\(\int (a+b \cosh (c+d x))^4 \, dx\) [63]

Optimal result

Integrand size = 12, antiderivative size = 137 \[ \int (a+b \cosh (c+d x))^4 \, dx=\frac {1}{8} \left (8 a^4+24 a^2 b^2+3 b^4\right ) x+\frac {a b \left (19 a^2+16 b^2\right ) \sinh (c+d x)}{6 d}+\frac {b^2 \left (26 a^2+9 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{24 d}+\frac {7 a b (a+b \cosh (c+d x))^2 \sinh (c+d x)}{12 d}+\frac {b (a+b \cosh (c+d x))^3 \sinh (c+d x)}{4 d} \]

[Out]

1/8*(8*a^4+24*a^2*b^2+3*b^4)*x+1/6*a*b*(19*a^2+16*b^2)*sinh(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*(a+b*cosh(d*x+c))^2*sinh(d*x+c)/d+1/4*b*(a+b*cosh(d*x+c))^3*sinh(d*x+c)/d

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2735, 2832, 2813} \[ \int (a+b \cosh (c+d x))^4 \, dx=\frac {a b \left (19 a^2+16 b^2\right ) \sinh (c+d x)}{6 d}+\frac {b^2 \left (26 a^2+9 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{24 d}+\frac {1}{8} x \left (8 a^4+24 a^2 b^2+3 b^4\right )+\frac {b \sinh (c+d x) (a+b \cosh (c+d x))^3}{4 d}+\frac {7 a b \sinh (c+d x) (a+b \cosh (c+d x))^2}{12 d} \]

[In]

Int[(a + b*Cosh[c + d*x])^4,x]

[Out]

((8*a^4 + 24*a^2*b^2 + 3*b^4)*x)/8 + (a*b*(19*a^2 + 16*b^2)*Sinh[c + d*x])/(6*d) + (b^2*(26*a^2 + 9*b^2)*Cosh[
c + d*x]*Sinh[c + d*x])/(24*d) + (7*a*b*(a + b*Cosh[c + d*x])^2*Sinh[c + d*x])/(12*d) + (b*(a + b*Cosh[c + d*x
])^3*Sinh[c + d*x])/(4*d)

Rule 2735

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] + Dist[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]

Rule 2813

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)*(Cos[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2832

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] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[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]

Rubi steps \begin{align*} \text {integral}& = \frac {b (a+b \cosh (c+d x))^3 \sinh (c+d x)}{4 d}+\frac {1}{4} \int (a+b \cosh (c+d x))^2 \left (4 a^2+3 b^2+7 a b \cosh (c+d x)\right ) \, dx \\ & = \frac {7 a b (a+b \cosh (c+d x))^2 \sinh (c+d x)}{12 d}+\frac {b (a+b \cosh (c+d x))^3 \sinh (c+d x)}{4 d}+\frac {1}{12} \int (a+b \cosh (c+d x)) \left (a \left (12 a^2+23 b^2\right )+b \left (26 a^2+9 b^2\right ) \cosh (c+d x)\right ) \, 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 ) \sinh (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 (a+b \cosh (c+d x))^2 \sinh (c+d x)}{12 d}+\frac {b (a+b \cosh (c+d x))^3 \sinh (c+d x)}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.76 \[ \int (a+b \cosh (c+d x))^4 \, dx=\frac {12 \left (8 a^4+24 a^2 b^2+3 b^4\right ) (c+d x)+96 a b \left (4 a^2+3 b^2\right ) \sinh (c+d x)+24 b^2 \left (6 a^2+b^2\right ) \sinh (2 (c+d x))+32 a b^3 \sinh (3 (c+d x))+3 b^4 \sinh (4 (c+d x))}{96 d} \]

[In]

Integrate[(a + b*Cosh[c + d*x])^4,x]

[Out]

(12*(8*a^4 + 24*a^2*b^2 + 3*b^4)*(c + d*x) + 96*a*b*(4*a^2 + 3*b^2)*Sinh[c + d*x] + 24*b^2*(6*a^2 + b^2)*Sinh[
2*(c + d*x)] + 32*a*b^3*Sinh[3*(c + d*x)] + 3*b^4*Sinh[4*(c + d*x)])/(96*d)

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.74

method	result	size
parallelrisch	\(\frac {24 \left (6 a^{2} b^{2}+b^{4}\right ) \sinh \left (2 d x +2 c \right )+32 a \,b^{3} \sinh \left (3 d x +3 c \right )+3 b^{4} \sinh \left (4 d x +4 c \right )+96 \left (4 a^{3} b +3 a \,b^{3}\right ) \sinh \left (d x +c \right )+96 d x \left (a^{4}+3 a^{2} b^{2}+\frac {3}{8} b^{4}\right )}{96 d}\)	\(101\)
derivativedivides	\(\frac {b^{4} \left (\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a \,b^{3} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )+6 a^{2} b^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{3} b \sinh \left (d x +c \right )+a^{4} \left (d x +c \right )}{d}\)	\(119\)
default	\(\frac {b^{4} \left (\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a \,b^{3} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )+6 a^{2} b^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{3} b \sinh \left (d x +c \right )+a^{4} \left (d x +c \right )}{d}\)	\(119\)
parts	\(x \,a^{4}+\frac {b^{4} \left (\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {4 a^{3} b \sinh \left (d x +c \right )}{d}+\frac {6 a^{2} b^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 a \,b^{3} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )}{d}\)	\(123\)
risch	\(x \,a^{4}+3 x \,a^{2} b^{2}+\frac {3 x \,b^{4}}{8}+\frac {b^{4} {\mathrm e}^{4 d x +4 c}}{64 d}+\frac {a \,b^{3} {\mathrm e}^{3 d x +3 c}}{6 d}+\frac {3 b^{2} {\mathrm e}^{2 d x +2 c} a^{2}}{4 d}+\frac {b^{4} {\mathrm e}^{2 d x +2 c}}{8 d}+\frac {2 a^{3} b \,{\mathrm e}^{d x +c}}{d}+\frac {3 a \,b^{3} {\mathrm e}^{d x +c}}{2 d}-\frac {2 a^{3} b \,{\mathrm e}^{-d x -c}}{d}-\frac {3 a \,b^{3} {\mathrm e}^{-d x -c}}{2 d}-\frac {3 b^{2} {\mathrm e}^{-2 d x -2 c} a^{2}}{4 d}-\frac {b^{4} {\mathrm e}^{-2 d x -2 c}}{8 d}-\frac {a \,b^{3} {\mathrm e}^{-3 d x -3 c}}{6 d}-\frac {b^{4} {\mathrm e}^{-4 d x -4 c}}{64 d}\)	\(232\)

[In]

int((a+b*cosh(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/96*(24*(6*a^2*b^2+b^4)*sinh(2*d*x+2*c)+32*a*b^3*sinh(3*d*x+3*c)+3*b^4*sinh(4*d*x+4*c)+96*(4*a^3*b+3*a*b^3)*s
inh(d*x+c)+96*d*x*(a^4+3*a^2*b^2+3/8*b^4))/d

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int (a+b \cosh (c+d x))^4 \, dx=\frac {{\left (3 \, b^{4} \cosh \left (d x + c\right ) + 8 \, a b^{3}\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{4} + 24 \, a^{2} b^{2} + 3 \, b^{4}\right )} d x + 3 \, {\left (b^{4} \cosh \left (d x + c\right )^{3} + 8 \, a b^{3} \cosh \left (d x + c\right )^{2} + 32 \, a^{3} b + 24 \, a b^{3} + 4 \, {\left (6 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, d} \]

[In]

integrate((a+b*cosh(d*x+c))^4,x, algorithm="fricas")

[Out]

1/24*((3*b^4*cosh(d*x + c) + 8*a*b^3)*sinh(d*x + c)^3 + 3*(8*a^4 + 24*a^2*b^2 + 3*b^4)*d*x + 3*(b^4*cosh(d*x +
c)^3 + 8*a*b^3*cosh(d*x + c)^2 + 32*a^3*b + 24*a*b^3 + 4*(6*a^2*b^2 + b^4)*cosh(d*x + c))*sinh(d*x + c))/d

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.75 \[ \int (a+b \cosh (c+d x))^4 \, dx=\begin {cases} a^{4} x + \frac {4 a^{3} b \sinh {\left (c + d x \right )}}{d} - 3 a^{2} b^{2} x \sinh ^{2}{\left (c + d x \right )} + 3 a^{2} b^{2} x \cosh ^{2}{\left (c + d x \right )} + \frac {3 a^{2} b^{2} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 a b^{3} \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac {4 a b^{3} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} + \frac {3 b^{4} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 b^{4} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 b^{4} x \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {3 b^{4} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} + \frac {5 b^{4} \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \cosh {\left (c \right )}\right )^{4} & \text {otherwise} \end {cases} \]

[In]

integrate((a+b*cosh(d*x+c))**4,x)

[Out]

Piecewise((a**4*x + 4*a**3*b*sinh(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 - 8*a*b**3*sinh(c + d*x)**3/(3*d) + 4*a*b**3*sinh(c + d*x)*cosh(c
+ d*x)**2/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 - 3*b**4*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) + 5*b**4*sinh(c + d*x)*cosh(c + d*x)**3/(8*d), Ne(d, 0
)), (x*(a + b*cosh(c))**4, True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.34 \[ \int (a+b \cosh (c+d x))^4 \, dx=\frac {1}{64} \, b^{4} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {3}{4} \, a^{2} b^{2} {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + a^{4} x + \frac {1}{6} \, a b^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {4 \, a^{3} b \sinh \left (d x + c\right )}{d} \]

[In]

integrate((a+b*cosh(d*x+c))^4,x, algorithm="maxima")

[Out]

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*sinh(d*x + c)/d

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.43 \[ \int (a+b \cosh (c+d x))^4 \, dx=\frac {b^{4} e^{\left (4 \, d x + 4 \, c\right )}}{64 \, d} + \frac {a b^{3} e^{\left (3 \, d x + 3 \, c\right )}}{6 \, d} - \frac {a b^{3} e^{\left (-3 \, d x - 3 \, c\right )}}{6 \, d} - \frac {b^{4} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} + \frac {1}{8} \, {\left (8 \, a^{4} + 24 \, a^{2} b^{2} + 3 \, b^{4}\right )} x + \frac {{\left (6 \, a^{2} b^{2} + b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} + \frac {{\left (4 \, a^{3} b + 3 \, a b^{3}\right )} e^{\left (d x + c\right )}}{2 \, d} - \frac {{\left (4 \, a^{3} b + 3 \, a b^{3}\right )} e^{\left (-d x - c\right )}}{2 \, d} - \frac {{\left (6 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} \]

[In]

integrate((a+b*cosh(d*x+c))^4,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.83 \[ \int (a+b \cosh (c+d x))^4 \, dx=\frac {6\,b^4\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {3\,b^4\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+8\,a\,b^3\,\mathrm {sinh}\left (3\,c+3\,d\,x\right )+36\,a^2\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+72\,a\,b^3\,\mathrm {sinh}\left (c+d\,x\right )+96\,a^3\,b\,\mathrm {sinh}\left (c+d\,x\right )+24\,a^4\,d\,x+9\,b^4\,d\,x+72\,a^2\,b^2\,d\,x}{24\,d} \]

[In]

int((a + b*cosh(c + d*x))^4,x)

[Out]

(6*b^4*sinh(2*c + 2*d*x) + (3*b^4*sinh(4*c + 4*d*x))/4 + 8*a*b^3*sinh(3*c + 3*d*x) + 36*a^2*b^2*sinh(2*c + 2*d
*x) + 72*a*b^3*sinh(c + d*x) + 96*a^3*b*sinh(c + d*x) + 24*a^4*d*x + 9*b^4*d*x + 72*a^2*b^2*d*x)/(24*d)