3.1.2 \(\int \sinh ^3(c+d x) (a+b \sinh ^2(c+d x)) \, dx\) [2]

Optimal. Leaf size=53 \[ -\frac {(a-b) \cosh (c+d x)}{d}+\frac {(a-2 b) \cosh ^3(c+d x)}{3 d}+\frac {b \cosh ^5(c+d x)}{5 d} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3092, 380} \begin {gather*} \frac {(a-2 b) \cosh ^3(c+d x)}{3 d}-\frac {(a-b) \cosh (c+d x)}{d}+\frac {b \cosh ^5(c+d x)}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^2),x]

[Out]

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

Rule 380

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]

Rule 3092

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[-f^(-1), Subst[I
nt[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2
, 0]

Rubi steps

\begin {align*} \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \left (a-b+b x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (a \left (1-\frac {b}{a}\right )-(a-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 {(a-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 = 77, normalized size = 1.45 \begin {gather*} -\frac {3 a \cosh (c+d x)}{4 d}+\frac {5 b \cosh (c+d x)}{8 d}+\frac {a \cosh (3 (c+d x))}{12 d}-\frac {5 b \cosh (3 (c+d x))}{48 d}+\frac {b \cosh (5 (c+d x))}{80 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^2),x]

[Out]

(-3*a*Cosh[c + d*x])/(4*d) + (5*b*Cosh[c + d*x])/(8*d) + (a*Cosh[3*(c + d*x)])/(12*d) - (5*b*Cosh[3*(c + d*x)]
)/(48*d) + (b*Cosh[5*(c + d*x)])/(80*d)

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Maple [A]
time = 0.65, size = 55, normalized size = 1.04

method	result	size
default	\(\frac {\left (-\frac {5 b}{16}+\frac {a}{4}\right ) \cosh \left (3 d x +3 c \right )}{3 d}+\frac {\left (\frac {5 b}{8}-\frac {3 a}{4}\right ) \cosh \left (d x +c \right )}{d}+\frac {b \cosh \left (5 d x +5 c \right )}{80 d}\)	\(55\)
risch	\(\frac {b \,{\mathrm e}^{5 d x +5 c}}{160 d}+\frac {{\mathrm e}^{3 d x +3 c} a}{24 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} b}{96 d}-\frac {3 \,{\mathrm e}^{d x +c} a}{8 d}+\frac {5 b \,{\mathrm e}^{d x +c}}{16 d}-\frac {3 \,{\mathrm e}^{-d x -c} a}{8 d}+\frac {5 \,{\mathrm e}^{-d x -c} b}{16 d}+\frac {{\mathrm e}^{-3 d x -3 c} a}{24 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} b}{96 d}+\frac {b \,{\mathrm e}^{-5 d x -5 c}}{160 d}\)	\(146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/3*(-5/16*b+1/4*a)/d*cosh(3*d*x+3*c)+(5/8*b-3/4*a)*cosh(d*x+c)/d+1/80*b*cosh(5*d*x+5*c)/d

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (49) = 98\).
time = 0.26, size = 141, normalized size = 2.66 \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 {1}{24} \, a {\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(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^2),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) + 1/24*a*(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 [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (49) = 98\).
time = 0.41, size = 102, normalized size = 1.92 \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} + 5 \, {\left (4 \, a - 5 \, b\right )} \cosh \left (d x + c\right )^{3} + 15 \, {\left (2 \, b \cosh \left (d x + c\right )^{3} + {\left (4 \, a - 5 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 30 \, {\left (6 \, 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)^3*(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

1/240*(3*b*cosh(d*x + c)^5 + 15*b*cosh(d*x + c)*sinh(d*x + c)^4 + 5*(4*a - 5*b)*cosh(d*x + c)^3 + 15*(2*b*cosh
(d*x + c)^3 + (4*a - 5*b)*cosh(d*x + c))*sinh(d*x + c)^2 - 30*(6*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. 105 vs. \(2 (42) = 84\).
time = 0.27, size = 105, normalized size = 1.98 \begin {gather*} \begin {cases} \frac {a \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 a \cosh ^{3}{\left (c + d x \right )}}{3 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 ^{2}{\left (c \right )}\right ) \sinh ^{3}{\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)**3*(a+b*sinh(d*x+c)**2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (49) = 98\).
time = 0.43, size = 112, normalized size = 2.11 \begin {gather*} \frac {b e^{\left (5 \, d x + 5 \, c\right )}}{160 \, d} + \frac {{\left (4 \, a - 5 \, b\right )} e^{\left (3 \, d x + 3 \, c\right )}}{96 \, d} - \frac {{\left (6 \, a - 5 \, b\right )} e^{\left (d x + c\right )}}{16 \, d} - \frac {{\left (6 \, a - 5 \, b\right )} e^{\left (-d x - c\right )}}{16 \, d} + \frac {{\left (4 \, a - 5 \, b\right )} 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)^3*(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

1/160*b*e^(5*d*x + 5*c)/d + 1/96*(4*a - 5*b)*e^(3*d*x + 3*c)/d - 1/16*(6*a - 5*b)*e^(d*x + c)/d - 1/16*(6*a -
5*b)*e^(-d*x - c)/d + 1/96*(4*a - 5*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 = 0.64, size = 57, normalized size = 1.08 \begin {gather*} \frac {15\,b\,\mathrm {cosh}\left (c+d\,x\right )-15\,a\,\mathrm {cosh}\left (c+d\,x\right )+5\,a\,{\mathrm {cosh}\left (c+d\,x\right )}^3-10\,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)^3*(a + b*sinh(c + d*x)^2),x)

[Out]

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

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