Integrand size = 23, antiderivative size = 85 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=-\frac {(a-b)^2 \cosh (c+d x)}{d}+\frac {(a-3 b) (a-b) \cosh ^3(c+d x)}{3 d}+\frac {(2 a-3 b) b \cosh ^5(c+d x)}{5 d}+\frac {b^2 \cosh ^7(c+d x)}{7 d} \]
-(a-b)^2*cosh(d*x+c)/d+1/3*(a-3*b)*(a-b)*cosh(d*x+c)^3/d+1/5*(2*a-3*b)*b*c osh(d*x+c)^5/d+1/7*b^2*cosh(d*x+c)^7/d
Time = 0.19 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.81 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=-\frac {3 a^2 \cosh (c+d x)}{4 d}+\frac {5 a b \cosh (c+d x)}{4 d}-\frac {35 b^2 \cosh (c+d x)}{64 d}+\frac {a^2 \cosh (3 (c+d x))}{12 d}-\frac {5 a b \cosh (3 (c+d x))}{24 d}+\frac {7 b^2 \cosh (3 (c+d x))}{64 d}+\frac {a b \cosh (5 (c+d x))}{40 d}-\frac {7 b^2 \cosh (5 (c+d x))}{320 d}+\frac {b^2 \cosh (7 (c+d x))}{448 d} \]
(-3*a^2*Cosh[c + d*x])/(4*d) + (5*a*b*Cosh[c + d*x])/(4*d) - (35*b^2*Cosh[ c + d*x])/(64*d) + (a^2*Cosh[3*(c + d*x)])/(12*d) - (5*a*b*Cosh[3*(c + d*x )])/(24*d) + (7*b^2*Cosh[3*(c + d*x)])/(64*d) + (a*b*Cosh[5*(c + d*x)])/(4 0*d) - (7*b^2*Cosh[5*(c + d*x)])/(320*d) + (b^2*Cosh[7*(c + d*x)])/(448*d)
Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 26, 3665, 290, 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.
\(\displaystyle \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \sin (i c+i d x)^3 \left (a-b \sin (i c+i d x)^2\right )^2dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \sin (i c+i d x)^3 \left (a-b \sin (i c+i d x)^2\right )^2dx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle -\frac {\int \left (1-\cosh ^2(c+d x)\right ) \left (b \cosh ^2(c+d x)+a-b\right )^2d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 290 |
\(\displaystyle -\frac {\int \left (-b^2 \cosh ^6(c+d x)-(2 a-3 b) b \cosh ^4(c+d x)+(a-3 b) (b-a) \cosh ^2(c+d x)+(a-b)^2\right )d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {1}{5} b (2 a-3 b) \cosh ^5(c+d x)-\frac {1}{3} (a-3 b) (a-b) \cosh ^3(c+d x)+(a-b)^2 \cosh (c+d x)-\frac {1}{7} b^2 \cosh ^7(c+d x)}{d}\) |
-(((a - b)^2*Cosh[c + d*x] - ((a - 3*b)*(a - b)*Cosh[c + d*x]^3)/3 - ((2*a - 3*b)*b*Cosh[c + d*x]^5)/5 - (b^2*Cosh[c + d*x]^7)/7)/d)
3.1.11.3.1 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_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d }, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.65 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.12
method | result | size |
parallelrisch | \(\frac {560 \left (a -\frac {7 b}{4}\right ) \left (a -\frac {3 b}{4}\right ) \cosh \left (3 d x +3 c \right )+168 b \left (a -\frac {7 b}{8}\right ) \cosh \left (5 d x +5 c \right )+15 b^{2} \cosh \left (7 d x +7 c \right )+\left (-5040 a^{2}+8400 a b -3675 b^{2}\right ) \cosh \left (d x +c \right )-4480 a^{2}+7168 a b -3072 b^{2}}{6720 d}\) | \(95\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+2 a b \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )+b^{2} \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}\) | \(102\) |
default | \(\frac {a^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+2 a b \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )+b^{2} \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}\) | \(102\) |
parts | \(\frac {a^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}+\frac {b^{2} \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}+\frac {2 a b \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )}{d}\) | \(107\) |
risch | \(\frac {b^{2} {\mathrm e}^{7 d x +7 c}}{896 d}+\frac {b \,{\mathrm e}^{5 d x +5 c} a}{80 d}-\frac {7 b^{2} {\mathrm e}^{5 d x +5 c}}{640 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2}}{24 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} a b}{48 d}+\frac {7 \,{\mathrm e}^{3 d x +3 c} b^{2}}{128 d}-\frac {3 \,{\mathrm e}^{d x +c} a^{2}}{8 d}+\frac {5 \,{\mathrm e}^{d x +c} a b}{8 d}-\frac {35 \,{\mathrm e}^{d x +c} b^{2}}{128 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2}}{8 d}+\frac {5 \,{\mathrm e}^{-d x -c} a b}{8 d}-\frac {35 \,{\mathrm e}^{-d x -c} b^{2}}{128 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{2}}{24 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} a b}{48 d}+\frac {7 \,{\mathrm e}^{-3 d x -3 c} b^{2}}{128 d}+\frac {b \,{\mathrm e}^{-5 d x -5 c} a}{80 d}-\frac {7 b^{2} {\mathrm e}^{-5 d x -5 c}}{640 d}+\frac {b^{2} {\mathrm e}^{-7 d x -7 c}}{896 d}\) | \(293\) |
1/6720*(560*(a-7/4*b)*(a-3/4*b)*cosh(3*d*x+3*c)+168*b*(a-7/8*b)*cosh(5*d*x +5*c)+15*b^2*cosh(7*d*x+7*c)+(-5040*a^2+8400*a*b-3675*b^2)*cosh(d*x+c)-448 0*a^2+7168*a*b-3072*b^2)/d
Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (79) = 158\).
Time = 0.29 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.51 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {15 \, b^{2} \cosh \left (d x + c\right )^{7} + 105 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + 21 \, {\left (8 \, a b - 7 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 105 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} + {\left (8 \, a b - 7 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 35 \, {\left (16 \, a^{2} - 40 \, a b + 21 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 105 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{5} + 2 \, {\left (8 \, a b - 7 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (16 \, a^{2} - 40 \, a b + 21 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 105 \, {\left (48 \, a^{2} - 80 \, a b + 35 \, b^{2}\right )} \cosh \left (d x + c\right )}{6720 \, d} \]
1/6720*(15*b^2*cosh(d*x + c)^7 + 105*b^2*cosh(d*x + c)*sinh(d*x + c)^6 + 2 1*(8*a*b - 7*b^2)*cosh(d*x + c)^5 + 105*(5*b^2*cosh(d*x + c)^3 + (8*a*b - 7*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 35*(16*a^2 - 40*a*b + 21*b^2)*cosh (d*x + c)^3 + 105*(3*b^2*cosh(d*x + c)^5 + 2*(8*a*b - 7*b^2)*cosh(d*x + c) ^3 + (16*a^2 - 40*a*b + 21*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 105*(48*a ^2 - 80*a*b + 35*b^2)*cosh(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (68) = 136\).
Time = 0.45 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.40 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\begin {cases} \frac {a^{2} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 a^{2} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {2 a b \sinh ^{4}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 a b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {16 a b \cosh ^{5}{\left (c + d x \right )}}{15 d} + \frac {b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {8 b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {16 b^{2} \cosh ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{2} \sinh ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((a**2*sinh(c + d*x)**2*cosh(c + d*x)/d - 2*a**2*cosh(c + d*x)**3 /(3*d) + 2*a*b*sinh(c + d*x)**4*cosh(c + d*x)/d - 8*a*b*sinh(c + d*x)**2*c osh(c + d*x)**3/(3*d) + 16*a*b*cosh(c + d*x)**5/(15*d) + b**2*sinh(c + d*x )**6*cosh(c + d*x)/d - 2*b**2*sinh(c + d*x)**4*cosh(c + d*x)**3/d + 8*b**2 *sinh(c + d*x)**2*cosh(c + d*x)**5/(5*d) - 16*b**2*cosh(c + d*x)**7/(35*d) , Ne(d, 0)), (x*(a + b*sinh(c)**2)**2*sinh(c)**3, True))
Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (79) = 158\).
Time = 0.20 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.91 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=-\frac {1}{4480} \, b^{2} {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{240} \, a b {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {1}{24} \, a^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]
-1/4480*b^2*((49*e^(-2*d*x - 2*c) - 245*e^(-4*d*x - 4*c) + 1225*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/d + (1225*e^(-d*x - c) - 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/d) + 1/240*a*b*(3*e^(5*d*x + 5 *c)/d - 25*e^(3*d*x + 3*c)/d + 150*e^(d*x + c)/d + 150*e^(-d*x - c)/d - 25 *e^(-3*d*x - 3*c)/d + 3*e^(-5*d*x - 5*c)/d) + 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. 196 vs. \(2 (79) = 158\).
Time = 0.27 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.31 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {b^{2} e^{\left (7 \, d x + 7 \, c\right )}}{896 \, d} + \frac {b^{2} e^{\left (-7 \, d x - 7 \, c\right )}}{896 \, d} + \frac {{\left (8 \, a b - 7 \, b^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {{\left (16 \, a^{2} - 40 \, a b + 21 \, b^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{384 \, d} - \frac {{\left (48 \, a^{2} - 80 \, a b + 35 \, b^{2}\right )} e^{\left (d x + c\right )}}{128 \, d} - \frac {{\left (48 \, a^{2} - 80 \, a b + 35 \, b^{2}\right )} e^{\left (-d x - c\right )}}{128 \, d} + \frac {{\left (16 \, a^{2} - 40 \, a b + 21 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{384 \, d} + \frac {{\left (8 \, a b - 7 \, b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} \]
1/896*b^2*e^(7*d*x + 7*c)/d + 1/896*b^2*e^(-7*d*x - 7*c)/d + 1/640*(8*a*b - 7*b^2)*e^(5*d*x + 5*c)/d + 1/384*(16*a^2 - 40*a*b + 21*b^2)*e^(3*d*x + 3 *c)/d - 1/128*(48*a^2 - 80*a*b + 35*b^2)*e^(d*x + c)/d - 1/128*(48*a^2 - 8 0*a*b + 35*b^2)*e^(-d*x - c)/d + 1/384*(16*a^2 - 40*a*b + 21*b^2)*e^(-3*d* x - 3*c)/d + 1/640*(8*a*b - 7*b^2)*e^(-5*d*x - 5*c)/d
Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.32 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {\frac {a^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}-a^2\,\mathrm {cosh}\left (c+d\,x\right )+\frac {2\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {4\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+2\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )+\frac {b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}-\frac {3\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}+b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3-b^2\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]