Integrand size = 21, antiderivative size = 99 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=-\frac {5 b x}{16}-\frac {a \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}+\frac {5 b \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {5 b \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b \cosh (c+d x) \sinh ^5(c+d x)}{6 d} \]
-5/16*b*x-a*cosh(d*x+c)/d+1/3*a*cosh(d*x+c)^3/d+5/16*b*cosh(d*x+c)*sinh(d* x+c)/d-5/24*b*cosh(d*x+c)*sinh(d*x+c)^3/d+1/6*b*cosh(d*x+c)*sinh(d*x+c)^5/ d
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.67 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=\frac {-144 a \cosh (c+d x)+16 a \cosh (3 (c+d x))+b (-60 c-60 d x+45 \sinh (2 (c+d x))-9 \sinh (4 (c+d x))+\sinh (6 (c+d x)))}{192 d} \]
(-144*a*Cosh[c + d*x] + 16*a*Cosh[3*(c + d*x)] + b*(-60*c - 60*d*x + 45*Si nh[2*(c + d*x)] - 9*Sinh[4*(c + d*x)] + Sinh[6*(c + d*x)]))/(192*d)
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 26, 3699, 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.
\(\displaystyle \int \sinh ^3(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \sin (i c+i d x)^3 \left (a+i b \sin (i c+i d x)^3\right )dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \sin (i c+i d x)^3 \left (i b \sin (i c+i d x)^3+a\right )dx\) |
\(\Big \downarrow \) 3699 |
\(\displaystyle i \int \left (-i b \sinh ^6(c+d x)-i a \sinh ^3(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i \left (-\frac {i a \cosh ^3(c+d x)}{3 d}+\frac {i a \cosh (c+d x)}{d}-\frac {i b \sinh ^5(c+d x) \cosh (c+d x)}{6 d}+\frac {5 i b \sinh ^3(c+d x) \cosh (c+d x)}{24 d}-\frac {5 i b \sinh (c+d x) \cosh (c+d x)}{16 d}+\frac {5 i b x}{16}\right )\) |
I*(((5*I)/16)*b*x + (I*a*Cosh[c + d*x])/d - ((I/3)*a*Cosh[c + d*x]^3)/d - (((5*I)/16)*b*Cosh[c + d*x]*Sinh[c + d*x])/d + (((5*I)/24)*b*Cosh[c + d*x] *Sinh[c + d*x]^3)/d - ((I/6)*b*Cosh[c + d*x]*Sinh[c + d*x]^5)/d)
3.2.42.3.1 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[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n) ^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4] || Gt Q[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
Time = 0.41 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.72
method | result | size |
parallelrisch | \(\frac {-60 b x d -144 \cosh \left (d x +c \right ) a +b \sinh \left (6 d x +6 c \right )-9 b \sinh \left (4 d x +4 c \right )+45 b \sinh \left (2 d x +2 c \right )+16 a \cosh \left (3 d x +3 c \right )-128 a}{192 d}\) | \(71\) |
derivativedivides | \(\frac {a \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+b \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d}\) | \(72\) |
default | \(\frac {a \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+b \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d}\) | \(72\) |
parts | \(\frac {a \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}+\frac {b \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d}\) | \(74\) |
risch | \(-\frac {5 b x}{16}+\frac {b \,{\mathrm e}^{6 d x +6 c}}{384 d}-\frac {3 b \,{\mathrm e}^{4 d x +4 c}}{128 d}+\frac {a \,{\mathrm e}^{3 d x +3 c}}{24 d}+\frac {15 b \,{\mathrm e}^{2 d x +2 c}}{128 d}-\frac {3 a \,{\mathrm e}^{d x +c}}{8 d}-\frac {3 a \,{\mathrm e}^{-d x -c}}{8 d}-\frac {15 b \,{\mathrm e}^{-2 d x -2 c}}{128 d}+\frac {a \,{\mathrm e}^{-3 d x -3 c}}{24 d}+\frac {3 b \,{\mathrm e}^{-4 d x -4 c}}{128 d}-\frac {b \,{\mathrm e}^{-6 d x -6 c}}{384 d}\) | \(153\) |
1/192*(-60*b*x*d-144*cosh(d*x+c)*a+b*sinh(6*d*x+6*c)-9*b*sinh(4*d*x+4*c)+4 5*b*sinh(2*d*x+2*c)+16*a*cosh(3*d*x+3*c)-128*a)/d
Time = 0.25 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.36 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=\frac {3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 8 \, a \cosh \left (d x + c\right )^{3} + 24 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 2 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 9 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 30 \, b d x - 72 \, a \cosh \left (d x + c\right ) + 3 \, {\left (b \cosh \left (d x + c\right )^{5} - 6 \, b \cosh \left (d x + c\right )^{3} + 15 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, d} \]
1/96*(3*b*cosh(d*x + c)*sinh(d*x + c)^5 + 8*a*cosh(d*x + c)^3 + 24*a*cosh( d*x + c)*sinh(d*x + c)^2 + 2*(5*b*cosh(d*x + c)^3 - 9*b*cosh(d*x + c))*sin h(d*x + c)^3 - 30*b*d*x - 72*a*cosh(d*x + c) + 3*(b*cosh(d*x + c)^5 - 6*b* cosh(d*x + c)^3 + 15*b*cosh(d*x + c))*sinh(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (92) = 184\).
Time = 0.33 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.96 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=\begin {cases} \frac {a \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 a \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 b x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {15 b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {15 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {5 b x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {11 b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac {5 b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right ) \sinh ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((a*sinh(c + d*x)**2*cosh(c + d*x)/d - 2*a*cosh(c + d*x)**3/(3*d) + 5*b*x*sinh(c + d*x)**6/16 - 15*b*x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 + 15*b*x*sinh(c + d*x)**2*cosh(c + d*x)**4/16 - 5*b*x*cosh(c + d*x)**6/16 + 11*b*sinh(c + d*x)**5*cosh(c + d*x)/(16*d) - 5*b*sinh(c + d*x)**3*cosh( c + d*x)**3/(6*d) + 5*b*sinh(c + d*x)*cosh(c + d*x)**5/(16*d), Ne(d, 0)), (x*(a + b*sinh(c)**3)*sinh(c)**3, True))
Time = 0.22 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.44 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=-\frac {1}{384} \, b {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} + \frac {1}{24} \, a {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]
-1/384*b*((9*e^(-2*d*x - 2*c) - 45*e^(-4*d*x - 4*c) - 1)*e^(6*d*x + 6*c)/d + 120*(d*x + c)/d + (45*e^(-2*d*x - 2*c) - 9*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c))/d) + 1/24*a*(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)
Time = 0.34 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.54 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=-\frac {5}{16} \, b x + \frac {b e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} - \frac {3 \, b e^{\left (4 \, d x + 4 \, c\right )}}{128 \, d} + \frac {a e^{\left (3 \, d x + 3 \, c\right )}}{24 \, d} + \frac {15 \, b e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {3 \, a e^{\left (d x + c\right )}}{8 \, d} - \frac {3 \, a e^{\left (-d x - c\right )}}{8 \, d} - \frac {15 \, b e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} + \frac {a e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} + \frac {3 \, b e^{\left (-4 \, d x - 4 \, c\right )}}{128 \, d} - \frac {b e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \]
-5/16*b*x + 1/384*b*e^(6*d*x + 6*c)/d - 3/128*b*e^(4*d*x + 4*c)/d + 1/24*a *e^(3*d*x + 3*c)/d + 15/128*b*e^(2*d*x + 2*c)/d - 3/8*a*e^(d*x + c)/d - 3/ 8*a*e^(-d*x - c)/d - 15/128*b*e^(-2*d*x - 2*c)/d + 1/24*a*e^(-3*d*x - 3*c) /d + 3/128*b*e^(-4*d*x - 4*c)/d - 1/384*b*e^(-6*d*x - 6*c)/d
Time = 0.49 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.68 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=\frac {\frac {a\,\mathrm {cosh}\left (3\,c+3\,d\,x\right )}{12}-\frac {3\,a\,\mathrm {cosh}\left (c+d\,x\right )}{4}+\frac {15\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{64}-\frac {3\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{64}+\frac {b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{192}}{d}-\frac {5\,b\,x}{16} \]