Integrand size = 8, antiderivative size = 67 \[ \int \cosh ^6(a+b x) \, dx=\frac {5 x}{16}+\frac {5 \cosh (a+b x) \sinh (a+b x)}{16 b}+\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{24 b}+\frac {\cosh ^5(a+b x) \sinh (a+b x)}{6 b} \] Output:
5/16*x+5/16*cosh(b*x+a)*sinh(b*x+a)/b+5/24*cosh(b*x+a)^3*sinh(b*x+a)/b+1/6 *cosh(b*x+a)^5*sinh(b*x+a)/b
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64 \[ \int \cosh ^6(a+b x) \, dx=\frac {60 a+60 b x+45 \sinh (2 (a+b x))+9 \sinh (4 (a+b x))+\sinh (6 (a+b x))}{192 b} \] Input:
Integrate[Cosh[a + b*x]^6,x]
Output:
(60*a + 60*b*x + 45*Sinh[2*(a + b*x)] + 9*Sinh[4*(a + b*x)] + Sinh[6*(a + b*x)])/(192*b)
Time = 0.37 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {3042, 3115, 3042, 3115, 3042, 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.
\(\displaystyle \int \cosh ^6(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^6dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {5}{6} \int \cosh ^4(a+b x)dx+\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}+\frac {5}{6} \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^4dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int \cosh ^2(a+b x)dx+\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )+\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}+\frac {5}{6} \left (\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}\right )+\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )+\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}+\frac {5}{6} \left (\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )\right )\) |
Input:
Int[Cosh[a + b*x]^6,x]
Output:
(Cosh[a + b*x]^5*Sinh[a + b*x])/(6*b) + (5*((Cosh[a + b*x]^3*Sinh[a + b*x] )/(4*b) + (3*(x/2 + (Cosh[a + b*x]*Sinh[a + b*x])/(2*b)))/4))/6
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]
Time = 2.83 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.63
method | result | size |
parallelrisch | \(\frac {60 b x +\sinh \left (6 b x +6 a \right )+9 \sinh \left (4 b x +4 a \right )+45 \sinh \left (2 b x +2 a \right )}{192 b}\) | \(42\) |
derivativedivides | \(\frac {\left (\frac {\cosh \left (b x +a \right )^{5}}{6}+\frac {5 \cosh \left (b x +a \right )^{3}}{24}+\frac {5 \cosh \left (b x +a \right )}{16}\right ) \sinh \left (b x +a \right )+\frac {5 b x}{16}+\frac {5 a}{16}}{b}\) | \(49\) |
default | \(\frac {\left (\frac {\cosh \left (b x +a \right )^{5}}{6}+\frac {5 \cosh \left (b x +a \right )^{3}}{24}+\frac {5 \cosh \left (b x +a \right )}{16}\right ) \sinh \left (b x +a \right )+\frac {5 b x}{16}+\frac {5 a}{16}}{b}\) | \(49\) |
risch | \(\frac {5 x}{16}+\frac {{\mathrm e}^{6 b x +6 a}}{384 b}+\frac {3 \,{\mathrm e}^{4 b x +4 a}}{128 b}+\frac {15 \,{\mathrm e}^{2 b x +2 a}}{128 b}-\frac {15 \,{\mathrm e}^{-2 b x -2 a}}{128 b}-\frac {3 \,{\mathrm e}^{-4 b x -4 a}}{128 b}-\frac {{\mathrm e}^{-6 b x -6 a}}{384 b}\) | \(89\) |
orering | \(x \cosh \left (b x +a \right )^{6}+\frac {49 \cosh \left (b x +a \right )^{5} \sinh \left (b x +a \right )}{24 b}-\frac {49 x \left (30 \cosh \left (b x +a \right )^{4} b^{2} \sinh \left (b x +a \right )^{2}+6 \cosh \left (b x +a \right )^{6} b^{2}\right )}{144 b^{2}}-\frac {7 \left (120 \cosh \left (b x +a \right )^{3} b^{3} \sinh \left (b x +a \right )^{3}+96 \cosh \left (b x +a \right )^{5} b^{3} \sinh \left (b x +a \right )\right )}{288 b^{4}}+\frac {7 x \left (360 \sinh \left (b x +a \right )^{4} \cosh \left (b x +a \right )^{2} b^{4}+840 \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{4} b^{4}+96 \cosh \left (b x +a \right )^{6} b^{4}\right )}{288 b^{4}}+\frac {4800 \sinh \left (b x +a \right )^{3} \cosh \left (b x +a \right )^{3} b^{5}+720 \sinh \left (b x +a \right )^{5} \cosh \left (b x +a \right ) b^{5}+2256 \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{5} b^{5}}{2304 b^{6}}-\frac {x \left (25680 \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{4} b^{6}+18000 \sinh \left (b x +a \right )^{4} \cosh \left (b x +a \right )^{2} b^{6}+720 \sinh \left (b x +a \right )^{6} b^{6}+2256 b^{6} \cosh \left (b x +a \right )^{6}\right )}{2304 b^{6}}\) | \(320\) |
Input:
int(cosh(b*x+a)^6,x,method=_RETURNVERBOSE)
Output:
1/192*(60*b*x+sinh(6*b*x+6*a)+9*sinh(4*b*x+4*a)+45*sinh(2*b*x+2*a))/b
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.34 \[ \int \cosh ^6(a+b x) \, dx=\frac {3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 2 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 30 \, b x + 3 \, {\left (\cosh \left (b x + a\right )^{5} + 6 \, \cosh \left (b x + a\right )^{3} + 15 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{96 \, b} \] Input:
integrate(cosh(b*x+a)^6,x, algorithm="fricas")
Output:
1/96*(3*cosh(b*x + a)*sinh(b*x + a)^5 + 2*(5*cosh(b*x + a)^3 + 9*cosh(b*x + a))*sinh(b*x + a)^3 + 30*b*x + 3*(cosh(b*x + a)^5 + 6*cosh(b*x + a)^3 + 15*cosh(b*x + a))*sinh(b*x + a))/b
Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (61) = 122\).
Time = 0.38 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.07 \[ \int \cosh ^6(a+b x) \, dx=\begin {cases} - \frac {5 x \sinh ^{6}{\left (a + b x \right )}}{16} + \frac {15 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} - \frac {15 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{16} + \frac {5 x \cosh ^{6}{\left (a + b x \right )}}{16} + \frac {5 \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{16 b} - \frac {5 \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{6 b} + \frac {11 \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x \cosh ^{6}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cosh(b*x+a)**6,x)
Output:
Piecewise((-5*x*sinh(a + b*x)**6/16 + 15*x*sinh(a + b*x)**4*cosh(a + b*x)* *2/16 - 15*x*sinh(a + b*x)**2*cosh(a + b*x)**4/16 + 5*x*cosh(a + b*x)**6/1 6 + 5*sinh(a + b*x)**5*cosh(a + b*x)/(16*b) - 5*sinh(a + b*x)**3*cosh(a + b*x)**3/(6*b) + 11*sinh(a + b*x)*cosh(a + b*x)**5/(16*b), Ne(b, 0)), (x*co sh(a)**6, True))
Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.28 \[ \int \cosh ^6(a+b x) \, dx=\frac {{\left (9 \, e^{\left (-2 \, b x - 2 \, a\right )} + 45 \, e^{\left (-4 \, b x - 4 \, a\right )} + 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} + \frac {5 \, {\left (b x + a\right )}}{16 \, b} - \frac {45 \, e^{\left (-2 \, b x - 2 \, a\right )} + 9 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \] Input:
integrate(cosh(b*x+a)^6,x, algorithm="maxima")
Output:
1/384*(9*e^(-2*b*x - 2*a) + 45*e^(-4*b*x - 4*a) + 1)*e^(6*b*x + 6*a)/b + 5 /16*(b*x + a)/b - 1/384*(45*e^(-2*b*x - 2*a) + 9*e^(-4*b*x - 4*a) + e^(-6* b*x - 6*a))/b
Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31 \[ \int \cosh ^6(a+b x) \, dx=\frac {5}{16} \, x + \frac {e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} + \frac {3 \, e^{\left (4 \, b x + 4 \, a\right )}}{128 \, b} + \frac {15 \, e^{\left (2 \, b x + 2 \, a\right )}}{128 \, b} - \frac {15 \, e^{\left (-2 \, b x - 2 \, a\right )}}{128 \, b} - \frac {3 \, e^{\left (-4 \, b x - 4 \, a\right )}}{128 \, b} - \frac {e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \] Input:
integrate(cosh(b*x+a)^6,x, algorithm="giac")
Output:
5/16*x + 1/384*e^(6*b*x + 6*a)/b + 3/128*e^(4*b*x + 4*a)/b + 15/128*e^(2*b *x + 2*a)/b - 15/128*e^(-2*b*x - 2*a)/b - 3/128*e^(-4*b*x - 4*a)/b - 1/384 *e^(-6*b*x - 6*a)/b
Time = 2.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.63 \[ \int \cosh ^6(a+b x) \, dx=\frac {5\,x}{16}+\frac {\frac {15\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{64}+\frac {3\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{64}+\frac {\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{192}}{b} \] Input:
int(cosh(a + b*x)^6,x)
Output:
(5*x)/16 + ((15*sinh(2*a + 2*b*x))/64 + (3*sinh(4*a + 4*b*x))/64 + sinh(6* a + 6*b*x)/192)/b
Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.36 \[ \int \cosh ^6(a+b x) \, dx=\frac {e^{12 b x +12 a}+9 e^{10 b x +10 a}+45 e^{8 b x +8 a}+120 e^{6 b x +6 a} b x -45 e^{4 b x +4 a}-9 e^{2 b x +2 a}-1}{384 e^{6 b x +6 a} b} \] Input:
int(cosh(b*x+a)^6,x)
Output:
(e**(12*a + 12*b*x) + 9*e**(10*a + 10*b*x) + 45*e**(8*a + 8*b*x) + 120*e** (6*a + 6*b*x)*b*x - 45*e**(4*a + 4*b*x) - 9*e**(2*a + 2*b*x) - 1)/(384*e** (6*a + 6*b*x)*b)