Integrand size = 23, antiderivative size = 72 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^2 \sinh ^3(c+d x) \, dx=-\frac {a (a-2 b) \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}+\frac {(2 a-b) b \text {sech}(c+d x)}{d}+\frac {b^2 \text {sech}^3(c+d x)}{3 d} \]
-a*(a-2*b)*cosh(d*x+c)/d+1/3*a^2*cosh(d*x+c)^3/d+(2*a-b)*b*sech(d*x+c)/d+1 /3*b^2*sech(d*x+c)^3/d
Time = 3.61 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.15 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^2 \sinh ^3(c+d x) \, dx=\frac {\left (-26 a^2+168 a b-16 b^2-3 \left (11 a^2-64 a b+16 b^2\right ) \cosh (2 (c+d x))-6 a (a-4 b) \cosh (4 (c+d x))+a^2 \cosh (6 (c+d x))\right ) \text {sech}^3(c+d x)}{96 d} \]
((-26*a^2 + 168*a*b - 16*b^2 - 3*(11*a^2 - 64*a*b + 16*b^2)*Cosh[2*(c + d* x)] - 6*a*(a - 4*b)*Cosh[4*(c + d*x)] + a^2*Cosh[6*(c + d*x)])*Sech[c + d* x]^3)/(96*d)
Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 26, 4621, 355, 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.
\(\displaystyle \int \sinh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \sin (i c+i d x)^3 \left (a+b \sec (i c+i d x)^2\right )^2dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \left (b \sec (i c+i d x)^2+a\right )^2 \sin (i c+i d x)^3dx\) |
\(\Big \downarrow \) 4621 |
\(\displaystyle -\frac {\int \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2 \text {sech}^4(c+d x)d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 355 |
\(\displaystyle -\frac {\int \left (b^2 \text {sech}^4(c+d x)+(2 a-b) b \text {sech}^2(c+d x)-a^2 \cosh ^2(c+d x)+a (a-2 b)\right )d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {1}{3} a^2 \cosh ^3(c+d x)+a (a-2 b) \cosh (c+d x)-b (2 a-b) \text {sech}(c+d x)-\frac {1}{3} b^2 \text {sech}^3(c+d x)}{d}\) |
-((a*(a - 2*b)*Cosh[c + d*x] - (a^2*Cosh[c + d*x]^3)/3 - (2*a - b)*b*Sech[ c + d*x] - (b^2*Sech[c + d*x]^3)/3)/d)
3.1.10.3.1 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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] & & IGtQ[q, 0]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/ 2] && IntegerQ[n] && IntegerQ[p]
Time = 27.89 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {b^{2} \operatorname {sech}\left (d x +c \right )^{3}}{3}+2 a b \,\operatorname {sech}\left (d x +c \right )-b^{2} \operatorname {sech}\left (d x +c \right )+\frac {a^{2}}{3 \operatorname {sech}\left (d x +c \right )^{3}}-\frac {a \left (a -2 b \right )}{\operatorname {sech}\left (d x +c \right )}}{d}\) | \(69\) |
default | \(\frac {\frac {b^{2} \operatorname {sech}\left (d x +c \right )^{3}}{3}+2 a b \,\operatorname {sech}\left (d x +c \right )-b^{2} \operatorname {sech}\left (d x +c \right )+\frac {a^{2}}{3 \operatorname {sech}\left (d x +c \right )^{3}}-\frac {a \left (a -2 b \right )}{\operatorname {sech}\left (d x +c \right )}}{d}\) | \(69\) |
parts | \(\frac {a^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}+\frac {b^{2} \left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{3}-\operatorname {sech}\left (d x +c \right )\right )}{d}+\frac {2 a b \left (\frac {\sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )}+\frac {2}{\cosh \left (d x +c \right )}\right )}{d}\) | \(88\) |
risch | \(\frac {a^{2} {\mathrm e}^{3 d x +3 c}}{24 d}-\frac {3 a^{2} {\mathrm e}^{d x +c}}{8 d}+\frac {{\mathrm e}^{d x +c} a b}{d}-\frac {3 a^{2} {\mathrm e}^{-d x -c}}{8 d}+\frac {a \,{\mathrm e}^{-d x -c} b}{d}+\frac {a^{2} {\mathrm e}^{-3 d x -3 c}}{24 d}+\frac {2 \,{\mathrm e}^{d x +c} b \left (6 a \,{\mathrm e}^{4 d x +4 c}-3 b \,{\mathrm e}^{4 d x +4 c}+12 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+6 a -3 b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3}}\) | \(174\) |
1/d*(1/3*b^2*sech(d*x+c)^3+2*a*b*sech(d*x+c)-b^2*sech(d*x+c)+1/3*a^2/sech( d*x+c)^3-a*(a-2*b)/sech(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (68) = 136\).
Time = 0.25 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.94 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^2 \sinh ^3(c+d x) \, dx=\frac {a^{2} \cosh \left (d x + c\right )^{6} + a^{2} \sinh \left (d x + c\right )^{6} - 6 \, {\left (a^{2} - 4 \, a b\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, a^{2} \cosh \left (d x + c\right )^{2} - 2 \, a^{2} + 8 \, a b\right )} \sinh \left (d x + c\right )^{4} - 3 \, {\left (11 \, a^{2} - 64 \, a b + 16 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, a^{2} \cosh \left (d x + c\right )^{4} - 12 \, {\left (a^{2} - 4 \, a b\right )} \cosh \left (d x + c\right )^{2} - 11 \, a^{2} + 64 \, a b - 16 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - 26 \, a^{2} + 168 \, a b - 16 \, b^{2}}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \]
1/24*(a^2*cosh(d*x + c)^6 + a^2*sinh(d*x + c)^6 - 6*(a^2 - 4*a*b)*cosh(d*x + c)^4 + 3*(5*a^2*cosh(d*x + c)^2 - 2*a^2 + 8*a*b)*sinh(d*x + c)^4 - 3*(1 1*a^2 - 64*a*b + 16*b^2)*cosh(d*x + c)^2 + 3*(5*a^2*cosh(d*x + c)^4 - 12*( a^2 - 4*a*b)*cosh(d*x + c)^2 - 11*a^2 + 64*a*b - 16*b^2)*sinh(d*x + c)^2 - 26*a^2 + 168*a*b - 16*b^2)/(d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c)*sinh(d* x + c)^2 + 3*d*cosh(d*x + c))
\[ \int \left (a+b \text {sech}^2(c+d x)\right )^2 \sinh ^3(c+d x) \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \sinh ^{3}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (68) = 136\).
Time = 0.20 (sec) , antiderivative size = 266, normalized size of antiderivative = 3.69 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^2 \sinh ^3(c+d x) \, dx=\frac {1}{24} \, a^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + a b {\left (\frac {e^{\left (-d x - c\right )}}{d} + \frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}\right )} - \frac {2}{3} \, b^{2} {\left (\frac {3 \, e^{\left (-d x - c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {2 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} \]
1/24*a^2*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d *x - 3*c)/d) + a*b*(e^(-d*x - c)/d + (5*e^(-2*d*x - 2*c) + 1)/(d*(e^(-d*x - c) + e^(-3*d*x - 3*c)))) - 2/3*b^2*(3*e^(-d*x - c)/(d*(3*e^(-2*d*x - 2*c ) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1)) + 2*e^(-3*d*x - 3*c)/(d*(3 *e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1)) + 3*e^(-5* d*x - 5*c)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1)))
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (68) = 136\).
Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.94 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^2 \sinh ^3(c+d x) \, dx=\frac {a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 12 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 24 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + \frac {16 \, {\left (6 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 3 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 4 \, b^{2}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}}}{24 \, d} \]
1/24*(a^2*(e^(d*x + c) + e^(-d*x - c))^3 - 12*a^2*(e^(d*x + c) + e^(-d*x - c)) + 24*a*b*(e^(d*x + c) + e^(-d*x - c)) + 16*(6*a*b*(e^(d*x + c) + e^(- d*x - c))^2 - 3*b^2*(e^(d*x + c) + e^(-d*x - c))^2 + 4*b^2)/(e^(d*x + c) + e^(-d*x - c))^3)/d
Time = 2.13 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.79 \[ \int \left (a+b \text {sech}^2(c+d x)\right )^2 \sinh ^3(c+d x) \, dx=\frac {{\mathrm {e}}^{c+d\,x}\,\left (8\,a\,b-3\,a^2\right )}{8\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (8\,a\,b-3\,a^2\right )}{8\,d}+\frac {a^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}+\frac {a^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a\,b-b^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
(exp(c + d*x)*(8*a*b - 3*a^2))/(8*d) + (exp(- c - d*x)*(8*a*b - 3*a^2))/(8 *d) + (a^2*exp(- 3*c - 3*d*x))/(24*d) + (a^2*exp(3*c + 3*d*x))/(24*d) - (8 *b^2*exp(c + d*x))/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (2*exp(c + d*x)*(2*a*b - b^2))/(d*(exp(2*c + 2*d*x) + 1) ) + (8*b^2*exp(c + d*x))/(3*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))