Integrand size = 21, antiderivative size = 89 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {1}{16} (6 a-5 b) x-\frac {(6 a-5 b) \cosh (c+d x) \sinh (c+d x)}{16 d}+\frac {(6 a-5 b) \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b \cosh (c+d x) \sinh ^5(c+d x)}{6 d} \]
1/16*(6*a-5*b)*x-1/16*(6*a-5*b)*cosh(d*x+c)*sinh(d*x+c)/d+1/24*(6*a-5*b)*c osh(d*x+c)*sinh(d*x+c)^3/d+1/6*b*cosh(d*x+c)*sinh(d*x+c)^5/d
Time = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {72 a c-60 b c+72 a d x-60 b d x+(-48 a+45 b) \sinh (2 (c+d x))+(6 a-9 b) \sinh (4 (c+d x))+b \sinh (6 (c+d x))}{192 d} \]
(72*a*c - 60*b*c + 72*a*d*x - 60*b*d*x + (-48*a + 45*b)*Sinh[2*(c + d*x)] + (6*a - 9*b)*Sinh[4*(c + d*x)] + b*Sinh[6*(c + d*x)])/(192*d)
Time = 0.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3493, 3042, 3115, 25, 3042, 25, 3115, 24}
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.
\(\displaystyle \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (i c+i d x)^4 \left (a-b \sin (i c+i d x)^2\right )dx\) |
\(\Big \downarrow \) 3493 |
\(\displaystyle \frac {1}{6} (6 a-5 b) \int \sinh ^4(c+d x)dx+\frac {b \sinh ^5(c+d x) \cosh (c+d x)}{6 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b \sinh ^5(c+d x) \cosh (c+d x)}{6 d}+\frac {1}{6} (6 a-5 b) \int \sin (i c+i d x)^4dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{6} (6 a-5 b) \left (\frac {3}{4} \int -\sinh ^2(c+d x)dx+\frac {\sinh ^3(c+d x) \cosh (c+d x)}{4 d}\right )+\frac {b \sinh ^5(c+d x) \cosh (c+d x)}{6 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{6} (6 a-5 b) \left (\frac {\sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac {3}{4} \int \sinh ^2(c+d x)dx\right )+\frac {b \sinh ^5(c+d x) \cosh (c+d x)}{6 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b \sinh ^5(c+d x) \cosh (c+d x)}{6 d}+\frac {1}{6} (6 a-5 b) \left (\frac {\sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac {3}{4} \int -\sin (i c+i d x)^2dx\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {b \sinh ^5(c+d x) \cosh (c+d x)}{6 d}+\frac {1}{6} (6 a-5 b) \left (\frac {\sinh ^3(c+d x) \cosh (c+d x)}{4 d}+\frac {3}{4} \int \sin (i c+i d x)^2dx\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{6} (6 a-5 b) \left (\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )+\frac {\sinh ^3(c+d x) \cosh (c+d x)}{4 d}\right )+\frac {b \sinh ^5(c+d x) \cosh (c+d x)}{6 d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{6} (6 a-5 b) \left (\frac {\sinh ^3(c+d x) \cosh (c+d x)}{4 d}+\frac {3}{4} \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )\right )+\frac {b \sinh ^5(c+d x) \cosh (c+d x)}{6 d}\) |
(b*Cosh[c + d*x]*Sinh[c + d*x]^5)/(6*d) + ((6*a - 5*b)*((Cosh[c + d*x]*Sin h[c + d*x]^3)/(4*d) + (3*(x/2 - (Cosh[c + d*x]*Sinh[c + d*x])/(2*d)))/4))/ 6
3.1.1.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*( x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f *(m + 2))), x] + Simp[(A*(m + 2) + C*(m + 1))/(m + 2) Int[(b*Sin[e + f*x] )^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && !LtQ[m, -1]
Time = 1.39 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(\frac {\left (-48 a +45 b \right ) \sinh \left (2 d x +2 c \right )+\left (6 a -9 b \right ) \sinh \left (4 d x +4 c \right )+b \sinh \left (6 d x +6 c \right )+72 x d \left (a -\frac {5 b}{6}\right )}{192 d}\) | \(61\) |
derivativedivides | \(\frac {a \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d}\) | \(88\) |
default | \(\frac {a \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d}\) | \(88\) |
parts | \(\frac {a \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {b \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d}\) | \(90\) |
risch | \(\frac {3 a x}{8}-\frac {5 b x}{16}+\frac {b \,{\mathrm e}^{6 d x +6 c}}{384 d}+\frac {{\mathrm e}^{4 d x +4 c} a}{64 d}-\frac {3 \,{\mathrm e}^{4 d x +4 c} b}{128 d}-\frac {{\mathrm e}^{2 d x +2 c} a}{8 d}+\frac {15 \,{\mathrm e}^{2 d x +2 c} b}{128 d}+\frac {{\mathrm e}^{-2 d x -2 c} a}{8 d}-\frac {15 \,{\mathrm e}^{-2 d x -2 c} b}{128 d}-\frac {{\mathrm e}^{-4 d x -4 c} a}{64 d}+\frac {3 \,{\mathrm e}^{-4 d x -4 c} b}{128 d}-\frac {b \,{\mathrm e}^{-6 d x -6 c}}{384 d}\) | \(160\) |
1/192*((-48*a+45*b)*sinh(2*d*x+2*c)+(6*a-9*b)*sinh(4*d*x+4*c)+b*sinh(6*d*x +6*c)+72*x*d*(a-5/6*b))/d
Time = 0.32 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.37 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} + 3 \, {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (6 \, a - 5 \, b\right )} d x + 3 \, {\left (b \cosh \left (d x + c\right )^{5} + 2 \, {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{3} - {\left (16 \, a - 15 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, d} \]
1/96*(3*b*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(5*b*cosh(d*x + c)^3 + 3*(2*a - 3*b)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(6*a - 5*b)*d*x + 3*(b*cosh(d*x + c)^5 + 2*(2*a - 3*b)*cosh(d*x + c)^3 - (16*a - 15*b)*cosh(d*x + c))*sinh (d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (82) = 164\).
Time = 0.33 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.90 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\begin {cases} \frac {3 a x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 a x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {5 a \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {3 a \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {5 b x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {15 b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {15 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {5 b x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {11 b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac {5 b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \sinh ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((3*a*x*sinh(c + d*x)**4/8 - 3*a*x*sinh(c + d*x)**2*cosh(c + d*x) **2/4 + 3*a*x*cosh(c + d*x)**4/8 + 5*a*sinh(c + d*x)**3*cosh(c + d*x)/(8*d ) - 3*a*sinh(c + d*x)*cosh(c + d*x)**3/(8*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*cos h(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)**2)*sinh(c)* *4, True))
Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.69 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {1}{64} \, a {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{384} \, b {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \]
1/64*a*(24*x + e^(4*d*x + 4*c)/d - 8*e^(2*d*x + 2*c)/d + 8*e^(-2*d*x - 2*c )/d - e^(-4*d*x - 4*c)/d) - 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)
Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.40 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {1}{16} \, {\left (6 \, a - 5 \, b\right )} x + \frac {b e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} + \frac {{\left (2 \, a - 3 \, b\right )} e^{\left (4 \, d x + 4 \, c\right )}}{128 \, d} - \frac {{\left (16 \, a - 15 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} + \frac {{\left (16 \, a - 15 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} - \frac {{\left (2 \, a - 3 \, b\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{128 \, d} - \frac {b e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \]
1/16*(6*a - 5*b)*x + 1/384*b*e^(6*d*x + 6*c)/d + 1/128*(2*a - 3*b)*e^(4*d* x + 4*c)/d - 1/128*(16*a - 15*b)*e^(2*d*x + 2*c)/d + 1/128*(16*a - 15*b)*e ^(-2*d*x - 2*c)/d - 1/128*(2*a - 3*b)*e^(-4*d*x - 4*c)/d - 1/384*b*e^(-6*d *x - 6*c)/d
Time = 1.46 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.85 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {\frac {3\,a\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{2}-12\,a\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {45\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}-\frac {9\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+\frac {b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}+18\,a\,d\,x-15\,b\,d\,x}{48\,d} \]