Integrand size = 23, antiderivative size = 77 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {(a+b) (a+3 b) \cosh (c+d x)}{d}+\frac {(a+b)^2 \cosh ^3(c+d x)}{3 d}-\frac {b (2 a+3 b) \text {sech}(c+d x)}{d}+\frac {b^2 \text {sech}^3(c+d x)}{3 d} \] Output:
-(a+b)*(a+3*b)*cosh(d*x+c)/d+1/3*(a+b)^2*cosh(d*x+c)^3/d-b*(2*a+3*b)*sech( d*x+c)/d+1/3*b^2*sech(d*x+c)^3/d
Time = 1.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {-3 \left (3 a^2+14 a b+11 b^2\right ) \cosh (c+d x)+(a+b)^2 \cosh (3 (c+d x))+4 b \text {sech}(c+d x) \left (-6 a-9 b+b \text {sech}^2(c+d x)\right )}{12 d} \] Input:
Integrate[Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2,x]
Output:
(-3*(3*a^2 + 14*a*b + 11*b^2)*Cosh[c + d*x] + (a + b)^2*Cosh[3*(c + d*x)] + 4*b*Sech[c + d*x]*(-6*a - 9*b + b*Sech[c + d*x]^2))/(12*d)
Time = 0.34 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 26, 4147, 25, 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.
| \(\displaystyle \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int i \sin (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )^2dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \sin (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )^2dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle \frac {\int -\cosh ^4(c+d x) \left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )^2d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \cosh ^4(c+d x) \left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )^2d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 355 |
| \(\displaystyle -\frac {\int \left ((a+b)^2 \cosh ^4(c+d x)+(-a-3 b) (a+b) \cosh ^2(c+d x)-b^2 \text {sech}^2(c+d x)+b (2 a+3 b)\right )d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{3} (a+b)^2 \cosh ^3(c+d x)-(a+b) (a+3 b) \cosh (c+d x)-b (2 a+3 b) \text {sech}(c+d x)+\frac {1}{3} b^2 \text {sech}^3(c+d x)}{d}\) |
Input:
Int[Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2,x]
Output:
(-((a + b)*(a + 3*b)*Cosh[c + d*x]) + ((a + b)^2*Cosh[c + d*x]^3)/3 - b*(2 *a + 3*b)*Sech[c + d*x] + (b^2*Sech[c + d*x]^3)/3)/d
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[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. \(147\) vs. \(2(73)=146\).
Time = 4.14 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.92
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+2 a b \left (\frac {\sinh \left (d x +c \right )^{4}}{3 \cosh \left (d x +c \right )}-\frac {4 \sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )}-\frac {8}{3 \cosh \left (d x +c \right )}\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{6}}{3 \cosh \left (d x +c \right )^{3}}-\frac {2 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{3}}-\frac {8 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{3}}-\frac {16}{3 \cosh \left (d x +c \right )^{3}}\right )}{d}\) | \(148\) |
| default | \(\frac {a^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+2 a b \left (\frac {\sinh \left (d x +c \right )^{4}}{3 \cosh \left (d x +c \right )}-\frac {4 \sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )}-\frac {8}{3 \cosh \left (d x +c \right )}\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{6}}{3 \cosh \left (d x +c \right )^{3}}-\frac {2 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{3}}-\frac {8 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{3}}-\frac {16}{3 \cosh \left (d x +c \right )^{3}}\right )}{d}\) | \(148\) |
| risch | \(\frac {{\mathrm e}^{3 d x +3 c} a^{2}}{24 d}+\frac {{\mathrm e}^{3 d x +3 c} a b}{12 d}+\frac {{\mathrm e}^{3 d x +3 c} b^{2}}{24 d}-\frac {3 \,{\mathrm e}^{d x +c} a^{2}}{8 d}-\frac {7 \,{\mathrm e}^{d x +c} a b}{4 d}-\frac {11 \,{\mathrm e}^{d x +c} b^{2}}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2}}{8 d}-\frac {7 \,{\mathrm e}^{-d x -c} a b}{4 d}-\frac {11 \,{\mathrm e}^{-d x -c} b^{2}}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{2}}{24 d}+\frac {{\mathrm e}^{-3 d x -3 c} a b}{12 d}+\frac {{\mathrm e}^{-3 d x -3 c} b^{2}}{24 d}-\frac {2 b \,{\mathrm e}^{d x +c} \left (6 \,{\mathrm e}^{4 d x +4 c} a +9 b \,{\mathrm e}^{4 d x +4 c}+12 \,{\mathrm e}^{2 d x +2 c} a +14 \,{\mathrm e}^{2 d x +2 c} b +6 a +9 b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3}}\) | \(273\) |
Input:
int(sinh(d*x+c)^3*(a+tanh(d*x+c)^2*b)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^2*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c)+2*a*b*(1/3*sinh(d*x+c)^4/cos h(d*x+c)-4/3*sinh(d*x+c)^2/cosh(d*x+c)-8/3/cosh(d*x+c))+b^2*(1/3*sinh(d*x+ c)^6/cosh(d*x+c)^3-2*sinh(d*x+c)^4/cosh(d*x+c)^3-8*sinh(d*x+c)^2/cosh(d*x+ c)^3-16/3/cosh(d*x+c)^3))
Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (73) = 146\).
Time = 0.11 (sec) , antiderivative size = 259, normalized size of antiderivative = 3.36 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{6} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{6} - 6 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} - 2 \, a^{2} - 12 \, a b - 10 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} - 3 \, {\left (11 \, a^{2} + 86 \, a b + 91 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} - 12 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} - 11 \, a^{2} - 86 \, a b - 91 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - 26 \, a^{2} - 220 \, a b - 210 \, b^{2}}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \] Input:
integrate(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")
Output:
1/24*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^6 - 6*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^4 + 3*(5*(a^2 + 2*a*b + b^2) *cosh(d*x + c)^2 - 2*a^2 - 12*a*b - 10*b^2)*sinh(d*x + c)^4 - 3*(11*a^2 + 86*a*b + 91*b^2)*cosh(d*x + c)^2 + 3*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^ 4 - 12*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^2 - 11*a^2 - 86*a*b - 91*b^2)*s inh(d*x + c)^2 - 26*a^2 - 220*a*b - 210*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))
\[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2} \sinh ^{3}{\left (c + d x \right )}\, dx \] Input:
integrate(sinh(d*x+c)**3*(a+b*tanh(d*x+c)**2)**2,x)
Output:
Integral((a + b*tanh(c + d*x)**2)**2*sinh(c + d*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (73) = 146\).
Time = 0.05 (sec) , antiderivative size = 265, normalized size of antiderivative = 3.44 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {1}{24} \, b^{2} {\left (\frac {33 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {30 \, e^{\left (-2 \, d x - 2 \, c\right )} + 240 \, e^{\left (-4 \, d x - 4 \, c\right )} + 322 \, e^{\left (-6 \, d x - 6 \, c\right )} + 177 \, e^{\left (-8 \, d x - 8 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )} + e^{\left (-9 \, d x - 9 \, c\right )}\right )}}\right )} - \frac {1}{12} \, a b {\left (\frac {21 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {20 \, e^{\left (-2 \, d x - 2 \, c\right )} + 69 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \frac {1}{24} \, a^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \] Input:
integrate(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")
Output:
-1/24*b^2*((33*e^(-d*x - c) - e^(-3*d*x - 3*c))/d + (30*e^(-2*d*x - 2*c) + 240*e^(-4*d*x - 4*c) + 322*e^(-6*d*x - 6*c) + 177*e^(-8*d*x - 8*c) - 1)/( d*(e^(-3*d*x - 3*c) + 3*e^(-5*d*x - 5*c) + 3*e^(-7*d*x - 7*c) + e^(-9*d*x - 9*c)))) - 1/12*a*b*((21*e^(-d*x - c) - e^(-3*d*x - 3*c))/d + (20*e^(-2*d *x - 2*c) + 69*e^(-4*d*x - 4*c) - 1)/(d*(e^(-3*d*x - 3*c) + e^(-5*d*x - 5* c)))) + 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)
Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (73) = 146\).
Time = 0.18 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.66 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 2 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 12 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 48 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 36 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - \frac {16 \, {\left (6 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 9 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4 \, b^{2}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}}}{24 \, d} \] Input:
integrate(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")
Output:
1/24*(a^2*(e^(d*x + c) + e^(-d*x - c))^3 + 2*a*b*(e^(d*x + c) + e^(-d*x - c))^3 + b^2*(e^(d*x + c) + e^(-d*x - c))^3 - 12*a^2*(e^(d*x + c) + e^(-d*x - c)) - 48*a*b*(e^(d*x + c) + e^(-d*x - c)) - 36*b^2*(e^(d*x + c) + e^(-d *x - c)) - 16*(6*a*b*(e^(d*x + c) + e^(-d*x - c))^2 + 9*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 = 0.34 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.79 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,{\left (a+b\right )}^2}{24\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a^2+14\,a\,b+11\,b^2\right )}{8\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,{\left (a+b\right )}^2}{24\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a^2+14\,a\,b+11\,b^2\right )}{8\,d}-\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (3\,b^2+2\,a\,b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \] Input:
int(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^2)^2,x)
Output:
(exp(- 3*c - 3*d*x)*(a + b)^2)/(24*d) - (exp(c + d*x)*(14*a*b + 3*a^2 + 11 *b^2))/(8*d) + (exp(3*c + 3*d*x)*(a + b)^2)/(24*d) - (exp(- c - d*x)*(14*a *b + 3*a^2 + 11*b^2))/(8*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 + 3*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))
Time = 0.26 (sec) , antiderivative size = 328, normalized size of antiderivative = 4.26 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {e^{12 d x +12 c} a^{2}+2 e^{12 d x +12 c} a b +e^{12 d x +12 c} b^{2}-6 e^{10 d x +10 c} a^{2}-36 e^{10 d x +10 c} a b -30 e^{10 d x +10 c} b^{2}-33 e^{8 d x +8 c} a^{2}-258 e^{8 d x +8 c} a b -273 e^{8 d x +8 c} b^{2}-52 e^{6 d x +6 c} a^{2}-440 e^{6 d x +6 c} a b -420 e^{6 d x +6 c} b^{2}-33 e^{4 d x +4 c} a^{2}-258 e^{4 d x +4 c} a b -273 e^{4 d x +4 c} b^{2}-6 e^{2 d x +2 c} a^{2}-36 e^{2 d x +2 c} a b -30 e^{2 d x +2 c} b^{2}+a^{2}+2 a b +b^{2}}{24 e^{3 d x +3 c} d \left (e^{6 d x +6 c}+3 e^{4 d x +4 c}+3 e^{2 d x +2 c}+1\right )} \] Input:
int(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^2)^2,x)
Output:
(e**(12*c + 12*d*x)*a**2 + 2*e**(12*c + 12*d*x)*a*b + e**(12*c + 12*d*x)*b **2 - 6*e**(10*c + 10*d*x)*a**2 - 36*e**(10*c + 10*d*x)*a*b - 30*e**(10*c + 10*d*x)*b**2 - 33*e**(8*c + 8*d*x)*a**2 - 258*e**(8*c + 8*d*x)*a*b - 273 *e**(8*c + 8*d*x)*b**2 - 52*e**(6*c + 6*d*x)*a**2 - 440*e**(6*c + 6*d*x)*a *b - 420*e**(6*c + 6*d*x)*b**2 - 33*e**(4*c + 4*d*x)*a**2 - 258*e**(4*c + 4*d*x)*a*b - 273*e**(4*c + 4*d*x)*b**2 - 6*e**(2*c + 2*d*x)*a**2 - 36*e**( 2*c + 2*d*x)*a*b - 30*e**(2*c + 2*d*x)*b**2 + a**2 + 2*a*b + b**2)/(24*e** (3*c + 3*d*x)*d*(e**(6*c + 6*d*x) + 3*e**(4*c + 4*d*x) + 3*e**(2*c + 2*d*x ) + 1))