*exp(4*c + 4*d*x)*(a^4*b + a^3*b^2))/(3*d) + (10*a^4*b*exp(6*c + 6*d*x))/(9*d))/(4*e xp(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) - ((80*exp(6*c + 6*d*x)*(8*a*b^4 + 7*a^4*b + 16*a^2*b^3 + 15*a^3*b ^2))/(9*d) + (80*exp(10*c + 10*d*x)*(8*a*b^4 + 7*a^4*b + 16*a^2*b^3 + 15*a ^3*b^2))/(9*d) + (4*exp(8*c + 8*d*x)*(320*a*b^4 + 175*a^4*b + 128*b^5 + 48 0*a^2*b^3 + 400*a^3*b^2))/(9*d) + (10*a^4*b)/(9*d) + (40*exp(4*c + 4*d*x)* (7*a^4*b + 8*a^2*b^3 + 12*a^3*b^2))/(9*d) + (40*exp(12*c + 12*d*x)*(7*a^4* b + 8*a^2*b^3 + 12*a^3*b^2))/(9*d) + (80*exp(2*c + 2*d*x)*(a^4*b + a^3*b^2 ))/(9*d) + (80*exp(14*c + 14*d*x)*(a^4*b + a^3*b^2))/(9*d) + (10*a^4*b*exp (16*c + 16*d*x))/(9*d))/(9*exp(2*c + 2*d*x) + 36*exp(4*c + 4*d*x) + 84*exp (6*c + 6*d*x) + 126*exp(8*c + 8*d*x) + 126*exp(10*c + 10*d*x) + 84*exp(12* c + 12*d*x) + 36*exp(14*c + 14*d*x) + 9*exp(16*c + 16*d*x) + exp(18*c + 18 *d*x) + 1) - ((10*(a^4*b + a^3*b^2))/(9*d) + (10*a^4*b*exp(2*c + 2*d*x))/( 9*d))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1) - ((10*(7*a^4*b + 8*a^2* b^3 + 12*a^3*b^2))/(63*d) + (20*exp(2*c + 2*d*x)*(8*a*b^4 + 7*a^4*b + 16*a ^2*b^3 + 15*a^3*b^2))/(21*d) + (200*exp(6*c + 6*d*x)*(8*a*b^4 + 7*a^4*b + 16*a^2*b^3 + 15*a^3*b^2))/(63*d) + (2*exp(4*c + 4*d*x)*(320*a*b^4 + 175*a^ 4*b + 128*b^5 + 480*a^2*b^3 + 400*a^3*b^2))/(21*d) + (50*exp(8*c + 8*d*...