Integrand size = 23, antiderivative size = 103 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=\frac {a^3 \log (\cosh (c+d x))}{d}-\frac {3 a^2 b \text {sech}^2(c+d x)}{2 d}-\frac {3 a b^2 \text {sech}^4(c+d x)}{4 d}-\frac {b^3 \text {sech}^6(c+d x)}{6 d}+\frac {\left (b+a \cosh ^2(c+d x)\right )^4 \text {sech}^8(c+d x)}{8 b d} \]
a^3*ln(cosh(d*x+c))/d-3/2*a^2*b*sech(d*x+c)^2/d-3/4*a*b^2*sech(d*x+c)^4/d- 1/6*b^3*sech(d*x+c)^6/d+1/8*(b+a*cosh(d*x+c)^2)^4*sech(d*x+c)^8/b/d
Time = 0.78 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.24 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=\frac {\cosh ^6(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \left (24 a^3 \log (\cosh (c+d x))+12 a^2 (a-3 b) \text {sech}^2(c+d x)+18 a (a-b) b \text {sech}^4(c+d x)+4 (3 a-b) b^2 \text {sech}^6(c+d x)+3 b^3 \text {sech}^8(c+d x)\right )}{3 d (a+2 b+a \cosh (2 c+2 d x))^3} \]
(Cosh[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3*(24*a^3*Log[Cosh[c + d*x]] + 12 *a^2*(a - 3*b)*Sech[c + d*x]^2 + 18*a*(a - b)*b*Sech[c + d*x]^4 + 4*(3*a - b)*b^2*Sech[c + d*x]^6 + 3*b^3*Sech[c + d*x]^8))/(3*d*(a + 2*b + a*Cosh[2 *c + 2*d*x])^3)
Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.91, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 26, 4626, 354, 87, 49, 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.
| \(\displaystyle \int \tanh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int i \tan (i c+i d x)^3 \left (a+b \sec (i c+i d x)^2\right )^3dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \left (b \sec (i c+i d x)^2+a\right )^3 \tan (i c+i d x)^3dx\) |
| \(\Big \downarrow \) 4626 |
| \(\displaystyle -\frac {\int \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^3 \text {sech}^9(c+d x)d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle -\frac {\int \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^3 \text {sech}^5(c+d x)d\cosh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {-\int \left (a \cosh ^2(c+d x)+b\right )^3 \text {sech}^4(c+d x)d\cosh ^2(c+d x)-\frac {\text {sech}^4(c+d x) \left (a \cosh ^2(c+d x)+b\right )^4}{4 b}}{2 d}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -\frac {-\int \left (b^3 \text {sech}^4(c+d x)+3 a b^2 \text {sech}^3(c+d x)+3 a^2 b \text {sech}^2(c+d x)+a^3 \text {sech}(c+d x)\right )d\cosh ^2(c+d x)-\frac {\text {sech}^4(c+d x) \left (a \cosh ^2(c+d x)+b\right )^4}{4 b}}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-a^3 \log \left (\cosh ^2(c+d x)\right )+3 a^2 b \text {sech}(c+d x)+\frac {3}{2} a b^2 \text {sech}^2(c+d x)-\frac {\text {sech}^4(c+d x) \left (a \cosh ^2(c+d x)+b\right )^4}{4 b}+\frac {1}{3} b^3 \text {sech}^3(c+d x)}{2 d}\) |
-1/2*(-(a^3*Log[Cosh[c + d*x]^2]) + 3*a^2*b*Sech[c + d*x] + (3*a*b^2*Sech[ c + d*x]^2)/2 + (b^3*Sech[c + d*x]^3)/3 - ((b + a*Cosh[c + d*x]^2)^4*Sech[ c + d*x]^4)/(4*b))/d
3.2.25.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> Module[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-(f *ff^(m + n*p - 1))^(-1) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff* x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n} , x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
Time = 111.36 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.10
| method | result | size |
| derivativedivides | \(\frac {\frac {b^{3} \operatorname {sech}\left (d x +c \right )^{8}}{8}+\frac {a \,b^{2} \operatorname {sech}\left (d x +c \right )^{6}}{2}-\frac {\operatorname {sech}\left (d x +c \right )^{6} b^{3}}{6}+\frac {3 a^{2} b \operatorname {sech}\left (d x +c \right )^{4}}{4}-\frac {3 \operatorname {sech}\left (d x +c \right )^{4} a \,b^{2}}{4}+\frac {a^{3} \operatorname {sech}\left (d x +c \right )^{2}}{2}-\frac {3 \operatorname {sech}\left (d x +c \right )^{2} a^{2} b}{2}-a^{3} \ln \left (\operatorname {sech}\left (d x +c \right )\right )}{d}\) | \(113\) |
| default | \(\frac {\frac {b^{3} \operatorname {sech}\left (d x +c \right )^{8}}{8}+\frac {a \,b^{2} \operatorname {sech}\left (d x +c \right )^{6}}{2}-\frac {\operatorname {sech}\left (d x +c \right )^{6} b^{3}}{6}+\frac {3 a^{2} b \operatorname {sech}\left (d x +c \right )^{4}}{4}-\frac {3 \operatorname {sech}\left (d x +c \right )^{4} a \,b^{2}}{4}+\frac {a^{3} \operatorname {sech}\left (d x +c \right )^{2}}{2}-\frac {3 \operatorname {sech}\left (d x +c \right )^{2} a^{2} b}{2}-a^{3} \ln \left (\operatorname {sech}\left (d x +c \right )\right )}{d}\) | \(113\) |
| parts | \(\frac {a^{3} \left (-\frac {\tanh \left (d x +c \right )^{2}}{2}-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}-\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {b^{3} \left (\frac {\operatorname {sech}\left (d x +c \right )^{8}}{8}-\frac {\operatorname {sech}\left (d x +c \right )^{6}}{6}\right )}{d}+\frac {3 a^{2} b \tanh \left (d x +c \right )^{4}}{4 d}+\frac {3 a \,b^{2} \left (-\frac {\tanh \left (d x +c \right )^{6}}{6}+\frac {\tanh \left (d x +c \right )^{4}}{4}\right )}{d}\) | \(117\) |
| risch | \(-a^{3} x -\frac {2 a^{3} c}{d}+\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (3 a^{3} {\mathrm e}^{12 d x +12 c}-9 a^{2} b \,{\mathrm e}^{12 d x +12 c}+18 a^{3} {\mathrm e}^{10 d x +10 c}-36 a^{2} b \,{\mathrm e}^{10 d x +10 c}-18 a \,b^{2} {\mathrm e}^{10 d x +10 c}+45 a^{3} {\mathrm e}^{8 d x +8 c}-63 a^{2} b \,{\mathrm e}^{8 d x +8 c}-24 a \,b^{2} {\mathrm e}^{8 d x +8 c}-16 b^{3} {\mathrm e}^{8 d x +8 c}+60 a^{3} {\mathrm e}^{6 d x +6 c}-72 a^{2} b \,{\mathrm e}^{6 d x +6 c}-12 a \,b^{2} {\mathrm e}^{6 d x +6 c}+16 \,{\mathrm e}^{6 d x +6 c} b^{3}+45 a^{3} {\mathrm e}^{4 d x +4 c}-63 a^{2} b \,{\mathrm e}^{4 d x +4 c}-24 a \,b^{2} {\mathrm e}^{4 d x +4 c}-16 \,{\mathrm e}^{4 d x +4 c} b^{3}+18 a^{3} {\mathrm e}^{2 d x +2 c}-36 a^{2} b \,{\mathrm e}^{2 d x +2 c}-18 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+3 a^{3}-9 a^{2} b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d}\) | \(366\) |
1/d*(1/8*b^3*sech(d*x+c)^8+1/2*a*b^2*sech(d*x+c)^6-1/6*sech(d*x+c)^6*b^3+3 /4*a^2*b*sech(d*x+c)^4-3/4*sech(d*x+c)^4*a*b^2+1/2*a^3*sech(d*x+c)^2-3/2*s ech(d*x+c)^2*a^2*b-a^3*ln(sech(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 4658 vs. \(2 (95) = 190\).
Time = 0.30 (sec) , antiderivative size = 4658, normalized size of antiderivative = 45.22 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=\text {Too large to display} \]
-1/3*(3*a^3*d*x*cosh(d*x + c)^16 + 48*a^3*d*x*cosh(d*x + c)*sinh(d*x + c)^ 15 + 3*a^3*d*x*sinh(d*x + c)^16 + 6*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^14 + 6*(60*a^3*d*x*cosh(d*x + c)^2 + 4*a^3*d*x - a^3 + 3*a^2*b)*sinh(d *x + c)^14 + 84*(20*a^3*d*x*cosh(d*x + c)^3 + (4*a^3*d*x - a^3 + 3*a^2*b)* cosh(d*x + c))*sinh(d*x + c)^13 + 12*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^ 2)*cosh(d*x + c)^12 + 6*(910*a^3*d*x*cosh(d*x + c)^4 + 14*a^3*d*x - 6*a^3 + 12*a^2*b + 6*a*b^2 + 91*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^2)*sin h(d*x + c)^12 + 24*(546*a^3*d*x*cosh(d*x + c)^5 + 91*(4*a^3*d*x - a^3 + 3* a^2*b)*cosh(d*x + c)^3 + 6*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d* x + c))*sinh(d*x + c)^11 + 2*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^10 + 2*(12012*a^3*d*x*cosh(d*x + c)^6 + 84*a^3*d*x + 3003*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^4 - 45*a^3 + 63*a^2*b + 24 *a*b^2 + 16*b^3 + 396*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c )^2)*sinh(d*x + c)^10 + 4*(8580*a^3*d*x*cosh(d*x + c)^7 + 3003*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^5 + 660*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a* b^2)*cosh(d*x + c)^3 + 5*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b ^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 2*(105*a^3*d*x - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3)*cosh(d*x + c)^8 + 2*(19305*a^3*d*x*cosh(d*x + c)^8 + 90 09*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^6 + 105*a^3*d*x + 2970*(7*a^3 *d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^4 - 60*a^3 + 72*a^2*b +...
Time = 1.27 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.73 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=\begin {cases} a^{3} x - \frac {a^{3} \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} - \frac {a^{3} \tanh ^{2}{\left (c + d x \right )}}{2 d} - \frac {3 a^{2} b \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}^{2}{\left (c + d x \right )}}{4 d} - \frac {3 a^{2} b \operatorname {sech}^{2}{\left (c + d x \right )}}{4 d} - \frac {a b^{2} \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}^{4}{\left (c + d x \right )}}{2 d} - \frac {a b^{2} \operatorname {sech}^{4}{\left (c + d x \right )}}{4 d} - \frac {b^{3} \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}^{6}{\left (c + d x \right )}}{8 d} - \frac {b^{3} \operatorname {sech}^{6}{\left (c + d x \right )}}{24 d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {sech}^{2}{\left (c \right )}\right )^{3} \tanh ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((a**3*x - a**3*log(tanh(c + d*x) + 1)/d - a**3*tanh(c + d*x)**2/ (2*d) - 3*a**2*b*tanh(c + d*x)**2*sech(c + d*x)**2/(4*d) - 3*a**2*b*sech(c + d*x)**2/(4*d) - a*b**2*tanh(c + d*x)**2*sech(c + d*x)**4/(2*d) - a*b**2 *sech(c + d*x)**4/(4*d) - b**3*tanh(c + d*x)**2*sech(c + d*x)**6/(8*d) - b **3*sech(c + d*x)**6/(24*d), Ne(d, 0)), (x*(a + b*sech(c)**2)**3*tanh(c)** 3, True))
Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (95) = 190\).
Time = 0.29 (sec) , antiderivative size = 652, normalized size of antiderivative = 6.33 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=\frac {3 \, a^{2} b \tanh \left (d x + c\right )^{4}}{4 \, d} + a^{3} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} - 4 \, a b^{2} {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, 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 )}} - \frac {2 \, e^{\left (-6 \, d x - 6 \, 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 )}} + \frac {3 \, e^{\left (-8 \, d x - 8 \, 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 {32}{3} \, b^{3} {\left (\frac {e^{\left (-6 \, d x - 6 \, 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 )}} - \frac {e^{\left (-8 \, d x - 8 \, 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 )}} + \frac {e^{\left (-10 \, d x - 10 \, 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 )} \]
3/4*a^2*b*tanh(d*x + c)^4/d + a^3*(x + c/d + log(e^(-2*d*x - 2*c) + 1)/d + 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1))) - 4*a *b^2*(3*e^(-4*d*x - 4*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)) - 2*e^(-6*d*x - 6*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*e^(-8*d*x - 8*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))) - 32/3*b^3*(e^(-6*d*x - 6*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)) - e^(-8*d*x - 8*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)) + e^(-10*d*x - 10*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 - 1 0*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)))
Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (95) = 190\).
Time = 0.38 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.76 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=-\frac {840 \, {\left (d x + c\right )} a^{3} - 840 \, a^{3} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac {2283 \, a^{3} e^{\left (16 \, d x + 16 \, c\right )} + 16584 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} + 5040 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 53844 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 20160 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 10080 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 102648 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 35280 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 13440 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 8960 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 126210 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 40320 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 6720 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 8960 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 102648 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 35280 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 13440 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 8960 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 53844 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 20160 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 10080 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 16584 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5040 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2283 \, a^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{840 \, d} \]
-1/840*(840*(d*x + c)*a^3 - 840*a^3*log(e^(2*d*x + 2*c) + 1) + (2283*a^3*e ^(16*d*x + 16*c) + 16584*a^3*e^(14*d*x + 14*c) + 5040*a^2*b*e^(14*d*x + 14 *c) + 53844*a^3*e^(12*d*x + 12*c) + 20160*a^2*b*e^(12*d*x + 12*c) + 10080* a*b^2*e^(12*d*x + 12*c) + 102648*a^3*e^(10*d*x + 10*c) + 35280*a^2*b*e^(10 *d*x + 10*c) + 13440*a*b^2*e^(10*d*x + 10*c) + 8960*b^3*e^(10*d*x + 10*c) + 126210*a^3*e^(8*d*x + 8*c) + 40320*a^2*b*e^(8*d*x + 8*c) + 6720*a*b^2*e^ (8*d*x + 8*c) - 8960*b^3*e^(8*d*x + 8*c) + 102648*a^3*e^(6*d*x + 6*c) + 35 280*a^2*b*e^(6*d*x + 6*c) + 13440*a*b^2*e^(6*d*x + 6*c) + 8960*b^3*e^(6*d* x + 6*c) + 53844*a^3*e^(4*d*x + 4*c) + 20160*a^2*b*e^(4*d*x + 4*c) + 10080 *a*b^2*e^(4*d*x + 4*c) + 16584*a^3*e^(2*d*x + 2*c) + 5040*a^2*b*e^(2*d*x + 2*c) + 2283*a^3)/(e^(2*d*x + 2*c) + 1)^8)/d
Time = 2.29 (sec) , antiderivative size = 573, normalized size of antiderivative = 5.56 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx=\frac {32\,\left (3\,a\,b^2-5\,b^3\right )}{d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-a^3\,x-\frac {128\,b^3}{d\,\left (7\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}+35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}+7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}+1\right )}-\frac {32\,\left (3\,a\,b^2-19\,b^3\right )}{3\,d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}+\frac {32\,b^3}{d\,\left (8\,{\mathrm {e}}^{2\,c+2\,d\,x}+28\,{\mathrm {e}}^{4\,c+4\,d\,x}+56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}+56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}+8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )}-\frac {8\,\left (9\,a^2\,b-21\,a\,b^2+4\,b^3\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {4\,\left (3\,a^2\,b-27\,a\,b^2+16\,b^3\right )}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {2\,\left (3\,a^2\,b-a^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {a^3\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1\right )}{d}-\frac {2\,\left (a^3-9\,a^2\,b+6\,a\,b^2\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
(32*(3*a*b^2 - 5*b^3))/(d*(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*e xp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1)) - a^3*x - (128*b^3)/(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)) - (32*(3*a*b^2 - 19*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*ex p(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1)) + (32*b^3)/(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(1 6*c + 16*d*x) + 1)) - (8*(9*a^2*b - 21*a*b^2 + 4*b^3))/(3*d*(3*exp(2*c + 2 *d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (4*(3*a^2*b - 27*a*b ^2 + 16*b^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)) - (2*(3*a^2*b - a^3))/(d*(exp(2*c + 2*d*x) + 1)) + (a^3*log(exp(2*c)*exp(2*d*x) + 1))/d - (2*(6*a*b^2 - 9*a^2*b + a^3) )/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))