Integrand size = 21, antiderivative size = 70 \[ \int \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {(a+b)^3 \cosh (c+d x)}{d}+\frac {3 b (a+b)^2 \text {sech}(c+d x)}{d}-\frac {b^2 (a+b) \text {sech}^3(c+d x)}{d}+\frac {b^3 \text {sech}^5(c+d x)}{5 d} \]
(a+b)^3*cosh(d*x+c)/d+3*b*(a+b)^2*sech(d*x+c)/d-b^2*(a+b)*sech(d*x+c)^3/d+ 1/5*b^3*sech(d*x+c)^5/d
Time = 0.55 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.89 \[ \int \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {(a+b)^3 \cosh (c+d x)+3 b (a+b)^2 \text {sech}(c+d x)-b^2 (a+b) \text {sech}^3(c+d x)+\frac {1}{5} b^3 \text {sech}^5(c+d x)}{d} \]
((a + b)^3*Cosh[c + d*x] + 3*b*(a + b)^2*Sech[c + d*x] - b^2*(a + b)*Sech[ c + d*x]^3 + (b^3*Sech[c + d*x]^5)/5)/d
Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 26, 4147, 244, 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.
| \(\displaystyle \int \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -i \sin (i c+i d x) \left (a-b \tan (i c+i d x)^2\right )^3dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \sin (i c+i d x) \left (a-b \tan (i c+i d x)^2\right )^3dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle -\frac {\int \cosh ^2(c+d x) \left (-b \text {sech}^2(c+d x)+a+b\right )^3d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -\frac {\int \left (-b^3 \text {sech}^4(c+d x)+3 b^2 (a+b) \text {sech}^2(c+d x)-3 b (a+b)^2+(a+b)^3 \cosh ^2(c+d x)\right )d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^2 (a+b) \text {sech}^3(c+d x)-(a+b)^3 \cosh (c+d x)-3 b (a+b)^2 \text {sech}(c+d x)-\frac {1}{5} b^3 \text {sech}^5(c+d x)}{d}\) |
-((-((a + b)^3*Cosh[c + d*x]) - 3*b*(a + b)^2*Sech[c + d*x] + b^2*(a + b)* Sech[c + d*x]^3 - (b^3*Sech[c + d*x]^5)/5)/d)
3.1.20.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Simp[1/(f*ff^ m) Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a - b + b*ff^2*x^2)^p/x^(m + 1 )), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[( m - 1)/2]
Leaf count of result is larger than twice the leaf count of optimal. \(169\) vs. \(2(68)=136\).
Time = 1.81 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.43
| method | result | size |
| derivativedivides | \(\frac {a^{3} \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )}+\frac {2}{\cosh \left (d x +c \right )}\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{3}}+\frac {4 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{3}}+\frac {8}{3 \cosh \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{6}}{\cosh \left (d x +c \right )^{5}}+\frac {6 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}+\frac {8 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{5}}+\frac {16}{5 \cosh \left (d x +c \right )^{5}}\right )}{d}\) | \(170\) |
| default | \(\frac {a^{3} \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )}+\frac {2}{\cosh \left (d x +c \right )}\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{3}}+\frac {4 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{3}}+\frac {8}{3 \cosh \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{6}}{\cosh \left (d x +c \right )^{5}}+\frac {6 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}+\frac {8 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{5}}+\frac {16}{5 \cosh \left (d x +c \right )^{5}}\right )}{d}\) | \(170\) |
| risch | \(\frac {{\mathrm e}^{d x +c} a^{3}}{2 d}+\frac {3 \,{\mathrm e}^{d x +c} a^{2} b}{2 d}+\frac {3 \,{\mathrm e}^{d x +c} a \,b^{2}}{2 d}+\frac {b^{3} {\mathrm e}^{d x +c}}{2 d}+\frac {{\mathrm e}^{-d x -c} a^{3}}{2 d}+\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{2 d}+\frac {3 \,{\mathrm e}^{-d x -c} a \,b^{2}}{2 d}+\frac {{\mathrm e}^{-d x -c} b^{3}}{2 d}+\frac {2 b \,{\mathrm e}^{d x +c} \left (15 a^{2} {\mathrm e}^{8 d x +8 c}+30 a b \,{\mathrm e}^{8 d x +8 c}+15 b^{2} {\mathrm e}^{8 d x +8 c}+60 a^{2} {\mathrm e}^{6 d x +6 c}+100 a b \,{\mathrm e}^{6 d x +6 c}+40 b^{2} {\mathrm e}^{6 d x +6 c}+90 a^{2} {\mathrm e}^{4 d x +4 c}+140 a b \,{\mathrm e}^{4 d x +4 c}+66 \,{\mathrm e}^{4 d x +4 c} b^{2}+60 a^{2} {\mathrm e}^{2 d x +2 c}+100 a b \,{\mathrm e}^{2 d x +2 c}+40 \,{\mathrm e}^{2 d x +2 c} b^{2}+15 a^{2}+30 a b +15 b^{2}\right )}{5 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{5}}\) | \(334\) |
1/d*(a^3*cosh(d*x+c)+3*a^2*b*(sinh(d*x+c)^2/cosh(d*x+c)+2/cosh(d*x+c))+3*a *b^2*(sinh(d*x+c)^4/cosh(d*x+c)^3+4*sinh(d*x+c)^2/cosh(d*x+c)^3+8/3/cosh(d *x+c)^3)+b^3*(sinh(d*x+c)^6/cosh(d*x+c)^5+6*sinh(d*x+c)^4/cosh(d*x+c)^5+8* sinh(d*x+c)^2/cosh(d*x+c)^5+16/5/cosh(d*x+c)^5))
Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (68) = 136\).
Time = 0.28 (sec) , antiderivative size = 383, normalized size of antiderivative = 5.47 \[ \int \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{6} + 5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{6} + 30 \, {\left (a^{3} + 5 \, a^{2} b + 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 15 \, {\left (2 \, a^{3} + 10 \, a^{2} b + 14 \, a b^{2} + 6 \, b^{3} + 5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} + 50 \, a^{3} + 330 \, a^{2} b + 430 \, a b^{2} + 182 \, b^{3} + 5 \, {\left (15 \, a^{3} + 93 \, a^{2} b + 125 \, a b^{2} + 47 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 5 \, {\left (15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{4} + 15 \, a^{3} + 93 \, a^{2} b + 125 \, a b^{2} + 47 \, b^{3} + 36 \, {\left (a^{3} + 5 \, a^{2} b + 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2}}{10 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )\right )}} \]
1/10*(5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^6 + 30*(a^3 + 5*a^2*b + 7*a*b^2 + 3*b^3)*c osh(d*x + c)^4 + 15*(2*a^3 + 10*a^2*b + 14*a*b^2 + 6*b^3 + 5*(a^3 + 3*a^2* b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 50*a^3 + 330*a^2*b + 430*a*b^2 + 182*b^3 + 5*(15*a^3 + 93*a^2*b + 125*a*b^2 + 47*b^3)*cosh(d*x + c)^2 + 5*(15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 15*a^3 + 93*a^2*b + 125*a*b^2 + 47*b^3 + 36*(a^3 + 5*a^2*b + 7*a*b^2 + 3*b^3)*cosh (d*x + c)^2)*sinh(d*x + c)^2)/(d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)*sinh( d*x + c)^4 + 5*d*cosh(d*x + c)^3 + 5*(2*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + 10*d*cosh(d*x + c))
\[ \int \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \sinh {\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (68) = 136\).
Time = 0.20 (sec) , antiderivative size = 321, normalized size of antiderivative = 4.59 \[ \int \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {1}{10} \, b^{3} {\left (\frac {5 \, e^{\left (-d x - c\right )}}{d} + \frac {85 \, e^{\left (-2 \, d x - 2 \, c\right )} + 210 \, e^{\left (-4 \, d x - 4 \, c\right )} + 314 \, e^{\left (-6 \, d x - 6 \, c\right )} + 185 \, e^{\left (-8 \, d x - 8 \, c\right )} + 65 \, e^{\left (-10 \, d x - 10 \, c\right )} + 5}{d {\left (e^{\left (-d x - c\right )} + 5 \, e^{\left (-3 \, d x - 3 \, c\right )} + 10 \, e^{\left (-5 \, d x - 5 \, c\right )} + 10 \, e^{\left (-7 \, d x - 7 \, c\right )} + 5 \, e^{\left (-9 \, d x - 9 \, c\right )} + e^{\left (-11 \, d x - 11 \, c\right )}\right )}}\right )} + \frac {1}{2} \, a b^{2} {\left (\frac {3 \, e^{\left (-d x - c\right )}}{d} + \frac {33 \, e^{\left (-2 \, d x - 2 \, c\right )} + 41 \, e^{\left (-4 \, d x - 4 \, c\right )} + 27 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3}{d {\left (e^{\left (-d x - c\right )} + 3 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )}\right )}}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {e^{\left (-d x - c\right )}}{d} + \frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}\right )} + \frac {a^{3} \cosh \left (d x + c\right )}{d} \]
1/10*b^3*(5*e^(-d*x - c)/d + (85*e^(-2*d*x - 2*c) + 210*e^(-4*d*x - 4*c) + 314*e^(-6*d*x - 6*c) + 185*e^(-8*d*x - 8*c) + 65*e^(-10*d*x - 10*c) + 5)/ (d*(e^(-d*x - c) + 5*e^(-3*d*x - 3*c) + 10*e^(-5*d*x - 5*c) + 10*e^(-7*d*x - 7*c) + 5*e^(-9*d*x - 9*c) + e^(-11*d*x - 11*c)))) + 1/2*a*b^2*(3*e^(-d* x - c)/d + (33*e^(-2*d*x - 2*c) + 41*e^(-4*d*x - 4*c) + 27*e^(-6*d*x - 6*c ) + 3)/(d*(e^(-d*x - c) + 3*e^(-3*d*x - 3*c) + 3*e^(-5*d*x - 5*c) + e^(-7* d*x - 7*c)))) + 3/2*a^2*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)))) + a^3*cosh(d*x + c)/d
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (68) = 136\).
Time = 0.41 (sec) , antiderivative size = 236, normalized size of antiderivative = 3.37 \[ \int \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {5 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 15 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 15 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 5 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + \frac {4 \, {\left (15 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} + 30 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} + 15 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} - 20 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 20 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 16 \, b^{3}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5}}}{10 \, d} \]
1/10*(5*a^3*(e^(d*x + c) + e^(-d*x - c)) + 15*a^2*b*(e^(d*x + c) + e^(-d*x - c)) + 15*a*b^2*(e^(d*x + c) + e^(-d*x - c)) + 5*b^3*(e^(d*x + c) + e^(- d*x - c)) + 4*(15*a^2*b*(e^(d*x + c) + e^(-d*x - c))^4 + 30*a*b^2*(e^(d*x + c) + e^(-d*x - c))^4 + 15*b^3*(e^(d*x + c) + e^(-d*x - c))^4 - 20*a*b^2* (e^(d*x + c) + e^(-d*x - c))^2 - 20*b^3*(e^(d*x + c) + e^(-d*x - c))^2 + 1 6*b^3)/(e^(d*x + c) + e^(-d*x - c))^5)/d
Time = 1.87 (sec) , antiderivative size = 308, normalized size of antiderivative = 4.40 \[ \int \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {{\mathrm {e}}^{c+d\,x}\,{\left (a+b\right )}^3}{2\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,{\left (a+b\right )}^3}{2\,d}+\frac {6\,{\mathrm {e}}^{c+d\,x}\,\left (a^2\,b+2\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {64\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (9\,b^3+5\,a\,b^2\right )}{5\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (b^3+a\,b^2\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
(exp(c + d*x)*(a + b)^3)/(2*d) + (exp(- c - d*x)*(a + b)^3)/(2*d) + (6*exp (c + d*x)*(2*a*b^2 + a^2*b + b^3))/(d*(exp(2*c + 2*d*x) + 1)) - (64*b^3*ex p(c + d*x))/(5*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)) + (8*exp(c + d*x)*(5*a*b^2 + 9*b^3))/(5*d*(3 *exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (32*b^3* exp(c + d*x))/(5*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)) - (8*exp(c + d*x) *(a*b^2 + b^3))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))