3.2.45 \(\int (a+b \sinh ^3(c+d x)) \, dx\) [145]

Optimal. Leaf size=32 \[ a x-\frac {b \cosh (c+d x)}{d}+\frac {b \cosh ^3(c+d x)}{3 d} \]

[Out]

a*x-b*cosh(d*x+c)/d+1/3*b*cosh(d*x+c)^3/d

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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2713} \begin {gather*} a x+\frac {b \cosh ^3(c+d x)}{3 d}-\frac {b \cosh (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*Sinh[c + d*x]^3,x]

[Out]

a*x - (b*Cosh[c + d*x])/d + (b*Cosh[c + d*x]^3)/(3*d)

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

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Mathematica [A]
time = 0.01, size = 34, normalized size = 1.06 \begin {gather*} a x-\frac {3 b \cosh (c+d x)}{4 d}+\frac {b \cosh (3 (c+d x))}{12 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sinh[c + d*x]^3,x]

[Out]

a*x - (3*b*Cosh[c + d*x])/(4*d) + (b*Cosh[3*(c + d*x)])/(12*d)

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Maple [A]
time = 0.85, size = 33, normalized size = 1.03

method	result	size
default	\(a x +b \left (-\frac {3 \cosh \left (d x +c \right )}{4 d}+\frac {\cosh \left (3 d x +3 c \right )}{12 d}\right )\)	\(33\)
risch	\(a x +\frac {b \,{\mathrm e}^{3 d x +3 c}}{24 d}-\frac {3 b \,{\mathrm e}^{d x +c}}{8 d}-\frac {3 b \,{\mathrm e}^{-d x -c}}{8 d}+\frac {b \,{\mathrm e}^{-3 d x -3 c}}{24 d}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*sinh(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

a*x+b*(-3/4/d*cosh(d*x+c)+1/12/d*cosh(3*d*x+3*c))

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Maxima [A]
time = 0.27, size = 59, normalized size = 1.84 \begin {gather*} a x + \frac {1}{24} \, b {\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sinh(d*x+c)^3,x, algorithm="maxima")

[Out]

a*x + 1/24*b*(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)

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Fricas [A]
time = 0.45, size = 47, normalized size = 1.47 \begin {gather*} \frac {b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 12 \, a d x - 9 \, b \cosh \left (d x + c\right )}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sinh(d*x+c)^3,x, algorithm="fricas")

[Out]

1/12*(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + 12*a*d*x - 9*b*cosh(d*x + c))/d

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Sympy [A]
time = 0.11, size = 41, normalized size = 1.28 \begin {gather*} a x + b \left (\begin {cases} \frac {\sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 \cosh ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \sinh ^{3}{\left (c \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sinh(d*x+c)**3,x)

[Out]

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

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Giac [A]
time = 0.41, size = 59, normalized size = 1.84 \begin {gather*} a x + \frac {1}{24} \, b {\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sinh(d*x+c)^3,x, algorithm="giac")

[Out]

a*x + 1/24*b*(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)

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Mupad [B]
time = 0.62, size = 29, normalized size = 0.91 \begin {gather*} a\,x-\frac {b\,\mathrm {cosh}\left (c+d\,x\right )-\frac {b\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*sinh(c + d*x)^3,x)

[Out]

a*x - (b*cosh(c + d*x) - (b*cosh(c + d*x)^3)/3)/d

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