3.4.5 \(\int \cosh ^3(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\) [305]

Optimal. Leaf size=98 \[ \frac {a^3 \sinh (c+d x)}{d}+\frac {a^2 (a+3 b) \sinh ^3(c+d x)}{3 d}+\frac {3 a b (a+b) \sinh ^5(c+d x)}{5 d}+\frac {b^2 (3 a+b) \sinh ^7(c+d x)}{7 d}+\frac {b^3 \sinh ^9(c+d x)}{9 d} \]

[Out]

a^3*sinh(d*x+c)/d+1/3*a^2*(a+3*b)*sinh(d*x+c)^3/d+3/5*a*b*(a+b)*sinh(d*x+c)^5/d+1/7*b^2*(3*a+b)*sinh(d*x+c)^7/
d+1/9*b^3*sinh(d*x+c)^9/d

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Rubi [A]
time = 0.06, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3269, 380} \begin {gather*} \frac {a^3 \sinh (c+d x)}{d}+\frac {a^2 (a+3 b) \sinh ^3(c+d x)}{3 d}+\frac {b^2 (3 a+b) \sinh ^7(c+d x)}{7 d}+\frac {3 a b (a+b) \sinh ^5(c+d x)}{5 d}+\frac {b^3 \sinh ^9(c+d x)}{9 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Sinh[c + d*x])/d + (a^2*(a + 3*b)*Sinh[c + d*x]^3)/(3*d) + (3*a*b*(a + b)*Sinh[c + d*x]^5)/(5*d) + (b^2*(
3*a + b)*Sinh[c + d*x]^7)/(7*d) + (b^3*Sinh[c + d*x]^9)/(9*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 )^3 \, dx &=\frac {\text {Subst}\left (\int \left (1+x^2\right ) \left (a+b x^2\right )^3 \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (a^3+a^2 (a+3 b) x^2+3 a b (a+b) x^4+b^2 (3 a+b) x^6+b^3 x^8\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^3 \sinh (c+d x)}{d}+\frac {a^2 (a+3 b) \sinh ^3(c+d x)}{3 d}+\frac {3 a b (a+b) \sinh ^5(c+d x)}{5 d}+\frac {b^2 (3 a+b) \sinh ^7(c+d x)}{7 d}+\frac {b^3 \sinh ^9(c+d x)}{9 d}\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 125, normalized size = 1.28 \begin {gather*} \frac {1890 \left (32 a^3-16 a^2 b+6 a b^2-b^3\right ) \sinh (c+d x)+420 \left (16 a^3+12 a^2 b-9 a b^2+2 b^3\right ) \sinh (3 (c+d x))+b (756 a (4 a-b) \sinh (5 (c+d x))+5 b (27 (4 a-b) \sinh (7 (c+d x))+7 b \sinh (9 (c+d x))))}{80640 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1890*(32*a^3 - 16*a^2*b + 6*a*b^2 - b^3)*Sinh[c + d*x] + 420*(16*a^3 + 12*a^2*b - 9*a*b^2 + 2*b^3)*Sinh[3*(c
+ d*x)] + b*(756*a*(4*a - b)*Sinh[5*(c + d*x)] + 5*b*(27*(4*a - b)*Sinh[7*(c + d*x)] + 7*b*Sinh[9*(c + d*x)]))
)/(80640*d)

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Maple [A]
time = 2.20, size = 142, normalized size = 1.45

method	result	size
default	\(\frac {\left (-\frac {3}{256} b^{3}+\frac {3}{64} a \,b^{2}\right ) \sinh \left (7 d x +7 c \right )}{7 d}+\frac {\left (-\frac {3}{64} a \,b^{2}+\frac {3}{16} a^{2} b \right ) \sinh \left (5 d x +5 c \right )}{5 d}+\frac {\left (-\frac {3}{128} b^{3}+\frac {9}{64} a \,b^{2}-\frac {3}{8} a^{2} b +\frac {3}{4} a^{3}\right ) \sinh \left (d x +c \right )}{d}+\frac {\left (\frac {1}{32} b^{3}-\frac {9}{64} a \,b^{2}+\frac {3}{16} a^{2} b +\frac {1}{4} a^{3}\right ) \sinh \left (3 d x +3 c \right )}{3 d}+\frac {b^{3} \sinh \left (9 d x +9 c \right )}{2304 d}\)	\(142\)
risch	\(-\frac {3 a \,b^{2} {\mathrm e}^{5 d x +5 c}}{640 d}-\frac {3 a \,b^{2} {\mathrm e}^{3 d x +3 c}}{128 d}-\frac {3 b \,{\mathrm e}^{d x +c} a^{2}}{16 d}+\frac {b^{3} {\mathrm e}^{9 d x +9 c}}{4608 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{3}}{24 d}+\frac {{\mathrm e}^{3 d x +3 c} b^{3}}{192 d}+\frac {3 a^{3} {\mathrm e}^{d x +c}}{8 d}-\frac {b^{3} {\mathrm e}^{-9 d x -9 c}}{4608 d}-\frac {9 a \,{\mathrm e}^{-d x -c} b^{2}}{128 d}+\frac {3 a \,b^{2} {\mathrm e}^{-3 d x -3 c}}{128 d}-\frac {3 b^{3} {\mathrm e}^{d x +c}}{256 d}+\frac {3 b^{3} {\mathrm e}^{-d x -c}}{256 d}-\frac {b^{3} {\mathrm e}^{-3 d x -3 c}}{192 d}-\frac {3 b^{3} {\mathrm e}^{7 d x +7 c}}{3584 d}+\frac {3 a \,b^{2} {\mathrm e}^{-5 d x -5 c}}{640 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{3}}{8 d}-\frac {{\mathrm e}^{-3 d x -3 c} a^{3}}{24 d}+\frac {3 b^{3} {\mathrm e}^{-7 d x -7 c}}{3584 d}+\frac {9 a \,{\mathrm e}^{d x +c} b^{2}}{128 d}+\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{16 d}-\frac {{\mathrm e}^{-3 d x -3 c} a^{2} b}{32 d}-\frac {3 \,{\mathrm e}^{-5 d x -5 c} a^{2} b}{160 d}+\frac {3 b^{2} {\mathrm e}^{7 d x +7 c} a}{896 d}+\frac {3 \,{\mathrm e}^{5 d x +5 c} a^{2} b}{160 d}-\frac {3 b^{2} {\mathrm e}^{-7 d x -7 c} a}{896 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2} b}{32 d}\)	\(446\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(-3/256*b^3+3/64*a*b^2)*sinh(7*d*x+7*c)/d+1/5*(-3/64*a*b^2+3/16*a^2*b)*sinh(5*d*x+5*c)/d+(-3/128*b^3+9/64*
a*b^2-3/8*a^2*b+3/4*a^3)*sinh(d*x+c)/d+1/3*(1/32*b^3-9/64*a*b^2+3/16*a^2*b+1/4*a^3)*sinh(3*d*x+3*c)/d+1/2304*b
^3*sinh(9*d*x+9*c)/d

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (90) = 180\).
time = 0.27, size = 349, normalized size = 3.56 \begin {gather*} -\frac {1}{32256} \, b^{3} {\left (\frac {{\left (27 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-6 \, d x - 6 \, c\right )} + 378 \, e^{\left (-8 \, d x - 8 \, c\right )} - 7\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac {378 \, e^{\left (-d x - c\right )} - 168 \, e^{\left (-3 \, d x - 3 \, c\right )} + 27 \, e^{\left (-7 \, d x - 7 \, c\right )} - 7 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d}\right )} - \frac {3}{4480} \, a b^{2} {\left (\frac {{\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} - 105 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {105 \, e^{\left (-d x - c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 7 \, e^{\left (-5 \, d x - 5 \, c\right )} + 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{160} \, a^{2} 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^{3} {\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)^3,x, algorithm="maxima")

[Out]

-1/32256*b^3*((27*e^(-2*d*x - 2*c) - 168*e^(-6*d*x - 6*c) + 378*e^(-8*d*x - 8*c) - 7)*e^(9*d*x + 9*c)/d - (378
*e^(-d*x - c) - 168*e^(-3*d*x - 3*c) + 27*e^(-7*d*x - 7*c) - 7*e^(-9*d*x - 9*c))/d) - 3/4480*a*b^2*((7*e^(-2*d
*x - 2*c) + 35*e^(-4*d*x - 4*c) - 105*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/d + (105*e^(-d*x - c) - 35*e^(-3*d
*x - 3*c) - 7*e^(-5*d*x - 5*c) + 5*e^(-7*d*x - 7*c))/d) + 1/160*a^2*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^3*(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. 324 vs. \(2 (90) = 180\).
time = 0.38, size = 324, normalized size = 3.31 \begin {gather*} \frac {35 \, b^{3} \sinh \left (d x + c\right )^{9} + 45 \, {\left (28 \, b^{3} \cosh \left (d x + c\right )^{2} + 12 \, a b^{2} - 3 \, b^{3}\right )} \sinh \left (d x + c\right )^{7} + 63 \, {\left (70 \, b^{3} \cosh \left (d x + c\right )^{4} + 48 \, a^{2} b - 12 \, a b^{2} + 45 \, {\left (4 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} + 105 \, {\left (28 \, b^{3} \cosh \left (d x + c\right )^{6} + 45 \, {\left (4 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{4} + 64 \, a^{3} + 48 \, a^{2} b - 36 \, a b^{2} + 8 \, b^{3} + 72 \, {\left (4 \, a^{2} b - a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 315 \, {\left (b^{3} \cosh \left (d x + c\right )^{8} + 3 \, {\left (4 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{6} + 12 \, {\left (4 \, a^{2} b - a b^{2}\right )} \cosh \left (d x + c\right )^{4} + 192 \, a^{3} - 96 \, a^{2} b + 36 \, a b^{2} - 6 \, b^{3} + 4 \, {\left (16 \, a^{3} + 12 \, a^{2} b - 9 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{80640 \, 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)^3,x, algorithm="fricas")

[Out]

1/80640*(35*b^3*sinh(d*x + c)^9 + 45*(28*b^3*cosh(d*x + c)^2 + 12*a*b^2 - 3*b^3)*sinh(d*x + c)^7 + 63*(70*b^3*
cosh(d*x + c)^4 + 48*a^2*b - 12*a*b^2 + 45*(4*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 105*(28*b^3*cosh
(d*x + c)^6 + 45*(4*a*b^2 - b^3)*cosh(d*x + c)^4 + 64*a^3 + 48*a^2*b - 36*a*b^2 + 8*b^3 + 72*(4*a^2*b - a*b^2)
*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 315*(b^3*cosh(d*x + c)^8 + 3*(4*a*b^2 - b^3)*cosh(d*x + c)^6 + 12*(4*a^2*b
- a*b^2)*cosh(d*x + c)^4 + 192*a^3 - 96*a^2*b + 36*a*b^2 - 6*b^3 + 4*(16*a^3 + 12*a^2*b - 9*a*b^2 + 2*b^3)*co
sh(d*x + c)^2)*sinh(d*x + c))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (87) = 174\).
time = 1.38, size = 182, normalized size = 1.86 \begin {gather*} \begin {cases} - \frac {2 a^{3} \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{3} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} - \frac {2 a^{2} b \sinh ^{5}{\left (c + d x \right )}}{5 d} + \frac {a^{2} b \sinh ^{3}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} - \frac {6 a b^{2} \sinh ^{7}{\left (c + d x \right )}}{35 d} + \frac {3 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{5 d} - \frac {2 b^{3} \sinh ^{9}{\left (c + d x \right )}}{63 d} + \frac {b^{3} \sinh ^{7}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{3} \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)**3,x)

[Out]

Piecewise((-2*a**3*sinh(c + d*x)**3/(3*d) + a**3*sinh(c + d*x)*cosh(c + d*x)**2/d - 2*a**2*b*sinh(c + d*x)**5/
(5*d) + a**2*b*sinh(c + d*x)**3*cosh(c + d*x)**2/d - 6*a*b**2*sinh(c + d*x)**7/(35*d) + 3*a*b**2*sinh(c + d*x)
**5*cosh(c + d*x)**2/(5*d) - 2*b**3*sinh(c + d*x)**9/(63*d) + b**3*sinh(c + d*x)**7*cosh(c + d*x)**2/(7*d), Ne
(d, 0)), (x*(a + b*sinh(c)**2)**3*cosh(c)**3, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (90) = 180\).
time = 0.46, size = 286, normalized size = 2.92 \begin {gather*} \frac {b^{3} e^{\left (9 \, d x + 9 \, c\right )}}{4608 \, d} - \frac {b^{3} e^{\left (-9 \, d x - 9 \, c\right )}}{4608 \, d} + \frac {3 \, {\left (4 \, a b^{2} - b^{3}\right )} e^{\left (7 \, d x + 7 \, c\right )}}{3584 \, d} + \frac {3 \, {\left (4 \, a^{2} b - a b^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {{\left (16 \, a^{3} + 12 \, a^{2} b - 9 \, a b^{2} + 2 \, b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{384 \, d} + \frac {3 \, {\left (32 \, a^{3} - 16 \, a^{2} b + 6 \, a b^{2} - b^{3}\right )} e^{\left (d x + c\right )}}{256 \, d} - \frac {3 \, {\left (32 \, a^{3} - 16 \, a^{2} b + 6 \, a b^{2} - b^{3}\right )} e^{\left (-d x - c\right )}}{256 \, d} - \frac {{\left (16 \, a^{3} + 12 \, a^{2} b - 9 \, a b^{2} + 2 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{384 \, d} - \frac {3 \, {\left (4 \, a^{2} b - a b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} - \frac {3 \, {\left (4 \, a b^{2} - b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{3584 \, 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)^3,x, algorithm="giac")

[Out]

1/4608*b^3*e^(9*d*x + 9*c)/d - 1/4608*b^3*e^(-9*d*x - 9*c)/d + 3/3584*(4*a*b^2 - b^3)*e^(7*d*x + 7*c)/d + 3/64
0*(4*a^2*b - a*b^2)*e^(5*d*x + 5*c)/d + 1/384*(16*a^3 + 12*a^2*b - 9*a*b^2 + 2*b^3)*e^(3*d*x + 3*c)/d + 3/256*
(32*a^3 - 16*a^2*b + 6*a*b^2 - b^3)*e^(d*x + c)/d - 3/256*(32*a^3 - 16*a^2*b + 6*a*b^2 - b^3)*e^(-d*x - c)/d -
1/384*(16*a^3 + 12*a^2*b - 9*a*b^2 + 2*b^3)*e^(-3*d*x - 3*c)/d - 3/640*(4*a^2*b - a*b^2)*e^(-5*d*x - 5*c)/d -
3/3584*(4*a*b^2 - b^3)*e^(-7*d*x - 7*c)/d

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Mupad [B]
time = 0.29, size = 112, normalized size = 1.14 \begin {gather*} \frac {105\,a^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3+315\,a^3\,\mathrm {sinh}\left (c+d\,x\right )+189\,a^2\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^5+315\,a^2\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^3+135\,a\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^7+189\,a\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^5+35\,b^3\,{\mathrm {sinh}\left (c+d\,x\right )}^9+45\,b^3\,{\mathrm {sinh}\left (c+d\,x\right )}^7}{315\,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)^3,x)

[Out]

(315*a^3*sinh(c + d*x) + 105*a^3*sinh(c + d*x)^3 + 45*b^3*sinh(c + d*x)^7 + 35*b^3*sinh(c + d*x)^9 + 315*a^2*b
*sinh(c + d*x)^3 + 189*a*b^2*sinh(c + d*x)^5 + 189*a^2*b*sinh(c + d*x)^5 + 135*a*b^2*sinh(c + d*x)^7)/(315*d)

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