Integrand size = 12, antiderivative size = 137 \[ \int (a+b \sinh (c+d x))^4 \, dx=\frac {1}{8} \left (8 a^4-24 a^2 b^2+3 b^4\right ) x+\frac {a b \left (19 a^2-16 b^2\right ) \cosh (c+d x)}{6 d}+\frac {b^2 \left (26 a^2-9 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{24 d}+\frac {7 a b \cosh (c+d x) (a+b \sinh (c+d x))^2}{12 d}+\frac {b \cosh (c+d x) (a+b \sinh (c+d x))^3}{4 d} \] Output:
1/8*(8*a^4-24*a^2*b^2+3*b^4)*x+1/6*a*b*(19*a^2-16*b^2)*cosh(d*x+c)/d+1/24* b^2*(26*a^2-9*b^2)*cosh(d*x+c)*sinh(d*x+c)/d+7/12*a*b*cosh(d*x+c)*(a+b*sin h(d*x+c))^2/d+1/4*b*cosh(d*x+c)*(a+b*sinh(d*x+c))^3/d
Time = 0.46 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79 \[ \int (a+b \sinh (c+d x))^4 \, dx=\frac {96 a b \left (4 a^2-3 b^2\right ) \cosh (c+d x)+32 a b^3 \cosh (3 (c+d x))+3 \left (4 \left (8 a^4-24 a^2 b^2+3 b^4\right ) (c+d x)+8 \left (6 a^2 b^2-b^4\right ) \sinh (2 (c+d x))+b^4 \sinh (4 (c+d x))\right )}{96 d} \] Input:
Integrate[(a + b*Sinh[c + d*x])^4,x]
Output:
(96*a*b*(4*a^2 - 3*b^2)*Cosh[c + d*x] + 32*a*b^3*Cosh[3*(c + d*x)] + 3*(4* (8*a^4 - 24*a^2*b^2 + 3*b^4)*(c + d*x) + 8*(6*a^2*b^2 - b^4)*Sinh[2*(c + d *x)] + b^4*Sinh[4*(c + d*x)]))/(96*d)
Time = 0.51 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3135, 3042, 3232, 3042, 3213}
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.
| \(\displaystyle \int (a+b \sinh (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a-i b \sin (i c+i d x))^4dx\) |
| \(\Big \downarrow \) 3135 |
| \(\displaystyle \frac {1}{4} \int (a+b \sinh (c+d x))^2 \left (4 a^2+7 b \sinh (c+d x) a-3 b^2\right )dx+\frac {b \cosh (c+d x) (a+b \sinh (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \cosh (c+d x) (a+b \sinh (c+d x))^3}{4 d}+\frac {1}{4} \int (a-i b \sin (i c+i d x))^2 \left (4 a^2-7 i b \sin (i c+i d x) a-3 b^2\right )dx\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int (a+b \sinh (c+d x)) \left (a \left (12 a^2-23 b^2\right )+b \left (26 a^2-9 b^2\right ) \sinh (c+d x)\right )dx+\frac {7 a b \cosh (c+d x) (a+b \sinh (c+d x))^2}{3 d}\right )+\frac {b \cosh (c+d x) (a+b \sinh (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \cosh (c+d x) (a+b \sinh (c+d x))^3}{4 d}+\frac {1}{4} \left (\frac {7 a b \cosh (c+d x) (a+b \sinh (c+d x))^2}{3 d}+\frac {1}{3} \int (a-i b \sin (i c+i d x)) \left (a \left (12 a^2-23 b^2\right )-i b \left (26 a^2-9 b^2\right ) \sin (i c+i d x)\right )dx\right )\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {2 a b \left (19 a^2-16 b^2\right ) \cosh (c+d x)}{d}+\frac {b^2 \left (26 a^2-9 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {3}{2} x \left (8 a^4-24 a^2 b^2+3 b^4\right )\right )+\frac {7 a b \cosh (c+d x) (a+b \sinh (c+d x))^2}{3 d}\right )+\frac {b \cosh (c+d x) (a+b \sinh (c+d x))^3}{4 d}\) |
Input:
Int[(a + b*Sinh[c + d*x])^4,x]
Output:
(b*Cosh[c + d*x]*(a + b*Sinh[c + d*x])^3)/(4*d) + ((7*a*b*Cosh[c + d*x]*(a + b*Sinh[c + d*x])^2)/(3*d) + ((3*(8*a^4 - 24*a^2*b^2 + 3*b^4)*x)/2 + (2* a*b*(19*a^2 - 16*b^2)*Cosh[c + d*x])/d + (b^2*(26*a^2 - 9*b^2)*Cosh[c + d* x]*Sinh[c + d*x])/(2*d))/3)/4
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[1/n Int[(a + b* Sin[c + d*x])^(n - 2)*Simp[a^2*n + b^2*(n - 1) + a*b*(2*n - 1)*Sin[c + d*x] , x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Time = 10.64 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.87
| method | result | size |
| derivativedivides | \(\frac {b^{4} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a \,b^{3} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+6 a^{2} b^{2} \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+4 a^{3} b \cosh \left (d x +c \right )+a^{4} \left (d x +c \right )}{d}\) | \(119\) |
| default | \(\frac {b^{4} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a \,b^{3} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+6 a^{2} b^{2} \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+4 a^{3} b \cosh \left (d x +c \right )+a^{4} \left (d x +c \right )}{d}\) | \(119\) |
| parts | \(x \,a^{4}+\frac {b^{4} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {4 a^{3} b \cosh \left (d x +c \right )}{d}+\frac {6 a^{2} b^{2} \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d}+\frac {4 a \,b^{3} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}\) | \(123\) |
| parallelrisch | \(\frac {96 a^{4} d x -288 a^{2} b^{2} d x +36 b^{4} d x +384 a^{3} b \cosh \left (d x +c \right )-288 a \,b^{3} \cosh \left (d x +c \right )+32 a \,b^{3} \cosh \left (3 d x +3 c \right )+3 b^{4} \sinh \left (4 d x +4 c \right )+144 a^{2} b^{2} \sinh \left (2 d x +2 c \right )-24 b^{4} \sinh \left (2 d x +2 c \right )+384 a^{3} b -256 a \,b^{3}}{96 d}\) | \(127\) |
| risch | \(x \,a^{4}-3 x \,a^{2} b^{2}+\frac {3 x \,b^{4}}{8}+\frac {b^{4} {\mathrm e}^{4 d x +4 c}}{64 d}+\frac {a \,b^{3} {\mathrm e}^{3 d x +3 c}}{6 d}+\frac {3 b^{2} {\mathrm e}^{2 d x +2 c} a^{2}}{4 d}-\frac {b^{4} {\mathrm e}^{2 d x +2 c}}{8 d}+\frac {2 a^{3} b \,{\mathrm e}^{d x +c}}{d}-\frac {3 a \,b^{3} {\mathrm e}^{d x +c}}{2 d}+\frac {2 a^{3} b \,{\mathrm e}^{-d x -c}}{d}-\frac {3 a \,b^{3} {\mathrm e}^{-d x -c}}{2 d}-\frac {3 b^{2} {\mathrm e}^{-2 d x -2 c} a^{2}}{4 d}+\frac {b^{4} {\mathrm e}^{-2 d x -2 c}}{8 d}+\frac {a \,b^{3} {\mathrm e}^{-3 d x -3 c}}{6 d}-\frac {b^{4} {\mathrm e}^{-4 d x -4 c}}{64 d}\) | \(232\) |
Input:
int((a+b*sinh(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(b^4*((1/4*sinh(d*x+c)^3-3/8*sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c)+4 *a*b^3*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c)+6*a^2*b^2*(1/2*sinh(d*x+c)*cos h(d*x+c)-1/2*d*x-1/2*c)+4*a^3*b*cosh(d*x+c)+a^4*(d*x+c))
Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.07 \[ \int (a+b \sinh (c+d x))^4 \, dx=\frac {3 \, b^{4} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 8 \, a b^{3} \cosh \left (d x + c\right )^{3} + 24 \, a b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, {\left (8 \, a^{4} - 24 \, a^{2} b^{2} + 3 \, b^{4}\right )} d x + 24 \, {\left (4 \, a^{3} b - 3 \, a b^{3}\right )} \cosh \left (d x + c\right ) + 3 \, {\left (b^{4} \cosh \left (d x + c\right )^{3} + 4 \, {\left (6 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, d} \] Input:
integrate((a+b*sinh(d*x+c))^4,x, algorithm="fricas")
Output:
1/24*(3*b^4*cosh(d*x + c)*sinh(d*x + c)^3 + 8*a*b^3*cosh(d*x + c)^3 + 24*a *b^3*cosh(d*x + c)*sinh(d*x + c)^2 + 3*(8*a^4 - 24*a^2*b^2 + 3*b^4)*d*x + 24*(4*a^3*b - 3*a*b^3)*cosh(d*x + c) + 3*(b^4*cosh(d*x + c)^3 + 4*(6*a^2*b ^2 - b^4)*cosh(d*x + c))*sinh(d*x + c))/d
Time = 0.33 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.75 \[ \int (a+b \sinh (c+d x))^4 \, dx=\begin {cases} a^{4} x + \frac {4 a^{3} b \cosh {\left (c + d x \right )}}{d} + 3 a^{2} b^{2} x \sinh ^{2}{\left (c + d x \right )} - 3 a^{2} b^{2} x \cosh ^{2}{\left (c + d x \right )} + \frac {3 a^{2} b^{2} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} + \frac {4 a b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 a b^{3} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {3 b^{4} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 b^{4} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 b^{4} x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {5 b^{4} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {3 b^{4} \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh {\left (c \right )}\right )^{4} & \text {otherwise} \end {cases} \] Input:
integrate((a+b*sinh(d*x+c))**4,x)
Output:
Piecewise((a**4*x + 4*a**3*b*cosh(c + d*x)/d + 3*a**2*b**2*x*sinh(c + d*x) **2 - 3*a**2*b**2*x*cosh(c + d*x)**2 + 3*a**2*b**2*sinh(c + d*x)*cosh(c + d*x)/d + 4*a*b**3*sinh(c + d*x)**2*cosh(c + d*x)/d - 8*a*b**3*cosh(c + d*x )**3/(3*d) + 3*b**4*x*sinh(c + d*x)**4/8 - 3*b**4*x*sinh(c + d*x)**2*cosh( c + d*x)**2/4 + 3*b**4*x*cosh(c + d*x)**4/8 + 5*b**4*sinh(c + d*x)**3*cosh (c + d*x)/(8*d) - 3*b**4*sinh(c + d*x)*cosh(c + d*x)**3/(8*d), Ne(d, 0)), (x*(a + b*sinh(c))**4, True))
Time = 0.04 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.33 \[ \int (a+b \sinh (c+d x))^4 \, dx=\frac {1}{64} \, b^{4} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {3}{4} \, a^{2} b^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + a^{4} x + \frac {1}{6} \, a b^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {4 \, a^{3} b \cosh \left (d x + c\right )}{d} \] Input:
integrate((a+b*sinh(d*x+c))^4,x, algorithm="maxima")
Output:
1/64*b^4*(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) - 3/4*a^2*b^2*(4*x - e^(2*d*x + 2*c)/d + e^(-2 *d*x - 2*c)/d) + a^4*x + 1/6*a*b^3*(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) + 4*a^3*b*cosh(d*x + c)/d
Time = 0.13 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.46 \[ \int (a+b \sinh (c+d x))^4 \, dx=\frac {b^{4} e^{\left (4 \, d x + 4 \, c\right )}}{64 \, d} + \frac {a b^{3} e^{\left (3 \, d x + 3 \, c\right )}}{6 \, d} + \frac {a b^{3} e^{\left (-3 \, d x - 3 \, c\right )}}{6 \, d} - \frac {b^{4} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} + \frac {1}{8} \, {\left (8 \, a^{4} - 24 \, a^{2} b^{2} + 3 \, b^{4}\right )} x + \frac {{\left (6 \, a^{2} b^{2} - b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} + \frac {{\left (4 \, a^{3} b - 3 \, a b^{3}\right )} e^{\left (d x + c\right )}}{2 \, d} + \frac {{\left (4 \, a^{3} b - 3 \, a b^{3}\right )} e^{\left (-d x - c\right )}}{2 \, d} - \frac {{\left (6 \, a^{2} b^{2} - b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} \] Input:
integrate((a+b*sinh(d*x+c))^4,x, algorithm="giac")
Output:
1/64*b^4*e^(4*d*x + 4*c)/d + 1/6*a*b^3*e^(3*d*x + 3*c)/d + 1/6*a*b^3*e^(-3 *d*x - 3*c)/d - 1/64*b^4*e^(-4*d*x - 4*c)/d + 1/8*(8*a^4 - 24*a^2*b^2 + 3* b^4)*x + 1/8*(6*a^2*b^2 - b^4)*e^(2*d*x + 2*c)/d + 1/2*(4*a^3*b - 3*a*b^3) *e^(d*x + c)/d + 1/2*(4*a^3*b - 3*a*b^3)*e^(-d*x - c)/d - 1/8*(6*a^2*b^2 - b^4)*e^(-2*d*x - 2*c)/d
Time = 0.37 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.83 \[ \int (a+b \sinh (c+d x))^4 \, dx=\frac {\frac {3\,b^4\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}-6\,b^4\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+8\,a\,b^3\,\mathrm {cosh}\left (3\,c+3\,d\,x\right )+36\,a^2\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-72\,a\,b^3\,\mathrm {cosh}\left (c+d\,x\right )+96\,a^3\,b\,\mathrm {cosh}\left (c+d\,x\right )+24\,a^4\,d\,x+9\,b^4\,d\,x-72\,a^2\,b^2\,d\,x}{24\,d} \] Input:
int((a + b*sinh(c + d*x))^4,x)
Output:
((3*b^4*sinh(4*c + 4*d*x))/4 - 6*b^4*sinh(2*c + 2*d*x) + 8*a*b^3*cosh(3*c + 3*d*x) + 36*a^2*b^2*sinh(2*c + 2*d*x) - 72*a*b^3*cosh(c + d*x) + 96*a^3* b*cosh(c + d*x) + 24*a^4*d*x + 9*b^4*d*x - 72*a^2*b^2*d*x)/(24*d)
Time = 0.17 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.83 \[ \int (a+b \sinh (c+d x))^4 \, dx=\frac {3 e^{8 d x +8 c} b^{4}+32 e^{7 d x +7 c} a \,b^{3}+144 e^{6 d x +6 c} a^{2} b^{2}-24 e^{6 d x +6 c} b^{4}+384 e^{5 d x +5 c} a^{3} b -288 e^{5 d x +5 c} a \,b^{3}+192 e^{4 d x +4 c} a^{4} d x -576 e^{4 d x +4 c} a^{2} b^{2} d x +72 e^{4 d x +4 c} b^{4} d x +384 e^{3 d x +3 c} a^{3} b -288 e^{3 d x +3 c} a \,b^{3}-144 e^{2 d x +2 c} a^{2} b^{2}+24 e^{2 d x +2 c} b^{4}+32 e^{d x +c} a \,b^{3}-3 b^{4}}{192 e^{4 d x +4 c} d} \] Input:
int((a+b*sinh(d*x+c))^4,x)
Output:
(3*e**(8*c + 8*d*x)*b**4 + 32*e**(7*c + 7*d*x)*a*b**3 + 144*e**(6*c + 6*d* x)*a**2*b**2 - 24*e**(6*c + 6*d*x)*b**4 + 384*e**(5*c + 5*d*x)*a**3*b - 28 8*e**(5*c + 5*d*x)*a*b**3 + 192*e**(4*c + 4*d*x)*a**4*d*x - 576*e**(4*c + 4*d*x)*a**2*b**2*d*x + 72*e**(4*c + 4*d*x)*b**4*d*x + 384*e**(3*c + 3*d*x) *a**3*b - 288*e**(3*c + 3*d*x)*a*b**3 - 144*e**(2*c + 2*d*x)*a**2*b**2 + 2 4*e**(2*c + 2*d*x)*b**4 + 32*e**(c + d*x)*a*b**3 - 3*b**4)/(192*e**(4*c + 4*d*x)*d)