Integrand size = 23, antiderivative size = 198 \[ \int \text {sech}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {\left (64 a^3+48 a^2 b+24 a b^2+5 b^3\right ) \arctan (\sinh (c+d x))}{128 d}+\frac {\left (64 a^3+48 a^2 b+24 a b^2+5 b^3\right ) \text {sech}(c+d x) \tanh (c+d x)}{128 d}-\frac {b \left (72 a^2+52 a b+15 b^2\right ) \text {sech}^3(c+d x) \tanh (c+d x)}{192 d}-\frac {b (12 a+5 b) \text {sech}^5(c+d x) \left (a+(a+b) \sinh ^2(c+d x)\right ) \tanh (c+d x)}{48 d}-\frac {b \text {sech}^7(c+d x) \left (a+(a+b) \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{8 d} \]
1/128*(64*a^3+48*a^2*b+24*a*b^2+5*b^3)*arctan(sinh(d*x+c))/d+1/128*(64*a^3 +48*a^2*b+24*a*b^2+5*b^3)*sech(d*x+c)*tanh(d*x+c)/d-1/192*b*(72*a^2+52*a*b +15*b^2)*sech(d*x+c)^3*tanh(d*x+c)/d-1/48*b*(12*a+5*b)*sech(d*x+c)^5*(a+(a +b)*sinh(d*x+c)^2)*tanh(d*x+c)/d-1/8*b*sech(d*x+c)^7*(a+(a+b)*sinh(d*x+c)^ 2)^2*tanh(d*x+c)/d
Time = 13.01 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.80 \[ \int \text {sech}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {6 \left (64 a^3+48 a^2 b+24 a b^2+5 b^3\right ) \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+3 \left (64 a^3+48 a^2 b+24 a b^2+5 b^3\right ) \text {sech}(c+d x) \tanh (c+d x)-2 b \left (144 a^2+168 a b+59 b^2\right ) \text {sech}^3(c+d x) \tanh (c+d x)+8 b^2 (24 a+17 b) \text {sech}^5(c+d x) \tanh (c+d x)-48 b^3 \text {sech}^7(c+d x) \tanh (c+d x)}{384 d} \]
(6*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*ArcTan[Tanh[(c + d*x)/2]] + 3*(6 4*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*Sech[c + d*x]*Tanh[c + d*x] - 2*b*(14 4*a^2 + 168*a*b + 59*b^2)*Sech[c + d*x]^3*Tanh[c + d*x] + 8*b^2*(24*a + 17 *b)*Sech[c + d*x]^5*Tanh[c + d*x] - 48*b^3*Sech[c + d*x]^7*Tanh[c + d*x])/ (384*d)
Time = 0.41 (sec) , antiderivative size = 197, 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, 4159, 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.
\(\displaystyle \int \text {sech}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sec (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )^3dx\) |
\(\Big \downarrow \) 4159 |
\(\displaystyle \frac {\int \frac {\left ((a+b) \sinh ^2(c+d x)+a\right )^3}{\left (\sinh ^2(c+d x)+1\right )^5}d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 315 |
\(\displaystyle \frac {\frac {1}{8} \int \frac {\left ((a+b) \sinh ^2(c+d x)+a\right ) \left ((a+b) (8 a+5 b) \sinh ^2(c+d x)+a (8 a+b)\right )}{\left (\sinh ^2(c+d x)+1\right )^4}d\sinh (c+d x)-\frac {b \sinh (c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )^2}{8 \left (\sinh ^2(c+d x)+1\right )^4}}{d}\) |
\(\Big \downarrow \) 401 |
\(\displaystyle \frac {\frac {1}{8} \left (-\frac {1}{6} \int -\frac {3 (a+b) \left (16 a^2+14 b a+5 b^2\right ) \sinh ^2(c+d x)+a \left (48 a^2+18 b a+5 b^2\right )}{\left (\sinh ^2(c+d x)+1\right )^3}d\sinh (c+d x)-\frac {b (12 a+5 b) \sinh (c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}\right )-\frac {b \sinh (c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )^2}{8 \left (\sinh ^2(c+d x)+1\right )^4}}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \int \frac {3 (a+b) \left (16 a^2+14 b a+5 b^2\right ) \sinh ^2(c+d x)+a \left (48 a^2+18 b a+5 b^2\right )}{\left (\sinh ^2(c+d x)+1\right )^3}d\sinh (c+d x)-\frac {b (12 a+5 b) \sinh (c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}\right )-\frac {b \sinh (c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )^2}{8 \left (\sinh ^2(c+d x)+1\right )^4}}{d}\) |
\(\Big \downarrow \) 298 |
\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {3}{4} \left (64 a^3+48 a^2 b+24 a b^2+5 b^3\right ) \int \frac {1}{\left (\sinh ^2(c+d x)+1\right )^2}d\sinh (c+d x)-\frac {b \left (72 a^2+52 a b+15 b^2\right ) \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )-\frac {b (12 a+5 b) \sinh (c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}\right )-\frac {b \sinh (c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )^2}{8 \left (\sinh ^2(c+d x)+1\right )^4}}{d}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {3}{4} \left (64 a^3+48 a^2 b+24 a b^2+5 b^3\right ) \left (\frac {1}{2} \int \frac {1}{\sinh ^2(c+d x)+1}d\sinh (c+d x)+\frac {\sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )-\frac {b \left (72 a^2+52 a b+15 b^2\right ) \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )-\frac {b (12 a+5 b) \sinh (c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}\right )-\frac {b \sinh (c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )^2}{8 \left (\sinh ^2(c+d x)+1\right )^4}}{d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {3}{4} \left (64 a^3+48 a^2 b+24 a b^2+5 b^3\right ) \left (\frac {1}{2} \arctan (\sinh (c+d x))+\frac {\sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )-\frac {b \left (72 a^2+52 a b+15 b^2\right ) \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )-\frac {b (12 a+5 b) \sinh (c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )}{6 \left (\sinh ^2(c+d x)+1\right )^3}\right )-\frac {b \sinh (c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )^2}{8 \left (\sinh ^2(c+d x)+1\right )^4}}{d}\) |
(-1/8*(b*Sinh[c + d*x]*(a + (a + b)*Sinh[c + d*x]^2)^2)/(1 + Sinh[c + d*x] ^2)^4 + (-1/6*(b*(12*a + 5*b)*Sinh[c + d*x]*(a + (a + b)*Sinh[c + d*x]^2)) /(1 + Sinh[c + d*x]^2)^3 + (-1/4*(b*(72*a^2 + 52*a*b + 15*b^2)*Sinh[c + d* x])/(1 + Sinh[c + d*x]^2)^2 + (3*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*(A rcTan[Sinh[c + d*x]]/2 + Sinh[c + d*x]/(2*(1 + Sinh[c + d*x]^2))))/4)/6)/8 )/d
3.2.3.3.1 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_.)*tan[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ExpandToSum[b*(ff*x)^n + 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] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
Time = 43.36 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.47
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (-\frac {\sinh \left (d x +c \right )}{3 \cosh \left (d x +c \right )^{4}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )}{3}+\frac {\arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+3 a \,b^{2} \left (-\frac {\sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{6}}-\frac {\sinh \left (d x +c \right )}{5 \cosh \left (d x +c \right )^{6}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )}{5}+\frac {\arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )}{7 \cosh \left (d x +c \right )^{8}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )}{7}+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\) | \(291\) |
default | \(\frac {a^{3} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (-\frac {\sinh \left (d x +c \right )}{3 \cosh \left (d x +c \right )^{4}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )}{3}+\frac {\arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+3 a \,b^{2} \left (-\frac {\sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{6}}-\frac {\sinh \left (d x +c \right )}{5 \cosh \left (d x +c \right )^{6}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )}{5}+\frac {\arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )}{7 \cosh \left (d x +c \right )^{8}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )}{7}+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\) | \(291\) |
risch | \(\frac {{\mathrm e}^{d x +c} \left (-432 a^{2} b \,{\mathrm e}^{12 d x +12 c}-72 a \,b^{2}-984 a \,b^{2} {\mathrm e}^{12 d x +12 c}-397 b^{3} {\mathrm e}^{12 d x +12 c}-192 a^{3}-895 \,{\mathrm e}^{4 d x +4 c} b^{3}+72 a \,b^{2} {\mathrm e}^{14 d x +14 c}+192 a^{3} {\mathrm e}^{14 d x +14 c}-144 a^{2} b +984 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+432 a^{2} b \,{\mathrm e}^{2 d x +2 c}+312 a \,b^{2} {\mathrm e}^{4 d x +4 c}+2160 a^{2} b \,{\mathrm e}^{4 d x +4 c}+1584 a^{2} b \,{\mathrm e}^{6 d x +6 c}-744 a \,b^{2} {\mathrm e}^{6 d x +6 c}-15 b^{3}+960 a^{3} {\mathrm e}^{12 d x +12 c}-1584 a^{2} b \,{\mathrm e}^{8 d x +8 c}+744 a \,b^{2} {\mathrm e}^{8 d x +8 c}+144 a^{2} b \,{\mathrm e}^{14 d x +14 c}-2160 a^{2} b \,{\mathrm e}^{10 d x +10 c}+895 b^{3} {\mathrm e}^{10 d x +10 c}+1728 a^{3} {\mathrm e}^{10 d x +10 c}-960 a^{3} {\mathrm e}^{6 d x +6 c}+1765 \,{\mathrm e}^{6 d x +6 c} b^{3}-1728 a^{3} {\mathrm e}^{4 d x +4 c}-960 a^{3} {\mathrm e}^{2 d x +2 c}+397 \,{\mathrm e}^{2 d x +2 c} b^{3}-1765 b^{3} {\mathrm e}^{8 d x +8 c}+960 a^{3} {\mathrm e}^{8 d x +8 c}-312 a \,b^{2} {\mathrm e}^{10 d x +10 c}+15 b^{3} {\mathrm e}^{14 d x +14 c}\right )}{192 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{2 d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2} b}{8 d}+\frac {3 i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{16 d}+\frac {5 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{128 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{2 d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{8 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a \,b^{2}}{16 d}-\frac {5 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{128 d}\) | \(582\) |
1/d*(a^3*(1/2*sech(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c)))+3*a^2*b*(-1/3/co sh(d*x+c)^4*sinh(d*x+c)+1/3*(1/4*sech(d*x+c)^3+3/8*sech(d*x+c))*tanh(d*x+c )+1/4*arctan(exp(d*x+c)))+3*a*b^2*(-1/3*sinh(d*x+c)^3/cosh(d*x+c)^6-1/5*si nh(d*x+c)/cosh(d*x+c)^6+1/5*(1/6*sech(d*x+c)^5+5/24*sech(d*x+c)^3+5/16*sec h(d*x+c))*tanh(d*x+c)+1/8*arctan(exp(d*x+c)))+b^3*(-1/3*sinh(d*x+c)^5/cosh (d*x+c)^8-1/3*sinh(d*x+c)^3/cosh(d*x+c)^8-1/7*sinh(d*x+c)/cosh(d*x+c)^8+1/ 7*(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)+5/64*arctan(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 6114 vs. \(2 (188) = 376\).
Time = 0.32 (sec) , antiderivative size = 6114, normalized size of antiderivative = 30.88 \[ \int \text {sech}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]
\[ \int \text {sech}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \operatorname {sech}^{3}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 553 vs. \(2 (188) = 376\).
Time = 0.30 (sec) , antiderivative size = 553, normalized size of antiderivative = 2.79 \[ \int \text {sech}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {1}{192} \, b^{3} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {15 \, e^{\left (-d x - c\right )} - 397 \, e^{\left (-3 \, d x - 3 \, c\right )} + 895 \, e^{\left (-5 \, d x - 5 \, c\right )} - 1765 \, e^{\left (-7 \, d x - 7 \, c\right )} + 1765 \, e^{\left (-9 \, d x - 9 \, c\right )} - 895 \, e^{\left (-11 \, d x - 11 \, c\right )} + 397 \, e^{\left (-13 \, d x - 13 \, c\right )} - 15 \, 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 {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} - 47 \, e^{\left (-3 \, d x - 3 \, c\right )} + 78 \, e^{\left (-5 \, d x - 5 \, c\right )} - 78 \, e^{\left (-7 \, d x - 7 \, c\right )} + 47 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3 \, 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 {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - 7 \, e^{\left (-3 \, d x - 3 \, c\right )} + 7 \, e^{\left (-5 \, d x - 5 \, c\right )} - 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*(15*arctan(e^(-d*x - c))/d - (15*e^(-d*x - c) - 397*e^(-3*d*x - 3*c) + 895*e^(-5*d*x - 5*c) - 1765*e^(-7*d*x - 7*c) + 1765*e^(-9*d*x - 9* c) - 895*e^(-11*d*x - 11*c) + 397*e^(-13*d*x - 13*c) - 15*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) + 7 0*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*(3*arctan(e^(-d*x - c))/d - (3*e^(-d*x - c) - 47*e^(-3*d*x - 3*c) + 78*e^(-5*d*x - 5*c) - 78 *e^(-7*d*x - 7*c) + 47*e^(-9*d*x - 9*c) - 3*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*(arcta n(e^(-d*x - c))/d - (e^(-d*x - c) - 7*e^(-3*d*x - 3*c) + 7*e^(-5*d*x - 5*c ) - 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. \(2 (188) = 376\).
Time = 0.41 (sec) , antiderivative size = 485, normalized size of antiderivative = 2.45 \[ \int \text {sech}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (64 \, a^{3} + 48 \, a^{2} b + 24 \, a b^{2} + 5 \, b^{3}\right )} + \frac {4 \, {\left (192 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 144 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 72 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 15 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 2304 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 576 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} - 480 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} - 292 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 9216 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 2304 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 4224 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 880 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 12288 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 9216 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 4608 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 960 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{4}}}{768 \, d} \]
1/768*(3*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3) + 4*(192*a^3*(e^(d*x + c) - e^(-d*x - c))^7 + 144*a^2*b*(e^(d*x + c) - e^(-d*x - c))^7 + 72*a*b^2*(e^(d*x + c) - e^(-d *x - c))^7 + 15*b^3*(e^(d*x + c) - e^(-d*x - c))^7 + 2304*a^3*(e^(d*x + c) - e^(-d*x - c))^5 + 576*a^2*b*(e^(d*x + c) - e^(-d*x - c))^5 - 480*a*b^2* (e^(d*x + c) - e^(-d*x - c))^5 - 292*b^3*(e^(d*x + c) - e^(-d*x - c))^5 + 9216*a^3*(e^(d*x + c) - e^(-d*x - c))^3 - 2304*a^2*b*(e^(d*x + c) - e^(-d* x - c))^3 - 4224*a*b^2*(e^(d*x + c) - e^(-d*x - c))^3 - 880*b^3*(e^(d*x + c) - e^(-d*x - c))^3 + 12288*a^3*(e^(d*x + c) - e^(-d*x - c)) - 9216*a^2*b *(e^(d*x + c) - e^(-d*x - c)) - 4608*a*b^2*(e^(d*x + c) - e^(-d*x - c)) - 960*b^3*(e^(d*x + c) - e^(-d*x - c)))/((e^(d*x + c) - e^(-d*x - c))^2 + 4) ^4)/d
Time = 1.99 (sec) , antiderivative size = 951, normalized size of antiderivative = 4.80 \[ \int \text {sech}^3(c+d x) \left (a+b \tanh ^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}+5\,b^3\,\sqrt {d^2}+24\,a\,b^2\,\sqrt {d^2}+48\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {4096\,a^6+6144\,a^5\,b+5376\,a^4\,b^2+2944\,a^3\,b^3+1056\,a^2\,b^4+240\,a\,b^5+25\,b^6}}\right )\,\sqrt {4096\,a^6+6144\,a^5\,b+5376\,a^4\,b^2+2944\,a^3\,b^3+1056\,a^2\,b^4+240\,a\,b^5+25\,b^6}}{64\,\sqrt {d^2}}-\frac {\frac {{\mathrm {e}}^{c+d\,x}\,{\left (a+b\right )}^3}{2\,d}+\frac {{\mathrm {e}}^{13\,c+13\,d\,x}\,{\left (a+b\right )}^3}{2\,d}-\frac {3\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (-5\,a^3+a^2\,b+a\,b^2-5\,b^3\right )}{2\,d}-\frac {3\,{\mathrm {e}}^{9\,c+9\,d\,x}\,\left (-5\,a^3+a^2\,b+a\,b^2-5\,b^3\right )}{2\,d}+\frac {2\,{\mathrm {e}}^{7\,c+7\,d\,x}\,\left (5\,a^3-3\,a^2\,b+3\,a\,b^2-5\,b^3\right )}{d}+\frac {3\,{\mathrm {e}}^{3\,c+3\,d\,x}\,{\left (a+b\right )}^2\,\left (a-b\right )}{d}+\frac {3\,{\mathrm {e}}^{11\,c+11\,d\,x}\,{\left (a+b\right )}^2\,\left (a-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 (85\,b^3+48\,a\,b^2\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 {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+48\,a^2\,b+24\,a\,b^2+5\,b^3\right )}{64\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {4\,{\mathrm {e}}^{c+d\,x}\,\left (35\,b^3+6\,a\,b^2\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+576\,a^2\,b+600\,a\,b^2+203\,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+600\,a\,b^2+305\,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 )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (24\,a^2\,b+168\,a\,b^2+145\,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 )} \]
(atan((exp(d*x)*exp(c)*(64*a^3*(d^2)^(1/2) + 5*b^3*(d^2)^(1/2) + 24*a*b^2* (d^2)^(1/2) + 48*a^2*b*(d^2)^(1/2)))/(d*(240*a*b^5 + 6144*a^5*b + 4096*a^6 + 25*b^6 + 1056*a^2*b^4 + 2944*a^3*b^3 + 5376*a^4*b^2)^(1/2)))*(240*a*b^5 + 6144*a^5*b + 4096*a^6 + 25*b^6 + 1056*a^2*b^4 + 2944*a^3*b^3 + 5376*a^4 *b^2)^(1/2))/(64*(d^2)^(1/2)) - ((exp(c + d*x)*(a + b)^3)/(2*d) + (exp(13* c + 13*d*x)*(a + b)^3)/(2*d) - (3*exp(5*c + 5*d*x)*(a*b^2 + a^2*b - 5*a^3 - 5*b^3))/(2*d) - (3*exp(9*c + 9*d*x)*(a*b^2 + a^2*b - 5*a^3 - 5*b^3))/(2* d) + (2*exp(7*c + 7*d*x)*(3*a*b^2 - 3*a^2*b + 5*a^3 - 5*b^3))/d + (3*exp(3 *c + 3*d*x)*(a + b)^2*(a - b))/d + (3*exp(11*c + 11*d*x)*(a + b)^2*(a - 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(1 4*c + 14*d*x) + exp(16*c + 16*d*x) + 1) + (2*exp(c + d*x)*(48*a*b^2 + 85*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)) + (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 + 1 4*d*x) + 1)) + (exp(c + d*x)*(24*a*b^2 + 48*a^2*b + 64*a^3 + 5*b^3))/(64*d *(exp(2*c + 2*d*x) + 1)) - (4*exp(c + d*x)*(6*a*b^2 + 35*b^3))/(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)) - (exp(c + d*x)...