3.2.87 \(\int \sinh (c+d x) (a+b \sinh ^4(c+d x)) \, dx\) [187]

Optimal. Leaf size=46 \[ \frac {(a+b) \cosh (c+d x)}{d}-\frac {2 b \cosh ^3(c+d x)}{3 d}+\frac {b \cosh ^5(c+d x)}{5 d} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3294} \begin {gather*} \frac {(a+b) \cosh (c+d x)}{d}+\frac {b \cosh ^5(c+d x)}{5 d}-\frac {2 b \cosh ^3(c+d x)}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^4),x]

[Out]

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

Rule 3294

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 69, normalized size = 1.50 \begin {gather*} \frac {a \cosh (c) \cosh (d x)}{d}+\frac {5 b \cosh (c+d x)}{8 d}-\frac {5 b \cosh (3 (c+d x))}{48 d}+\frac {b \cosh (5 (c+d x))}{80 d}+\frac {a \sinh (c) \sinh (d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^4),x]

[Out]

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

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Maple [A]
time = 0.71, size = 47, normalized size = 1.02

method	result	size
default	\(\frac {\left (\frac {5 b}{8}+a \right ) \cosh \left (d x +c \right )}{d}-\frac {5 b \cosh \left (3 d x +3 c \right )}{48 d}+\frac {b \cosh \left (5 d x +5 c \right )}{80 d}\)	\(47\)
risch	\(\frac {b \,{\mathrm e}^{5 d x +5 c}}{160 d}-\frac {5 b \,{\mathrm e}^{3 d x +3 c}}{96 d}+\frac {a \,{\mathrm e}^{d x +c}}{2 d}+\frac {5 b \,{\mathrm e}^{d x +c}}{16 d}+\frac {{\mathrm e}^{-d x -c} a}{2 d}+\frac {5 \,{\mathrm e}^{-d x -c} b}{16 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} b}{96 d}+\frac {b \,{\mathrm e}^{-5 d x -5 c}}{160 d}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (42) = 84\).
time = 0.26, size = 97, normalized size = 2.11 \begin {gather*} \frac {1}{480} \, b {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {a \cosh \left (d x + c\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*b*(3*e^(5*d*x + 5*c)/d - 25*e^(3*d*x + 3*c)/d + 150*e^(d*x + c)/d + 150*e^(-d*x - c)/d - 25*e^(-3*d*x -
3*c)/d + 3*e^(-5*d*x - 5*c)/d) + a*cosh(d*x + c)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (42) = 84\).
time = 0.57, size = 91, normalized size = 1.98 \begin {gather*} \frac {3 \, b \cosh \left (d x + c\right )^{5} + 15 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - 25 \, b \cosh \left (d x + c\right )^{3} + 15 \, {\left (2 \, b \cosh \left (d x + c\right )^{3} - 5 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 30 \, {\left (8 \, a + 5 \, b\right )} \cosh \left (d x + c\right )}{240 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(3*b*cosh(d*x + c)^5 + 15*b*cosh(d*x + c)*sinh(d*x + c)^4 - 25*b*cosh(d*x + c)^3 + 15*(2*b*cosh(d*x + c)
^3 - 5*b*cosh(d*x + c))*sinh(d*x + c)^2 + 30*(8*a + 5*b)*cosh(d*x + c))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (39) = 78\).
time = 0.28, size = 80, normalized size = 1.74 \begin {gather*} \begin {cases} \frac {a \cosh {\left (c + d x \right )}}{d} + \frac {b \sinh ^{4}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {8 b \cosh ^{5}{\left (c + d x \right )}}{15 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right ) \sinh {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (42) = 84\).
time = 0.42, size = 100, normalized size = 2.17 \begin {gather*} \frac {b e^{\left (5 \, d x + 5 \, c\right )}}{160 \, d} - \frac {5 \, b e^{\left (3 \, d x + 3 \, c\right )}}{96 \, d} + \frac {{\left (8 \, a + 5 \, b\right )} e^{\left (d x + c\right )}}{16 \, d} + \frac {{\left (8 \, a + 5 \, b\right )} e^{\left (-d x - c\right )}}{16 \, d} - \frac {5 \, b e^{\left (-3 \, d x - 3 \, c\right )}}{96 \, d} + \frac {b e^{\left (-5 \, d x - 5 \, c\right )}}{160 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/160*b*e^(5*d*x + 5*c)/d - 5/96*b*e^(3*d*x + 3*c)/d + 1/16*(8*a + 5*b)*e^(d*x + c)/d + 1/16*(8*a + 5*b)*e^(-d
*x - c)/d - 5/96*b*e^(-3*d*x - 3*c)/d + 1/160*b*e^(-5*d*x - 5*c)/d

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Mupad [B]
time = 0.10, size = 46, normalized size = 1.00 \begin {gather*} \frac {15\,a\,\mathrm {cosh}\left (c+d\,x\right )+15\,b\,\mathrm {cosh}\left (c+d\,x\right )-10\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3+3\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{15\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)*(a + b*sinh(c + d*x)^4),x)

[Out]

(15*a*cosh(c + d*x) + 15*b*cosh(c + d*x) - 10*b*cosh(c + d*x)^3 + 3*b*cosh(c + d*x)^5)/(15*d)

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