\(\int \sinh ^6(a+b x) \, dx\) [6]

Optimal result

Integrand size = 8, antiderivative size = 67 \[ \int \sinh ^6(a+b x) \, dx=-\frac {5 x}{16}+\frac {5 \cosh (a+b x) \sinh (a+b x)}{16 b}-\frac {5 \cosh (a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh (a+b x) \sinh ^5(a+b x)}{6 b} \]

[Out]

-5/16*x+5/16*cosh(b*x+a)*sinh(b*x+a)/b-5/24*cosh(b*x+a)*sinh(b*x+a)^3/b+1/6*cosh(b*x+a)*sinh(b*x+a)^5/b

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 8} \[ \int \sinh ^6(a+b x) \, dx=\frac {\sinh ^5(a+b x) \cosh (a+b x)}{6 b}-\frac {5 \sinh ^3(a+b x) \cosh (a+b x)}{24 b}+\frac {5 \sinh (a+b x) \cosh (a+b x)}{16 b}-\frac {5 x}{16} \]

[In]

Int[Sinh[a + b*x]^6,x]

[Out]

(-5*x)/16 + (5*Cosh[a + b*x]*Sinh[a + b*x])/(16*b) - (5*Cosh[a + b*x]*Sinh[a + b*x]^3)/(24*b) + (Cosh[a + b*x]
*Sinh[a + b*x]^5)/(6*b)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rubi steps \begin{align*} \text {integral}& = \frac {\cosh (a+b x) \sinh ^5(a+b x)}{6 b}-\frac {5}{6} \int \sinh ^4(a+b x) \, dx \\ & = -\frac {5 \cosh (a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh (a+b x) \sinh ^5(a+b x)}{6 b}+\frac {5}{8} \int \sinh ^2(a+b x) \, dx \\ & = \frac {5 \cosh (a+b x) \sinh (a+b x)}{16 b}-\frac {5 \cosh (a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh (a+b x) \sinh ^5(a+b x)}{6 b}-\frac {5 \int 1 \, dx}{16} \\ & = -\frac {5 x}{16}+\frac {5 \cosh (a+b x) \sinh (a+b x)}{16 b}-\frac {5 \cosh (a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh (a+b x) \sinh ^5(a+b x)}{6 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64 \[ \int \sinh ^6(a+b x) \, dx=\frac {-60 a-60 b x+45 \sinh (2 (a+b x))-9 \sinh (4 (a+b x))+\sinh (6 (a+b x))}{192 b} \]

[In]

Integrate[Sinh[a + b*x]^6,x]

[Out]

(-60*a - 60*b*x + 45*Sinh[2*(a + b*x)] - 9*Sinh[4*(a + b*x)] + Sinh[6*(a + b*x)])/(192*b)

Maple [A] (verified)

Time = 1.64 (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	\(\frac {\left (\frac {\sinh \left (b x +a \right )^{5}}{6}-\frac {5 \sinh \left (b x +a \right )^{3}}{24}+\frac {5 \sinh \left (b x +a \right )}{16}\right ) \cosh \left (b x +a \right )-\frac {5 b x}{16}-\frac {5 a}{16}}{b}\)	\(49\)
default	\(\frac {\left (\frac {\sinh \left (b x +a \right )^{5}}{6}-\frac {5 \sinh \left (b x +a \right )^{3}}{24}+\frac {5 \sinh \left (b x +a \right )}{16}\right ) \cosh \left (b x +a \right )-\frac {5 b x}{16}-\frac {5 a}{16}}{b}\)	\(49\)
risch	\(-\frac {5 x}{16}+\frac {{\mathrm e}^{6 b x +6 a}}{384 b}-\frac {3 \,{\mathrm e}^{4 b x +4 a}}{128 b}+\frac {15 \,{\mathrm e}^{2 b x +2 a}}{128 b}-\frac {15 \,{\mathrm e}^{-2 b x -2 a}}{128 b}+\frac {3 \,{\mathrm e}^{-4 b x -4 a}}{128 b}-\frac {{\mathrm e}^{-6 b x -6 a}}{384 b}\)	\(89\)

[In]

int(sinh(b*x+a)^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.34 \[ \int \sinh ^6(a+b x) \, dx=\frac {3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 2 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - 9 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 30 \, b x + 3 \, {\left (\cosh \left (b x + a\right )^{5} - 6 \, \cosh \left (b x + a\right )^{3} + 15 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{96 \, b} \]

[In]

integrate(sinh(b*x+a)^6,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (61) = 122\).

Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.07 \[ \int \sinh ^6(a+b x) \, dx=\begin {cases} \frac {5 x \sinh ^{6}{\left (a + b x \right )}}{16} - \frac {15 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} + \frac {15 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{16} - \frac {5 x \cosh ^{6}{\left (a + b x \right )}}{16} + \frac {11 \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{16 b} - \frac {5 \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{6 b} + \frac {5 \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x \sinh ^{6}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(sinh(b*x+a)**6,x)

[Out]

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)**6/16 + 11*sinh(a + b*x)**5*cosh(a + b*x)/(16*b) - 5*sinh(a + b*x)**3*cosh(a
+ b*x)**3/(6*b) + 5*sinh(a + b*x)*cosh(a + b*x)**5/(16*b), Ne(b, 0)), (x*sinh(a)**6, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.28 \[ \int \sinh ^6(a+b x) \, dx=-\frac {{\left (9 \, e^{\left (-2 \, b x - 2 \, a\right )} - 45 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} - \frac {5 \, {\left (b x + a\right )}}{16 \, b} - \frac {45 \, e^{\left (-2 \, b x - 2 \, a\right )} - 9 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \]

[In]

integrate(sinh(b*x+a)^6,x, algorithm="maxima")

[Out]

-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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31 \[ \int \sinh ^6(a+b x) \, dx=-\frac {5}{16} \, x + \frac {e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} - \frac {3 \, e^{\left (4 \, b x + 4 \, a\right )}}{128 \, b} + \frac {15 \, e^{\left (2 \, b x + 2 \, a\right )}}{128 \, b} - \frac {15 \, e^{\left (-2 \, b x - 2 \, a\right )}}{128 \, b} + \frac {3 \, e^{\left (-4 \, b x - 4 \, a\right )}}{128 \, b} - \frac {e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \]

[In]

integrate(sinh(b*x+a)^6,x, algorithm="giac")

[Out]

-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

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.63 \[ \int \sinh ^6(a+b x) \, dx=\frac {\frac {15\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{64}-\frac {3\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{64}+\frac {\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{192}}{b}-\frac {5\,x}{16} \]

[In]

int(sinh(a + b*x)^6,x)

[Out]

((15*sinh(2*a + 2*b*x))/64 - (3*sinh(4*a + 4*b*x))/64 + sinh(6*a + 6*b*x)/192)/b - (5*x)/16