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

Optimal. Leaf size=267 \[ \frac {a^3 \cosh (c+d x)}{d}+\frac {3 a^2 b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac {9 a^2 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {9}{8} a^2 b x+\frac {3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac {9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac {3 a b^2 \cosh ^3(c+d x)}{d}-\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {b^3 \sinh ^9(c+d x) \cosh (c+d x)}{10 d}-\frac {9 b^3 \sinh ^7(c+d x) \cosh (c+d x)}{80 d}+\frac {21 b^3 \sinh ^5(c+d x) \cosh (c+d x)}{160 d}-\frac {21 b^3 \sinh ^3(c+d x) \cosh (c+d x)}{128 d}+\frac {63 b^3 \sinh (c+d x) \cosh (c+d x)}{256 d}-\frac {63 b^3 x}{256} \]

[Out]

9/8*a^2*b*x-63/256*b^3*x+a^3*cosh(d*x+c)/d-3*a*b^2*cosh(d*x+c)/d+3*a*b^2*cosh(d*x+c)^3/d-9/5*a*b^2*cosh(d*x+c)
^5/d+3/7*a*b^2*cosh(d*x+c)^7/d-9/8*a^2*b*cosh(d*x+c)*sinh(d*x+c)/d+63/256*b^3*cosh(d*x+c)*sinh(d*x+c)/d+3/4*a^
2*b*cosh(d*x+c)*sinh(d*x+c)^3/d-21/128*b^3*cosh(d*x+c)*sinh(d*x+c)^3/d+21/160*b^3*cosh(d*x+c)*sinh(d*x+c)^5/d-
9/80*b^3*cosh(d*x+c)*sinh(d*x+c)^7/d+1/10*b^3*cosh(d*x+c)*sinh(d*x+c)^9/d

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Rubi [A]  time = 0.22, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3220, 2638, 2635, 8, 2633} \[ \frac {3 a^2 b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac {9 a^2 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {9}{8} a^2 b x+\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac {9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac {3 a b^2 \cosh ^3(c+d x)}{d}-\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {b^3 \sinh ^9(c+d x) \cosh (c+d x)}{10 d}-\frac {9 b^3 \sinh ^7(c+d x) \cosh (c+d x)}{80 d}+\frac {21 b^3 \sinh ^5(c+d x) \cosh (c+d x)}{160 d}-\frac {21 b^3 \sinh ^3(c+d x) \cosh (c+d x)}{128 d}+\frac {63 b^3 \sinh (c+d x) \cosh (c+d x)}{256 d}-\frac {63 b^3 x}{256} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*a^2*b*x)/8 - (63*b^3*x)/256 + (a^3*Cosh[c + d*x])/d - (3*a*b^2*Cosh[c + d*x])/d + (3*a*b^2*Cosh[c + d*x]^3)
/d - (9*a*b^2*Cosh[c + d*x]^5)/(5*d) + (3*a*b^2*Cosh[c + d*x]^7)/(7*d) - (9*a^2*b*Cosh[c + d*x]*Sinh[c + d*x])
/(8*d) + (63*b^3*Cosh[c + d*x]*Sinh[c + d*x])/(256*d) + (3*a^2*b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d) - (21*b^
3*Cosh[c + d*x]*Sinh[c + d*x]^3)/(128*d) + (21*b^3*Cosh[c + d*x]*Sinh[c + d*x]^5)/(160*d) - (9*b^3*Cosh[c + d*
x]*Sinh[c + d*x]^7)/(80*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x]^9)/(10*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3220

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rubi steps

\begin {align*} \int \sinh (c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx &=-\left (i \int \left (i a^3 \sinh (c+d x)+3 i a^2 b \sinh ^4(c+d x)+3 i a b^2 \sinh ^7(c+d x)+i b^3 \sinh ^{10}(c+d x)\right ) \, dx\right )\\ &=a^3 \int \sinh (c+d x) \, dx+\left (3 a^2 b\right ) \int \sinh ^4(c+d x) \, dx+\left (3 a b^2\right ) \int \sinh ^7(c+d x) \, dx+b^3 \int \sinh ^{10}(c+d x) \, dx\\ &=\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a^2 b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}+\frac {b^3 \cosh (c+d x) \sinh ^9(c+d x)}{10 d}-\frac {1}{4} \left (9 a^2 b\right ) \int \sinh ^2(c+d x) \, dx-\frac {1}{10} \left (9 b^3\right ) \int \sinh ^8(c+d x) \, dx-\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {a^3 \cosh (c+d x)}{d}-\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh ^3(c+d x)}{d}-\frac {9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac {3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac {9 a^2 b \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {3 a^2 b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac {9 b^3 \cosh (c+d x) \sinh ^7(c+d x)}{80 d}+\frac {b^3 \cosh (c+d x) \sinh ^9(c+d x)}{10 d}+\frac {1}{8} \left (9 a^2 b\right ) \int 1 \, dx+\frac {1}{80} \left (63 b^3\right ) \int \sinh ^6(c+d x) \, dx\\ &=\frac {9}{8} a^2 b x+\frac {a^3 \cosh (c+d x)}{d}-\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh ^3(c+d x)}{d}-\frac {9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac {3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac {9 a^2 b \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {3 a^2 b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}+\frac {21 b^3 \cosh (c+d x) \sinh ^5(c+d x)}{160 d}-\frac {9 b^3 \cosh (c+d x) \sinh ^7(c+d x)}{80 d}+\frac {b^3 \cosh (c+d x) \sinh ^9(c+d x)}{10 d}-\frac {1}{32} \left (21 b^3\right ) \int \sinh ^4(c+d x) \, dx\\ &=\frac {9}{8} a^2 b x+\frac {a^3 \cosh (c+d x)}{d}-\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh ^3(c+d x)}{d}-\frac {9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac {3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac {9 a^2 b \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {3 a^2 b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac {21 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{128 d}+\frac {21 b^3 \cosh (c+d x) \sinh ^5(c+d x)}{160 d}-\frac {9 b^3 \cosh (c+d x) \sinh ^7(c+d x)}{80 d}+\frac {b^3 \cosh (c+d x) \sinh ^9(c+d x)}{10 d}+\frac {1}{128} \left (63 b^3\right ) \int \sinh ^2(c+d x) \, dx\\ &=\frac {9}{8} a^2 b x+\frac {a^3 \cosh (c+d x)}{d}-\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh ^3(c+d x)}{d}-\frac {9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac {3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac {9 a^2 b \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {63 b^3 \cosh (c+d x) \sinh (c+d x)}{256 d}+\frac {3 a^2 b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac {21 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{128 d}+\frac {21 b^3 \cosh (c+d x) \sinh ^5(c+d x)}{160 d}-\frac {9 b^3 \cosh (c+d x) \sinh ^7(c+d x)}{80 d}+\frac {b^3 \cosh (c+d x) \sinh ^9(c+d x)}{10 d}-\frac {1}{256} \left (63 b^3\right ) \int 1 \, dx\\ &=\frac {9}{8} a^2 b x-\frac {63 b^3 x}{256}+\frac {a^3 \cosh (c+d x)}{d}-\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh ^3(c+d x)}{d}-\frac {9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac {3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac {9 a^2 b \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {63 b^3 \cosh (c+d x) \sinh (c+d x)}{256 d}+\frac {3 a^2 b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac {21 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{128 d}+\frac {21 b^3 \cosh (c+d x) \sinh ^5(c+d x)}{160 d}-\frac {9 b^3 \cosh (c+d x) \sinh ^7(c+d x)}{80 d}+\frac {b^3 \cosh (c+d x) \sinh ^9(c+d x)}{10 d}\\ \end {align*}

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Mathematica [A]  time = 0.45, size = 184, normalized size = 0.69 \[ \frac {1120 a \left (64 a^2-105 b^2\right ) \cosh (c+d x)+b \left (-53760 a^2 \sinh (2 (c+d x))+6720 a^2 \sinh (4 (c+d x))+80640 a^2 c+80640 a^2 d x+23520 a b \cosh (3 (c+d x))-4704 a b \cosh (5 (c+d x))+480 a b \cosh (7 (c+d x))+14700 b^2 \sinh (2 (c+d x))-4200 b^2 \sinh (4 (c+d x))+1050 b^2 \sinh (6 (c+d x))-175 b^2 \sinh (8 (c+d x))+14 b^2 \sinh (10 (c+d x))-17640 b^2 c-17640 b^2 d x\right )}{71680 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1120*a*(64*a^2 - 105*b^2)*Cosh[c + d*x] + b*(80640*a^2*c - 17640*b^2*c + 80640*a^2*d*x - 17640*b^2*d*x + 2352
0*a*b*Cosh[3*(c + d*x)] - 4704*a*b*Cosh[5*(c + d*x)] + 480*a*b*Cosh[7*(c + d*x)] - 53760*a^2*Sinh[2*(c + d*x)]
+ 14700*b^2*Sinh[2*(c + d*x)] + 6720*a^2*Sinh[4*(c + d*x)] - 4200*b^2*Sinh[4*(c + d*x)] + 1050*b^2*Sinh[6*(c
+ d*x)] - 175*b^2*Sinh[8*(c + d*x)] + 14*b^2*Sinh[10*(c + d*x)]))/(71680*d)

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fricas [A]  time = 0.56, size = 453, normalized size = 1.70 \[ \frac {35 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{9} + 120 \, a b^{2} \cosh \left (d x + c\right )^{7} + 840 \, a b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - 1176 \, a b^{2} \cosh \left (d x + c\right )^{5} + 70 \, {\left (6 \, b^{3} \cosh \left (d x + c\right )^{3} - 5 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{7} + 5880 \, a b^{2} \cosh \left (d x + c\right )^{3} + 7 \, {\left (126 \, b^{3} \cosh \left (d x + c\right )^{5} - 350 \, b^{3} \cosh \left (d x + c\right )^{3} + 225 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 840 \, {\left (5 \, a b^{2} \cosh \left (d x + c\right )^{3} - 7 \, a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 70 \, {\left (6 \, b^{3} \cosh \left (d x + c\right )^{7} - 35 \, b^{3} \cosh \left (d x + c\right )^{5} + 75 \, b^{3} \cosh \left (d x + c\right )^{3} + 12 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 630 \, {\left (32 \, a^{2} b - 7 \, b^{3}\right )} d x + 840 \, {\left (3 \, a b^{2} \cosh \left (d x + c\right )^{5} - 14 \, a b^{2} \cosh \left (d x + c\right )^{3} + 21 \, a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 280 \, {\left (64 \, a^{3} - 105 \, a b^{2}\right )} \cosh \left (d x + c\right ) + 35 \, {\left (b^{3} \cosh \left (d x + c\right )^{9} - 10 \, b^{3} \cosh \left (d x + c\right )^{7} + 45 \, b^{3} \cosh \left (d x + c\right )^{5} + 24 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 6 \, {\left (128 \, a^{2} b - 35 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{17920 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17920*(35*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + 120*a*b^2*cosh(d*x + c)^7 + 840*a*b^2*cosh(d*x + c)*sinh(d*x +
c)^6 - 1176*a*b^2*cosh(d*x + c)^5 + 70*(6*b^3*cosh(d*x + c)^3 - 5*b^3*cosh(d*x + c))*sinh(d*x + c)^7 + 5880*a
*b^2*cosh(d*x + c)^3 + 7*(126*b^3*cosh(d*x + c)^5 - 350*b^3*cosh(d*x + c)^3 + 225*b^3*cosh(d*x + c))*sinh(d*x
+ c)^5 + 840*(5*a*b^2*cosh(d*x + c)^3 - 7*a*b^2*cosh(d*x + c))*sinh(d*x + c)^4 + 70*(6*b^3*cosh(d*x + c)^7 - 3
5*b^3*cosh(d*x + c)^5 + 75*b^3*cosh(d*x + c)^3 + 12*(8*a^2*b - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 630*(32
*a^2*b - 7*b^3)*d*x + 840*(3*a*b^2*cosh(d*x + c)^5 - 14*a*b^2*cosh(d*x + c)^3 + 21*a*b^2*cosh(d*x + c))*sinh(d
*x + c)^2 + 280*(64*a^3 - 105*a*b^2)*cosh(d*x + c) + 35*(b^3*cosh(d*x + c)^9 - 10*b^3*cosh(d*x + c)^7 + 45*b^3
*cosh(d*x + c)^5 + 24*(8*a^2*b - 5*b^3)*cosh(d*x + c)^3 - 6*(128*a^2*b - 35*b^3)*cosh(d*x + c))*sinh(d*x + c))
/d

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giac [A]  time = 0.25, size = 379, normalized size = 1.42 \[ \frac {b^{3} e^{\left (10 \, d x + 10 \, c\right )}}{10240 \, d} - \frac {5 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )}}{4096 \, d} + \frac {3 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )}}{896 \, d} + \frac {15 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )}}{2048 \, d} - \frac {21 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {21 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )}}{128 \, d} + \frac {21 \, a b^{2} e^{\left (-3 \, d x - 3 \, c\right )}}{128 \, d} - \frac {21 \, a b^{2} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} - \frac {15 \, b^{3} e^{\left (-6 \, d x - 6 \, c\right )}}{2048 \, d} + \frac {3 \, a b^{2} e^{\left (-7 \, d x - 7 \, c\right )}}{896 \, d} + \frac {5 \, b^{3} e^{\left (-8 \, d x - 8 \, c\right )}}{4096 \, d} - \frac {b^{3} e^{\left (-10 \, d x - 10 \, c\right )}}{10240 \, d} + \frac {9}{256} \, {\left (32 \, a^{2} b - 7 \, b^{3}\right )} x + \frac {3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{512 \, d} - \frac {3 \, {\left (128 \, a^{2} b - 35 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{1024 \, d} + \frac {{\left (64 \, a^{3} - 105 \, a b^{2}\right )} e^{\left (d x + c\right )}}{128 \, d} + \frac {{\left (64 \, a^{3} - 105 \, a b^{2}\right )} e^{\left (-d x - c\right )}}{128 \, d} + \frac {3 \, {\left (128 \, a^{2} b - 35 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{1024 \, d} - \frac {3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{512 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10240*b^3*e^(10*d*x + 10*c)/d - 5/4096*b^3*e^(8*d*x + 8*c)/d + 3/896*a*b^2*e^(7*d*x + 7*c)/d + 15/2048*b^3*e
^(6*d*x + 6*c)/d - 21/640*a*b^2*e^(5*d*x + 5*c)/d + 21/128*a*b^2*e^(3*d*x + 3*c)/d + 21/128*a*b^2*e^(-3*d*x -
3*c)/d - 21/640*a*b^2*e^(-5*d*x - 5*c)/d - 15/2048*b^3*e^(-6*d*x - 6*c)/d + 3/896*a*b^2*e^(-7*d*x - 7*c)/d + 5
/4096*b^3*e^(-8*d*x - 8*c)/d - 1/10240*b^3*e^(-10*d*x - 10*c)/d + 9/256*(32*a^2*b - 7*b^3)*x + 3/512*(8*a^2*b
- 5*b^3)*e^(4*d*x + 4*c)/d - 3/1024*(128*a^2*b - 35*b^3)*e^(2*d*x + 2*c)/d + 1/128*(64*a^3 - 105*a*b^2)*e^(d*x
+ c)/d + 1/128*(64*a^3 - 105*a*b^2)*e^(-d*x - c)/d + 3/1024*(128*a^2*b - 35*b^3)*e^(-2*d*x - 2*c)/d - 3/512*(
8*a^2*b - 5*b^3)*e^(-4*d*x - 4*c)/d

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maple [A]  time = 0.05, size = 168, normalized size = 0.63 \[ \frac {b^{3} \left (\left (\frac {\left (\sinh ^{9}\left (d x +c \right )\right )}{10}-\frac {9 \left (\sinh ^{7}\left (d x +c \right )\right )}{80}+\frac {21 \left (\sinh ^{5}\left (d x +c \right )\right )}{160}-\frac {21 \left (\sinh ^{3}\left (d x +c \right )\right )}{128}+\frac {63 \sinh \left (d x +c \right )}{256}\right ) \cosh \left (d x +c \right )-\frac {63 d x}{256}-\frac {63 c}{256}\right )+3 a \,b^{2} \left (-\frac {16}{35}+\frac {\left (\sinh ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sinh ^{4}\left (d x +c \right )\right )}{35}+\frac {8 \left (\sinh ^{2}\left (d x +c \right )\right )}{35}\right ) \cosh \left (d x +c \right )+3 a^{2} b \left (\left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} \cosh \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^3*((1/10*sinh(d*x+c)^9-9/80*sinh(d*x+c)^7+21/160*sinh(d*x+c)^5-21/128*sinh(d*x+c)^3+63/256*sinh(d*x+c))
*cosh(d*x+c)-63/256*d*x-63/256*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)*cos
h(d*x+c)+3*a^2*b*((1/4*sinh(d*x+c)^3-3/8*sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c)+a^3*cosh(d*x+c))

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maxima [A]  time = 0.33, size = 318, normalized size = 1.19 \[ \frac {3}{64} \, a^{2} b {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{20480} \, b^{3} {\left (\frac {{\left (25 \, e^{\left (-2 \, d x - 2 \, c\right )} - 150 \, e^{\left (-4 \, d x - 4 \, c\right )} + 600 \, e^{\left (-6 \, d x - 6 \, c\right )} - 2100 \, e^{\left (-8 \, d x - 8 \, c\right )} - 2\right )} e^{\left (10 \, d x + 10 \, c\right )}}{d} + \frac {5040 \, {\left (d x + c\right )}}{d} + \frac {2100 \, e^{\left (-2 \, d x - 2 \, c\right )} - 600 \, e^{\left (-4 \, d x - 4 \, c\right )} + 150 \, e^{\left (-6 \, d x - 6 \, c\right )} - 25 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d}\right )} - \frac {3}{4480} \, a b^{2} {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {a^{3} \cosh \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/64*a^2*b*(24*x + e^(4*d*x + 4*c)/d - 8*e^(2*d*x + 2*c)/d + 8*e^(-2*d*x - 2*c)/d - e^(-4*d*x - 4*c)/d) - 1/20
480*b^3*((25*e^(-2*d*x - 2*c) - 150*e^(-4*d*x - 4*c) + 600*e^(-6*d*x - 6*c) - 2100*e^(-8*d*x - 8*c) - 2)*e^(10
*d*x + 10*c)/d + 5040*(d*x + c)/d + (2100*e^(-2*d*x - 2*c) - 600*e^(-4*d*x - 4*c) + 150*e^(-6*d*x - 6*c) - 25*
e^(-8*d*x - 8*c) + 2*e^(-10*d*x - 10*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) + a^3*cosh(d*x + c)/d

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mupad [B]  time = 2.95, size = 189, normalized size = 0.71 \[ \frac {8960\,a^3\,\mathrm {cosh}\left (c+d\,x\right )+\frac {3675\,b^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{2}-525\,b^3\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+\frac {525\,b^3\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}-\frac {175\,b^3\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}+\frac {7\,b^3\,\mathrm {sinh}\left (10\,c+10\,d\,x\right )}{4}+2940\,a\,b^2\,\mathrm {cosh}\left (3\,c+3\,d\,x\right )-588\,a\,b^2\,\mathrm {cosh}\left (5\,c+5\,d\,x\right )+60\,a\,b^2\,\mathrm {cosh}\left (7\,c+7\,d\,x\right )-6720\,a^2\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+840\,a^2\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-14700\,a\,b^2\,\mathrm {cosh}\left (c+d\,x\right )-2205\,b^3\,d\,x+10080\,a^2\,b\,d\,x}{8960\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8960*a^3*cosh(c + d*x) + (3675*b^3*sinh(2*c + 2*d*x))/2 - 525*b^3*sinh(4*c + 4*d*x) + (525*b^3*sinh(6*c + 6*d
*x))/4 - (175*b^3*sinh(8*c + 8*d*x))/8 + (7*b^3*sinh(10*c + 10*d*x))/4 + 2940*a*b^2*cosh(3*c + 3*d*x) - 588*a*
b^2*cosh(5*c + 5*d*x) + 60*a*b^2*cosh(7*c + 7*d*x) - 6720*a^2*b*sinh(2*c + 2*d*x) + 840*a^2*b*sinh(4*c + 4*d*x
) - 14700*a*b^2*cosh(c + d*x) - 2205*b^3*d*x + 10080*a^2*b*d*x)/(8960*d)

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sympy [A]  time = 21.92, size = 496, normalized size = 1.86 \[ \begin {cases} \frac {a^{3} \cosh {\left (c + d x \right )}}{d} + \frac {9 a^{2} b x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {9 a^{2} b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {9 a^{2} b x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {15 a^{2} b \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {9 a^{2} b \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {3 a b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {6 a b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {24 a b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {48 a b^{2} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {63 b^{3} x \sinh ^{10}{\left (c + d x \right )}}{256} - \frac {315 b^{3} x \sinh ^{8}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{256} + \frac {315 b^{3} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{128} - \frac {315 b^{3} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{128} + \frac {315 b^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{8}{\left (c + d x \right )}}{256} - \frac {63 b^{3} x \cosh ^{10}{\left (c + d x \right )}}{256} + \frac {193 b^{3} \sinh ^{9}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{256 d} - \frac {237 b^{3} \sinh ^{7}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac {21 b^{3} \sinh ^{5}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{10 d} - \frac {147 b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} + \frac {63 b^{3} \sinh {\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{256 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\relax (c )}\right )^{3} \sinh {\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*cosh(c + d*x)/d + 9*a**2*b*x*sinh(c + d*x)**4/8 - 9*a**2*b*x*sinh(c + d*x)**2*cosh(c + d*x)**2
/4 + 9*a**2*b*x*cosh(c + d*x)**4/8 + 15*a**2*b*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) - 9*a**2*b*sinh(c + d*x)*c
osh(c + d*x)**3/(8*d) + 3*a*b**2*sinh(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) + 63*b**3*x*sinh(c
+ d*x)**10/256 - 315*b**3*x*sinh(c + d*x)**8*cosh(c + d*x)**2/256 + 315*b**3*x*sinh(c + d*x)**6*cosh(c + d*x)*
*4/128 - 315*b**3*x*sinh(c + d*x)**4*cosh(c + d*x)**6/128 + 315*b**3*x*sinh(c + d*x)**2*cosh(c + d*x)**8/256 -
63*b**3*x*cosh(c + d*x)**10/256 + 193*b**3*sinh(c + d*x)**9*cosh(c + d*x)/(256*d) - 237*b**3*sinh(c + d*x)**7
*cosh(c + d*x)**3/(128*d) + 21*b**3*sinh(c + d*x)**5*cosh(c + d*x)**5/(10*d) - 147*b**3*sinh(c + d*x)**3*cosh(
c + d*x)**7/(128*d) + 63*b**3*sinh(c + d*x)*cosh(c + d*x)**9/(256*d), Ne(d, 0)), (x*(a + b*sinh(c)**3)**3*sinh
(c), True))

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