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

Optimal. Leaf size=115 \[ \frac {b^2 (3 a-4 b) \cosh ^7(c+d x)}{7 d}+\frac {3 b (a-2 b) (a-b) \cosh ^5(c+d x)}{5 d}+\frac {(a-4 b) (a-b)^2 \cosh ^3(c+d x)}{3 d}-\frac {(a-b)^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^9(c+d x)}{9 d} \]

[Out]

-(a-b)^3*cosh(d*x+c)/d+1/3*(a-4*b)*(a-b)^2*cosh(d*x+c)^3/d+3/5*(a-2*b)*(a-b)*b*cosh(d*x+c)^5/d+1/7*(3*a-4*b)*b
^2*cosh(d*x+c)^7/d+1/9*b^3*cosh(d*x+c)^9/d

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Rubi [A]  time = 0.13, antiderivative size = 115, 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 = {3186, 373} \[ \frac {b^2 (3 a-4 b) \cosh ^7(c+d x)}{7 d}+\frac {3 b (a-2 b) (a-b) \cosh ^5(c+d x)}{5 d}+\frac {(a-4 b) (a-b)^2 \cosh ^3(c+d x)}{3 d}-\frac {(a-b)^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^9(c+d x)}{9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a - b)^3*Cosh[c + d*x])/d) + ((a - 4*b)*(a - b)^2*Cosh[c + d*x]^3)/(3*d) + (3*(a - 2*b)*(a - b)*b*Cosh[c +
d*x]^5)/(5*d) + ((3*a - 4*b)*b^2*Cosh[c + d*x]^7)/(7*d) + (b^3*Cosh[c + d*x]^9)/(9*d)

Rule 373

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 3186

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(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 - b*ff^2*x^2)^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 ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \left (a-b+b x^2\right )^3 \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left ((a-b)^3-(a-4 b) (a-b)^2 x^2+3 (a-2 b) b (-a+b) x^4-(3 a-4 b) b^2 x^6-b^3 x^8\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {(a-b)^3 \cosh (c+d x)}{d}+\frac {(a-4 b) (a-b)^2 \cosh ^3(c+d x)}{3 d}+\frac {3 (a-2 b) (a-b) b \cosh ^5(c+d x)}{5 d}+\frac {(3 a-4 b) b^2 \cosh ^7(c+d x)}{7 d}+\frac {b^3 \cosh ^9(c+d x)}{9 d}\\ \end {align*}

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Mathematica [A]  time = 0.80, size = 127, normalized size = 1.10 \[ \frac {-1890 (4 a-3 b) \left (8 a^2-14 a b+7 b^2\right ) \cosh (c+d x)+420 \left (16 a^3-60 a^2 b+63 a b^2-21 b^3\right ) \cosh (3 (c+d x))+135 b^2 (4 a-3 b) \cosh (7 (c+d x))+756 b (4 a-3 b) (a-b) \cosh (5 (c+d x))+35 b^3 \cosh (9 (c+d x))}{80640 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1890*(4*a - 3*b)*(8*a^2 - 14*a*b + 7*b^2)*Cosh[c + d*x] + 420*(16*a^3 - 60*a^2*b + 63*a*b^2 - 21*b^3)*Cosh[3
*(c + d*x)] + 756*(4*a - 3*b)*(a - b)*b*Cosh[5*(c + d*x)] + 135*(4*a - 3*b)*b^2*Cosh[7*(c + d*x)] + 35*b^3*Cos
h[9*(c + d*x)])/(80640*d)

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fricas [B]  time = 0.51, size = 373, normalized size = 3.24 \[ \frac {35 \, b^{3} \cosh \left (d x + c\right )^{9} + 315 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} + 135 \, {\left (4 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 105 \, {\left (28 \, b^{3} \cosh \left (d x + c\right )^{3} + 9 \, {\left (4 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 756 \, {\left (4 \, a^{2} b - 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 315 \, {\left (14 \, b^{3} \cosh \left (d x + c\right )^{5} + 15 \, {\left (4 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 12 \, {\left (4 \, a^{2} b - 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 420 \, {\left (16 \, a^{3} - 60 \, a^{2} b + 63 \, a b^{2} - 21 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 315 \, {\left (4 \, b^{3} \cosh \left (d x + c\right )^{7} + 9 \, {\left (4 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 24 \, {\left (4 \, a^{2} b - 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 4 \, {\left (16 \, a^{3} - 60 \, a^{2} b + 63 \, a b^{2} - 21 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 1890 \, {\left (32 \, a^{3} - 80 \, a^{2} b + 70 \, a b^{2} - 21 \, b^{3}\right )} \cosh \left (d x + c\right )}{80640 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80640*(35*b^3*cosh(d*x + c)^9 + 315*b^3*cosh(d*x + c)*sinh(d*x + c)^8 + 135*(4*a*b^2 - 3*b^3)*cosh(d*x + c)^
7 + 105*(28*b^3*cosh(d*x + c)^3 + 9*(4*a*b^2 - 3*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 756*(4*a^2*b - 7*a*b^2
+ 3*b^3)*cosh(d*x + c)^5 + 315*(14*b^3*cosh(d*x + c)^5 + 15*(4*a*b^2 - 3*b^3)*cosh(d*x + c)^3 + 12*(4*a^2*b -
7*a*b^2 + 3*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 420*(16*a^3 - 60*a^2*b + 63*a*b^2 - 21*b^3)*cosh(d*x + c)^3
+ 315*(4*b^3*cosh(d*x + c)^7 + 9*(4*a*b^2 - 3*b^3)*cosh(d*x + c)^5 + 24*(4*a^2*b - 7*a*b^2 + 3*b^3)*cosh(d*x +
c)^3 + 4*(16*a^3 - 60*a^2*b + 63*a*b^2 - 21*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 1890*(32*a^3 - 80*a^2*b + 7
0*a*b^2 - 21*b^3)*cosh(d*x + c))/d

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giac [B]  time = 0.20, size = 296, normalized size = 2.57 \[ \frac {b^{3} e^{\left (9 \, d x + 9 \, c\right )}}{4608 \, d} + \frac {b^{3} e^{\left (-9 \, d x - 9 \, c\right )}}{4608 \, d} + \frac {3 \, {\left (4 \, a b^{2} - 3 \, b^{3}\right )} e^{\left (7 \, d x + 7 \, c\right )}}{3584 \, d} + \frac {3 \, {\left (4 \, a^{2} b - 7 \, a b^{2} + 3 \, b^{3}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {{\left (16 \, a^{3} - 60 \, a^{2} b + 63 \, a b^{2} - 21 \, b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{384 \, d} - \frac {3 \, {\left (32 \, a^{3} - 80 \, a^{2} b + 70 \, a b^{2} - 21 \, b^{3}\right )} e^{\left (d x + c\right )}}{256 \, d} - \frac {3 \, {\left (32 \, a^{3} - 80 \, a^{2} b + 70 \, a b^{2} - 21 \, b^{3}\right )} e^{\left (-d x - c\right )}}{256 \, d} + \frac {{\left (16 \, a^{3} - 60 \, a^{2} b + 63 \, a b^{2} - 21 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{384 \, d} + \frac {3 \, {\left (4 \, a^{2} b - 7 \, a b^{2} + 3 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} + \frac {3 \, {\left (4 \, a b^{2} - 3 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{3584 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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 - 3*b^3)*e^(7*d*x + 7*c)/d + 3/
640*(4*a^2*b - 7*a*b^2 + 3*b^3)*e^(5*d*x + 5*c)/d + 1/384*(16*a^3 - 60*a^2*b + 63*a*b^2 - 21*b^3)*e^(3*d*x + 3
*c)/d - 3/256*(32*a^3 - 80*a^2*b + 70*a*b^2 - 21*b^3)*e^(d*x + c)/d - 3/256*(32*a^3 - 80*a^2*b + 70*a*b^2 - 21
*b^3)*e^(-d*x - c)/d + 1/384*(16*a^3 - 60*a^2*b + 63*a*b^2 - 21*b^3)*e^(-3*d*x - 3*c)/d + 3/640*(4*a^2*b - 7*a
*b^2 + 3*b^3)*e^(-5*d*x - 5*c)/d + 3/3584*(4*a*b^2 - 3*b^3)*e^(-7*d*x - 7*c)/d

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maple [A]  time = 0.14, size = 158, normalized size = 1.37 \[ \frac {b^{3} \left (\frac {128}{315}+\frac {\left (\sinh ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sinh ^{6}\left (d x +c \right )\right )}{63}+\frac {16 \left (\sinh ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sinh ^{2}\left (d x +c \right )\right )}{315}\right ) \cosh \left (d x +c \right )+3 a \,b^{2} \left (-\frac {16}{35}+\frac {\left (\sinh ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sinh ^{4}\left (d x +c \right )\right )}{35}+\frac {8 \left (\sinh ^{2}\left (d x +c \right )\right )}{35}\right ) \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {8}{15}+\frac {\left (\sinh ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sinh ^{2}\left (d x +c \right )\right )}{15}\right ) \cosh \left (d x +c \right )+a^{3} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^3*(128/315+1/9*sinh(d*x+c)^8-8/63*sinh(d*x+c)^6+16/105*sinh(d*x+c)^4-64/315*sinh(d*x+c)^2)*cosh(d*x+c)+
3*a*b^2*(-16/35+1/7*sinh(d*x+c)^6-6/35*sinh(d*x+c)^4+8/35*sinh(d*x+c)^2)*cosh(d*x+c)+3*a^2*b*(8/15+1/5*sinh(d*
x+c)^4-4/15*sinh(d*x+c)^2)*cosh(d*x+c)+a^3*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c))

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maxima [B]  time = 0.49, size = 376, normalized size = 3.27 \[ -\frac {1}{161280} \, b^{3} {\left (\frac {{\left (405 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2268 \, e^{\left (-4 \, d x - 4 \, c\right )} + 8820 \, e^{\left (-6 \, d x - 6 \, c\right )} - 39690 \, e^{\left (-8 \, d x - 8 \, c\right )} - 35\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac {39690 \, e^{\left (-d x - c\right )} - 8820 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2268 \, e^{\left (-5 \, d x - 5 \, c\right )} - 405 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d}\right )} - \frac {3}{4480} \, a b^{2} {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{160} \, a^{2} b {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/161280*b^3*((405*e^(-2*d*x - 2*c) - 2268*e^(-4*d*x - 4*c) + 8820*e^(-6*d*x - 6*c) - 39690*e^(-8*d*x - 8*c)
- 35)*e^(9*d*x + 9*c)/d - (39690*e^(-d*x - c) - 8820*e^(-3*d*x - 3*c) + 2268*e^(-5*d*x - 5*c) - 405*e^(-7*d*x
- 7*c) + 35*e^(-9*d*x - 9*c))/d) - 3/4480*a*b^2*((49*e^(-2*d*x - 2*c) - 245*e^(-4*d*x - 4*c) + 1225*e^(-6*d*x
- 6*c) - 5)*e^(7*d*x + 7*c)/d + (1225*e^(-d*x - c) - 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x
- 7*c))/d) + 1/160*a^2*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^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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mupad [B]  time = 0.43, size = 185, normalized size = 1.61 \[ \frac {\frac {a^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}-a^3\,\mathrm {cosh}\left (c+d\,x\right )+\frac {3\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-2\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3+3\,a^2\,b\,\mathrm {cosh}\left (c+d\,x\right )+\frac {3\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}-\frac {9\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}+3\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3-3\,a\,b^2\,\mathrm {cosh}\left (c+d\,x\right )+\frac {b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{9}-\frac {4\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+\frac {6\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {4\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+b^3\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*cosh(c + d*x) - a^3*cosh(c + d*x) + (a^3*cosh(c + d*x)^3)/3 - (4*b^3*cosh(c + d*x)^3)/3 + (6*b^3*cosh(c +
d*x)^5)/5 - (4*b^3*cosh(c + d*x)^7)/7 + (b^3*cosh(c + d*x)^9)/9 + 3*a*b^2*cosh(c + d*x)^3 - 2*a^2*b*cosh(c +
d*x)^3 - (9*a*b^2*cosh(c + d*x)^5)/5 + (3*a^2*b*cosh(c + d*x)^5)/5 + (3*a*b^2*cosh(c + d*x)^7)/7 - 3*a*b^2*cos
h(c + d*x) + 3*a^2*b*cosh(c + d*x))/d

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sympy [A]  time = 14.69, size = 330, normalized size = 2.87 \[ \begin {cases} \frac {a^{3} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 a^{3} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {3 a^{2} b \sinh ^{4}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {8 a^{2} b \cosh ^{5}{\left (c + d x \right )}}{5 d} + \frac {3 a b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {6 a b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {24 a b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {48 a b^{2} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {b^{3} \sinh ^{8}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {16 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {64 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {128 b^{3} \cosh ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\relax (c )}\right )^{3} \sinh ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*sinh(c + d*x)**2*cosh(c + d*x)/d - 2*a**3*cosh(c + d*x)**3/(3*d) + 3*a**2*b*sinh(c + d*x)**4*c
osh(c + d*x)/d - 4*a**2*b*sinh(c + d*x)**2*cosh(c + d*x)**3/d + 8*a**2*b*cosh(c + d*x)**5/(5*d) + 3*a*b**2*sin
h(c + d*x)**6*cosh(c + d*x)/d - 6*a*b**2*sinh(c + d*x)**4*cosh(c + d*x)**3/d + 24*a*b**2*sinh(c + d*x)**2*cosh
(c + d*x)**5/(5*d) - 48*a*b**2*cosh(c + d*x)**7/(35*d) + b**3*sinh(c + d*x)**8*cosh(c + d*x)/d - 8*b**3*sinh(c
+ d*x)**6*cosh(c + d*x)**3/(3*d) + 16*b**3*sinh(c + d*x)**4*cosh(c + d*x)**5/(5*d) - 64*b**3*sinh(c + d*x)**2
*cosh(c + d*x)**7/(35*d) + 128*b**3*cosh(c + d*x)**9/(315*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)**3*sinh(c)**3,
True))

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