3.3.84 \(\int \cosh ^3(c+d x) (a+b \sinh ^2(c+d x)) \, dx\) [284]

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

[Out]

a*sinh(d*x+c)/d+1/3*(a+b)*sinh(d*x+c)^3/d+1/5*b*sinh(d*x+c)^5/d

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Rubi [A]
time = 0.03, antiderivative size = 46, 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 = {3269, 380} \begin {gather*} \frac {(a+b) \sinh ^3(c+d x)}{3 d}+\frac {a \sinh (c+d x)}{d}+\frac {b \sinh ^5(c+d x)}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sinh[c + d*x])/d + ((a + b)*Sinh[c + d*x]^3)/(3*d) + (b*Sinh[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 3269

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

Rubi steps

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

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Mathematica [A]
time = 0.11, size = 48, normalized size = 1.04 \begin {gather*} \frac {(100 a-11 b+4 (5 a+2 b) \cosh (2 (c+d x))+3 b \cosh (4 (c+d x))) \sinh (c+d x)}{120 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((100*a - 11*b + 4*(5*a + 2*b)*Cosh[2*(c + d*x)] + 3*b*Cosh[4*(c + d*x)])*Sinh[c + d*x])/(120*d)

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Maple [A]
time = 1.64, size = 55, normalized size = 1.20

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/8*b+3/4*a)*sinh(d*x+c)/d+1/3*(1/16*b+1/4*a)*sinh(3*d*x+3*c)/d+1/80*b*sinh(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. 136 vs. \(2 (42) = 84\).
time = 0.26, size = 136, normalized size = 2.96 \begin {gather*} \frac {1}{480} \, b {\left (\frac {{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 30 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3\right )} e^{\left (5 \, d x + 5 \, c\right )}}{d} + \frac {30 \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-3 \, d x - 3 \, c\right )} - 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(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/480*b*((5*e^(-2*d*x - 2*c) - 30*e^(-4*d*x - 4*c) + 3)*e^(5*d*x + 5*c)/d + (30*e^(-d*x - c) - 5*e^(-3*d*x - 3
*c) - 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 [A]
time = 0.38, size = 82, normalized size = 1.78 \begin {gather*} \frac {3 \, b \sinh \left (d x + c\right )^{5} + 5 \, {\left (6 \, b \cosh \left (d x + c\right )^{2} + 4 \, a + b\right )} \sinh \left (d x + c\right )^{3} + 15 \, {\left (b \cosh \left (d x + c\right )^{4} + {\left (4 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 12 \, a - 2 \, b\right )} \sinh \left (d x + c\right )}{240 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

1/240*(3*b*sinh(d*x + c)^5 + 5*(6*b*cosh(d*x + c)^2 + 4*a + b)*sinh(d*x + c)^3 + 15*(b*cosh(d*x + c)^4 + (4*a
+ b)*cosh(d*x + c)^2 + 12*a - 2*b)*sinh(d*x + c))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (39) = 78\).
time = 0.28, size = 85, normalized size = 1.85 \begin {gather*} \begin {cases} - \frac {2 a \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac {a \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} - \frac {2 b \sinh ^{5}{\left (c + d x \right )}}{15 d} + \frac {b \sinh ^{3}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \cosh ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)**3*(a+b*sinh(d*x+c)**2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (42) = 84\).
time = 0.42, size = 108, normalized size = 2.35 \begin {gather*} \frac {b e^{\left (5 \, d x + 5 \, c\right )}}{160 \, d} + \frac {{\left (4 \, a + b\right )} e^{\left (3 \, d x + 3 \, c\right )}}{96 \, d} + \frac {{\left (6 \, a - b\right )} e^{\left (d x + c\right )}}{16 \, d} - \frac {{\left (6 \, a - b\right )} e^{\left (-d x - c\right )}}{16 \, d} - \frac {{\left (4 \, a + 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(cosh(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 + b)*e^(3*d*x + 3*c)/d + 1/16*(6*a - b)*e^(d*x + c)/d - 1/16*(6*a - b)*e
^(-d*x - c)/d - 1/96*(4*a + 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 = 1.34, size = 48, normalized size = 1.04 \begin {gather*} \frac {15\,a\,\mathrm {sinh}\left (c+d\,x\right )+5\,a\,{\mathrm {sinh}\left (c+d\,x\right )}^3+5\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^3+3\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^5}{15\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^3*(a + b*sinh(c + d*x)^2),x)

[Out]

(15*a*sinh(c + d*x) + 5*a*sinh(c + d*x)^3 + 5*b*sinh(c + d*x)^3 + 3*b*sinh(c + d*x)^5)/(15*d)

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