3.208 \(\int \sinh ^3(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)
Optimal. Leaf size=183 \[ \frac{b \left (3 a^2+30 a b+35 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac{3 b^2 (a+7 b) \cosh ^{11}(c+d x)}{11 d}-\frac{5 b^2 (3 a+7 b) \cosh ^9(c+d x)}{9 d}-\frac{3 b (a+b) (3 a+7 b) \cosh ^5(c+d x)}{5 d}+\frac{(a+b)^2 (a+7 b) \cosh ^3(c+d x)}{3 d}-\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^{15}(c+d x)}{15 d}-\frac{7 b^3 \cosh ^{13}(c+d x)}{13 d} \]
[Out]
-(((a + b)^3*Cosh[c + d*x])/d) + ((a + b)^2*(a + 7*b)*Cosh[c + d*x]^3)/(3*d) - (3*b*(a + b)*(3*a + 7*b)*Cosh[c
+ d*x]^5)/(5*d) + (b*(3*a^2 + 30*a*b + 35*b^2)*Cosh[c + d*x]^7)/(7*d) - (5*b^2*(3*a + 7*b)*Cosh[c + d*x]^9)/(
9*d) + (3*b^2*(a + 7*b)*Cosh[c + d*x]^11)/(11*d) - (7*b^3*Cosh[c + d*x]^13)/(13*d) + (b^3*Cosh[c + d*x]^15)/(1
5*d)
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Rubi [A] time = 0.175103, antiderivative size = 183, normalized size of antiderivative = 1.,
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 =
{3215, 1153} \[ \frac{b \left (3 a^2+30 a b+35 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac{3 b^2 (a+7 b) \cosh ^{11}(c+d x)}{11 d}-\frac{5 b^2 (3 a+7 b) \cosh ^9(c+d x)}{9 d}-\frac{3 b (a+b) (3 a+7 b) \cosh ^5(c+d x)}{5 d}+\frac{(a+b)^2 (a+7 b) \cosh ^3(c+d x)}{3 d}-\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^{15}(c+d x)}{15 d}-\frac{7 b^3 \cosh ^{13}(c+d x)}{13 d} \]
Antiderivative was successfully verified.
[In]
Int[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
-(((a + b)^3*Cosh[c + d*x])/d) + ((a + b)^2*(a + 7*b)*Cosh[c + d*x]^3)/(3*d) - (3*b*(a + b)*(3*a + 7*b)*Cosh[c
+ d*x]^5)/(5*d) + (b*(3*a^2 + 30*a*b + 35*b^2)*Cosh[c + d*x]^7)/(7*d) - (5*b^2*(3*a + 7*b)*Cosh[c + d*x]^9)/(
9*d) + (3*b^2*(a + 7*b)*Cosh[c + d*x]^11)/(11*d) - (7*b^3*Cosh[c + d*x]^13)/(13*d) + (b^3*Cosh[c + d*x]^15)/(1
5*d)
Rule 3215
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(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 - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Rule 1153
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Rubi steps
\begin{align*} \int \sinh ^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (a+b-2 b x^2+b x^4\right )^3 \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left ((a+b)^3-(a+b)^2 (a+7 b) x^2+3 b (a+b) (3 a+7 b) x^4-b \left (3 a^2+30 a b+35 b^2\right ) x^6+5 b^2 (3 a+7 b) x^8-3 b^2 (a+7 b) x^{10}+7 b^3 x^{12}-b^3 x^{14}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{(a+b)^2 (a+7 b) \cosh ^3(c+d x)}{3 d}-\frac{3 b (a+b) (3 a+7 b) \cosh ^5(c+d x)}{5 d}+\frac{b \left (3 a^2+30 a b+35 b^2\right ) \cosh ^7(c+d x)}{7 d}-\frac{5 b^2 (3 a+7 b) \cosh ^9(c+d x)}{9 d}+\frac{3 b^2 (a+7 b) \cosh ^{11}(c+d x)}{11 d}-\frac{7 b^3 \cosh ^{13}(c+d x)}{13 d}+\frac{b^3 \cosh ^{15}(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 2.47, size = 185, normalized size = 1.01 \[ \frac{-135135 \left (8960 a^2 b+4096 a^3+7392 a b^2+2145 b^3\right ) \cosh (c+d x)+15015 \left (16128 a^2 b+4096 a^3+15840 a b^2+5005 b^3\right ) \cosh (3 (c+d x))+b \left (-27027 \left (1792 a^2+2640 a b+1001 b^2\right ) \cosh (5 (c+d x))+19305 \left (256 a^2+880 a b+455 b^2\right ) \cosh (7 (c+d x))-7 b (715 (528 a+455 b) \cosh (9 (c+d x))-1755 (16 a+35 b) \cosh (11 (c+d x))+7425 b \cosh (13 (c+d x))-429 b \cosh (15 (c+d x)))\right )}{738017280 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
(-135135*(4096*a^3 + 8960*a^2*b + 7392*a*b^2 + 2145*b^3)*Cosh[c + d*x] + 15015*(4096*a^3 + 16128*a^2*b + 15840
*a*b^2 + 5005*b^3)*Cosh[3*(c + d*x)] + b*(-27027*(1792*a^2 + 2640*a*b + 1001*b^2)*Cosh[5*(c + d*x)] + 19305*(2
56*a^2 + 880*a*b + 455*b^2)*Cosh[7*(c + d*x)] - 7*b*(715*(528*a + 455*b)*Cosh[9*(c + d*x)] - 1755*(16*a + 35*b
)*Cosh[11*(c + d*x)] + 7425*b*Cosh[13*(c + d*x)] - 429*b*Cosh[15*(c + d*x)])))/(738017280*d)
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Maple [A] time = 0.059, size = 218, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ( -{\frac{2048}{6435}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{14}}{15}}-{\frac{14\, \left ( \sinh \left ( dx+c \right ) \right ) ^{12}}{195}}+{\frac{56\, \left ( \sinh \left ( dx+c \right ) \right ) ^{10}}{715}}-{\frac{112\, \left ( \sinh \left ( dx+c \right ) \right ) ^{8}}{1287}}+{\frac{128\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{1287}}-{\frac{256\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{2145}}+{\frac{1024\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{6435}} \right ) \cosh \left ( dx+c \right ) +3\,a{b}^{2} \left ( -{\frac{256}{693}}+1/11\, \left ( \sinh \left ( dx+c \right ) \right ) ^{10}-{\frac{10\, \left ( \sinh \left ( dx+c \right ) \right ) ^{8}}{99}}+{\frac{80\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{693}}-{\frac{32\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{231}}+{\frac{128\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{693}} \right ) \cosh \left ( dx+c \right ) +3\,{a}^{2}b \left ( -{\frac{16}{35}}+1/7\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}-{\frac{6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \cosh \left ( dx+c \right ) +{a}^{3} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4)^3,x)
[Out]
1/d*(b^3*(-2048/6435+1/15*sinh(d*x+c)^14-14/195*sinh(d*x+c)^12+56/715*sinh(d*x+c)^10-112/1287*sinh(d*x+c)^8+12
8/1287*sinh(d*x+c)^6-256/2145*sinh(d*x+c)^4+1024/6435*sinh(d*x+c)^2)*cosh(d*x+c)+3*a*b^2*(-256/693+1/11*sinh(d
*x+c)^10-10/99*sinh(d*x+c)^8+80/693*sinh(d*x+c)^6-32/231*sinh(d*x+c)^4+128/693*sinh(d*x+c)^2)*cosh(d*x+c)+3*a^
2*b*(-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)+a^3*(-2/3+1/3*sinh(d*x+c)^2)*
cosh(d*x+c))
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Maxima [B] time = 1.07978, size = 676, normalized size = 3.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")
[Out]
-1/210862080*b^3*((7425*e^(-2*d*x - 2*c) - 61425*e^(-4*d*x - 4*c) + 325325*e^(-6*d*x - 6*c) - 1254825*e^(-8*d*
x - 8*c) + 3864861*e^(-10*d*x - 10*c) - 10735725*e^(-12*d*x - 12*c) + 41409225*e^(-14*d*x - 14*c) - 429)*e^(15
*d*x + 15*c)/d + (41409225*e^(-d*x - c) - 10735725*e^(-3*d*x - 3*c) + 3864861*e^(-5*d*x - 5*c) - 1254825*e^(-7
*d*x - 7*c) + 325325*e^(-9*d*x - 9*c) - 61425*e^(-11*d*x - 11*c) + 7425*e^(-13*d*x - 13*c) - 429*e^(-15*d*x -
15*c))/d) - 1/473088*a*b^2*((847*e^(-2*d*x - 2*c) - 5445*e^(-4*d*x - 4*c) + 22869*e^(-6*d*x - 6*c) - 76230*e^(
-8*d*x - 8*c) + 320166*e^(-10*d*x - 10*c) - 63)*e^(11*d*x + 11*c)/d + (320166*e^(-d*x - c) - 76230*e^(-3*d*x -
3*c) + 22869*e^(-5*d*x - 5*c) - 5445*e^(-7*d*x - 7*c) + 847*e^(-9*d*x - 9*c) - 63*e^(-11*d*x - 11*c))/d) - 3/
4480*a^2*b*((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) + 1/24*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)
________________________________________________________________________________________
Fricas [B] time = 1.73061, size = 2207, normalized size = 12.06 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")
[Out]
1/738017280*(3003*b^3*cosh(d*x + c)^15 + 45045*b^3*cosh(d*x + c)*sinh(d*x + c)^14 - 51975*b^3*cosh(d*x + c)^13
+ 15015*(91*b^3*cosh(d*x + c)^3 - 45*b^3*cosh(d*x + c))*sinh(d*x + c)^12 + 12285*(16*a*b^2 + 35*b^3)*cosh(d*x
+ c)^11 + 9009*(1001*b^3*cosh(d*x + c)^5 - 1650*b^3*cosh(d*x + c)^3 + 15*(16*a*b^2 + 35*b^3)*cosh(d*x + c))*s
inh(d*x + c)^10 - 5005*(528*a*b^2 + 455*b^3)*cosh(d*x + c)^9 + 45045*(429*b^3*cosh(d*x + c)^7 - 1485*b^3*cosh(
d*x + c)^5 + 45*(16*a*b^2 + 35*b^3)*cosh(d*x + c)^3 - (528*a*b^2 + 455*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 + 1
9305*(256*a^2*b + 880*a*b^2 + 455*b^3)*cosh(d*x + c)^7 + 15015*(1001*b^3*cosh(d*x + c)^9 - 5940*b^3*cosh(d*x +
c)^7 + 378*(16*a*b^2 + 35*b^3)*cosh(d*x + c)^5 - 28*(528*a*b^2 + 455*b^3)*cosh(d*x + c)^3 + 9*(256*a^2*b + 88
0*a*b^2 + 455*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 27027*(1792*a^2*b + 2640*a*b^2 + 1001*b^3)*cosh(d*x + c)^5
+ 45045*(91*b^3*cosh(d*x + c)^11 - 825*b^3*cosh(d*x + c)^9 + 90*(16*a*b^2 + 35*b^3)*cosh(d*x + c)^7 - 14*(528
*a*b^2 + 455*b^3)*cosh(d*x + c)^5 + 15*(256*a^2*b + 880*a*b^2 + 455*b^3)*cosh(d*x + c)^3 - 3*(1792*a^2*b + 264
0*a*b^2 + 1001*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 15015*(4096*a^3 + 16128*a^2*b + 15840*a*b^2 + 5005*b^3)*c
osh(d*x + c)^3 + 45045*(7*b^3*cosh(d*x + c)^13 - 90*b^3*cosh(d*x + c)^11 + 15*(16*a*b^2 + 35*b^3)*cosh(d*x + c
)^9 - 4*(528*a*b^2 + 455*b^3)*cosh(d*x + c)^7 + 9*(256*a^2*b + 880*a*b^2 + 455*b^3)*cosh(d*x + c)^5 - 6*(1792*
a^2*b + 2640*a*b^2 + 1001*b^3)*cosh(d*x + c)^3 + (4096*a^3 + 16128*a^2*b + 15840*a*b^2 + 5005*b^3)*cosh(d*x +
c))*sinh(d*x + c)^2 - 135135*(4096*a^3 + 8960*a^2*b + 7392*a*b^2 + 2145*b^3)*cosh(d*x + c))/d
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)**3*(a+b*sinh(d*x+c)**4)**3,x)
[Out]
Timed out
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Giac [B] time = 1.72125, size = 779, normalized size = 4.26 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")
[Out]
1/1476034560*(3003*b^3*e^(15*d*x + 15*c) - 51975*b^3*e^(13*d*x + 13*c) + 196560*a*b^2*e^(11*d*x + 11*c) + 4299
75*b^3*e^(11*d*x + 11*c) - 2642640*a*b^2*e^(9*d*x + 9*c) - 2277275*b^3*e^(9*d*x + 9*c) + 4942080*a^2*b*e^(7*d*
x + 7*c) + 16988400*a*b^2*e^(7*d*x + 7*c) + 8783775*b^3*e^(7*d*x + 7*c) - 48432384*a^2*b*e^(5*d*x + 5*c) - 713
51280*a*b^2*e^(5*d*x + 5*c) - 27054027*b^3*e^(5*d*x + 5*c) + 61501440*a^3*e^(3*d*x + 3*c) + 242161920*a^2*b*e^
(3*d*x + 3*c) + 237837600*a*b^2*e^(3*d*x + 3*c) + 75150075*b^3*e^(3*d*x + 3*c) - 553512960*a^3*e^(d*x + c) - 1
210809600*a^2*b*e^(d*x + c) - 998917920*a*b^2*e^(d*x + c) - 289864575*b^3*e^(d*x + c) - (553512960*a^3*e^(14*d
*x + 14*c) + 1210809600*a^2*b*e^(14*d*x + 14*c) + 998917920*a*b^2*e^(14*d*x + 14*c) + 289864575*b^3*e^(14*d*x
+ 14*c) - 61501440*a^3*e^(12*d*x + 12*c) - 242161920*a^2*b*e^(12*d*x + 12*c) - 237837600*a*b^2*e^(12*d*x + 12*
c) - 75150075*b^3*e^(12*d*x + 12*c) + 48432384*a^2*b*e^(10*d*x + 10*c) + 71351280*a*b^2*e^(10*d*x + 10*c) + 27
054027*b^3*e^(10*d*x + 10*c) - 4942080*a^2*b*e^(8*d*x + 8*c) - 16988400*a*b^2*e^(8*d*x + 8*c) - 8783775*b^3*e^
(8*d*x + 8*c) + 2642640*a*b^2*e^(6*d*x + 6*c) + 2277275*b^3*e^(6*d*x + 6*c) - 196560*a*b^2*e^(4*d*x + 4*c) - 4
29975*b^3*e^(4*d*x + 4*c) + 51975*b^3*e^(2*d*x + 2*c) - 3003*b^3)*e^(-15*d*x - 15*c))/d