3.1.5 \(\int \sinh ^5(a+b x) \, dx\) [5]

Optimal. Leaf size=41 \[ \frac {\cosh (a+b x)}{b}-\frac {2 \cosh ^3(a+b x)}{3 b}+\frac {\cosh ^5(a+b x)}{5 b} \]

[Out]

cosh(b*x+a)/b-2/3*cosh(b*x+a)^3/b+1/5*cosh(b*x+a)^5/b

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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2713} \begin {gather*} \frac {\cosh ^5(a+b x)}{5 b}-\frac {2 \cosh ^3(a+b x)}{3 b}+\frac {\cosh (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]^5,x]

[Out]

Cosh[a + b*x]/b - (2*Cosh[a + b*x]^3)/(3*b) + Cosh[a + b*x]^5/(5*b)

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin {align*} \int \sinh ^5(a+b x) \, dx &=\frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac {\cosh (a+b x)}{b}-\frac {2 \cosh ^3(a+b x)}{3 b}+\frac {\cosh ^5(a+b x)}{5 b}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 44, normalized size = 1.07 \begin {gather*} \frac {5 \cosh (a+b x)}{8 b}-\frac {5 \cosh (3 (a+b x))}{48 b}+\frac {\cosh (5 (a+b x))}{80 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x]^5,x]

[Out]

(5*Cosh[a + b*x])/(8*b) - (5*Cosh[3*(a + b*x)])/(48*b) + Cosh[5*(a + b*x)]/(80*b)

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Maple [A]
time = 0.44, size = 41, normalized size = 1.00

method	result	size
default	\(\frac {5 \cosh \left (b x +a \right )}{8 b}-\frac {5 \cosh \left (3 b x +3 a \right )}{48 b}+\frac {\cosh \left (5 b x +5 a \right )}{80 b}\)	\(41\)
risch	\(\frac {{\mathrm e}^{5 b x +5 a}}{160 b}-\frac {5 \,{\mathrm e}^{3 b x +3 a}}{96 b}+\frac {5 \,{\mathrm e}^{b x +a}}{16 b}+\frac {5 \,{\mathrm e}^{-b x -a}}{16 b}-\frac {5 \,{\mathrm e}^{-3 b x -3 a}}{96 b}+\frac {{\mathrm e}^{-5 b x -5 a}}{160 b}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

5/8*cosh(b*x+a)/b-5/48/b*cosh(3*b*x+3*a)+1/80/b*cosh(5*b*x+5*a)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (37) = 74\).
time = 0.27, size = 82, normalized size = 2.00 \begin {gather*} \frac {e^{\left (5 \, b x + 5 \, a\right )}}{160 \, b} - \frac {5 \, e^{\left (3 \, b x + 3 \, a\right )}}{96 \, b} + \frac {5 \, e^{\left (b x + a\right )}}{16 \, b} + \frac {5 \, e^{\left (-b x - a\right )}}{16 \, b} - \frac {5 \, e^{\left (-3 \, b x - 3 \, a\right )}}{96 \, b} + \frac {e^{\left (-5 \, b x - 5 \, a\right )}}{160 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)^5,x, algorithm="maxima")

[Out]

1/160*e^(5*b*x + 5*a)/b - 5/96*e^(3*b*x + 3*a)/b + 5/16*e^(b*x + a)/b + 5/16*e^(-b*x - a)/b - 5/96*e^(-3*b*x -
3*a)/b + 1/160*e^(-5*b*x - 5*a)/b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (37) = 74\).
time = 0.43, size = 79, normalized size = 1.93 \begin {gather*} \frac {3 \, \cosh \left (b x + a\right )^{5} + 15 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 25 \, \cosh \left (b x + a\right )^{3} + 15 \, {\left (2 \, \cosh \left (b x + a\right )^{3} - 5 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 150 \, \cosh \left (b x + a\right )}{240 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)^5,x, algorithm="fricas")

[Out]

1/240*(3*cosh(b*x + a)^5 + 15*cosh(b*x + a)*sinh(b*x + a)^4 - 25*cosh(b*x + a)^3 + 15*(2*cosh(b*x + a)^3 - 5*c
osh(b*x + a))*sinh(b*x + a)^2 + 150*cosh(b*x + a))/b

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Sympy [A]
time = 0.26, size = 58, normalized size = 1.41 \begin {gather*} \begin {cases} \frac {\sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {4 \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {8 \cosh ^{5}{\left (a + b x \right )}}{15 b} & \text {for}\: b \neq 0 \\x \sinh ^{5}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)**5,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (37) = 74\).
time = 0.41, size = 82, normalized size = 2.00 \begin {gather*} \frac {e^{\left (5 \, b x + 5 \, a\right )}}{160 \, b} - \frac {5 \, e^{\left (3 \, b x + 3 \, a\right )}}{96 \, b} + \frac {5 \, e^{\left (b x + a\right )}}{16 \, b} + \frac {5 \, e^{\left (-b x - a\right )}}{16 \, b} - \frac {5 \, e^{\left (-3 \, b x - 3 \, a\right )}}{96 \, b} + \frac {e^{\left (-5 \, b x - 5 \, a\right )}}{160 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)^5,x, algorithm="giac")

[Out]

1/160*e^(5*b*x + 5*a)/b - 5/96*e^(3*b*x + 3*a)/b + 5/16*e^(b*x + a)/b + 5/16*e^(-b*x - a)/b - 5/96*e^(-3*b*x -
3*a)/b + 1/160*e^(-5*b*x - 5*a)/b

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Mupad [B]
time = 0.41, size = 31, normalized size = 0.76 \begin {gather*} \frac {\frac {{\mathrm {cosh}\left (a+b\,x\right )}^5}{5}-\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3}+\mathrm {cosh}\left (a+b\,x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)^5,x)

[Out]

(cosh(a + b*x) - (2*cosh(a + b*x)^3)/3 + cosh(a + b*x)^5/5)/b

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