3.207 \(\int \sinh ^5(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)
Optimal. Leaf size=220 \[ \frac{b \left (3 a^2+45 a b+70 b^2\right ) \cosh ^9(c+d x)}{9 d}-\frac{4 b \left (3 a^2+15 a b+14 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac{(a+b) \left (a^2+17 a b+28 b^2\right ) \cosh ^5(c+d x)}{5 d}+\frac{b^2 (3 a+28 b) \cosh ^{13}(c+d x)}{13 d}-\frac{2 b^2 (9 a+28 b) \cosh ^{11}(c+d x)}{11 d}-\frac{2 (a+b)^2 (a+4 b) \cosh ^3(c+d x)}{3 d}+\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^{17}(c+d x)}{17 d}-\frac{8 b^3 \cosh ^{15}(c+d x)}{15 d} \]
[Out]
((a + b)^3*Cosh[c + d*x])/d - (2*(a + b)^2*(a + 4*b)*Cosh[c + d*x]^3)/(3*d) + ((a + b)*(a^2 + 17*a*b + 28*b^2)
*Cosh[c + d*x]^5)/(5*d) - (4*b*(3*a^2 + 15*a*b + 14*b^2)*Cosh[c + d*x]^7)/(7*d) + (b*(3*a^2 + 45*a*b + 70*b^2)
*Cosh[c + d*x]^9)/(9*d) - (2*b^2*(9*a + 28*b)*Cosh[c + d*x]^11)/(11*d) + (b^2*(3*a + 28*b)*Cosh[c + d*x]^13)/(
13*d) - (8*b^3*Cosh[c + d*x]^15)/(15*d) + (b^3*Cosh[c + d*x]^17)/(17*d)
________________________________________________________________________________________
Rubi [A] time = 0.218072, antiderivative size = 220, 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} \[ \frac{b \left (3 a^2+45 a b+70 b^2\right ) \cosh ^9(c+d x)}{9 d}-\frac{4 b \left (3 a^2+15 a b+14 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac{(a+b) \left (a^2+17 a b+28 b^2\right ) \cosh ^5(c+d x)}{5 d}+\frac{b^2 (3 a+28 b) \cosh ^{13}(c+d x)}{13 d}-\frac{2 b^2 (9 a+28 b) \cosh ^{11}(c+d x)}{11 d}-\frac{2 (a+b)^2 (a+4 b) \cosh ^3(c+d x)}{3 d}+\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^{17}(c+d x)}{17 d}-\frac{8 b^3 \cosh ^{15}(c+d x)}{15 d} \]
Antiderivative was successfully verified.
[In]
Int[Sinh[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
((a + b)^3*Cosh[c + d*x])/d - (2*(a + b)^2*(a + 4*b)*Cosh[c + d*x]^3)/(3*d) + ((a + b)*(a^2 + 17*a*b + 28*b^2)
*Cosh[c + d*x]^5)/(5*d) - (4*b*(3*a^2 + 15*a*b + 14*b^2)*Cosh[c + d*x]^7)/(7*d) + (b*(3*a^2 + 45*a*b + 70*b^2)
*Cosh[c + d*x]^9)/(9*d) - (2*b^2*(9*a + 28*b)*Cosh[c + d*x]^11)/(11*d) + (b^2*(3*a + 28*b)*Cosh[c + d*x]^13)/(
13*d) - (8*b^3*Cosh[c + d*x]^15)/(15*d) + (b^3*Cosh[c + d*x]^17)/(17*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 ^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (1-x^2\right )^2 \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)^3-2 (a+b)^2 (a+4 b) x^2+(a+b) \left (a^2+17 a b+28 b^2\right ) x^4-4 b \left (3 a^2+15 a b+14 b^2\right ) x^6+b \left (3 a^2+45 a b+70 b^2\right ) x^8-2 b^2 (9 a+28 b) x^{10}+b^2 (3 a+28 b) x^{12}-8 b^3 x^{14}+b^3 x^{16}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{(a+b)^3 \cosh (c+d x)}{d}-\frac{2 (a+b)^2 (a+4 b) \cosh ^3(c+d x)}{3 d}+\frac{(a+b) \left (a^2+17 a b+28 b^2\right ) \cosh ^5(c+d x)}{5 d}-\frac{4 b \left (3 a^2+15 a b+14 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac{b \left (3 a^2+45 a b+70 b^2\right ) \cosh ^9(c+d x)}{9 d}-\frac{2 b^2 (9 a+28 b) \cosh ^{11}(c+d x)}{11 d}+\frac{b^2 (3 a+28 b) \cosh ^{13}(c+d x)}{13 d}-\frac{8 b^3 \cosh ^{15}(c+d x)}{15 d}+\frac{b^3 \cosh ^{17}(c+d x)}{17 d}\\ \end{align*}
Mathematica [A] time = 2.26629, size = 288, normalized size = 1.31 \[ \frac{1531530 \left (48384 a^2 b+20480 a^3+41184 a b^2+12155 b^3\right ) \cosh (c+d x)-2042040 \left (8064 a^2 b+2560 a^3+7722 a b^2+2431 b^3\right ) \cosh (3 (c+d x))+4234374144 a^2 b \cosh (5 (c+d x))-756138240 a^2 b \cosh (7 (c+d x))+65345280 a^2 b \cosh (9 (c+d x))+627314688 a^3 \cosh (5 (c+d x))+5256210960 a b^2 \cosh (5 (c+d x))-1501774560 a b^2 \cosh (7 (c+d x))+318558240 a b^2 \cosh (9 (c+d x))-43439760 a b^2 \cosh (11 (c+d x))+2827440 a b^2 \cosh (13 (c+d x))+1895421528 b^3 \cosh (5 (c+d x))-676936260 b^3 \cosh (7 (c+d x))+202502300 b^3 \cosh (9 (c+d x))-47338200 b^3 \cosh (11 (c+d x))+8011080 b^3 \cosh (13 (c+d x))-867867 b^3 \cosh (15 (c+d x))+45045 b^3 \cosh (17 (c+d x))}{50185175040 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Sinh[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
(1531530*(20480*a^3 + 48384*a^2*b + 41184*a*b^2 + 12155*b^3)*Cosh[c + d*x] - 2042040*(2560*a^3 + 8064*a^2*b +
7722*a*b^2 + 2431*b^3)*Cosh[3*(c + d*x)] + 627314688*a^3*Cosh[5*(c + d*x)] + 4234374144*a^2*b*Cosh[5*(c + d*x)
] + 5256210960*a*b^2*Cosh[5*(c + d*x)] + 1895421528*b^3*Cosh[5*(c + d*x)] - 756138240*a^2*b*Cosh[7*(c + d*x)]
- 1501774560*a*b^2*Cosh[7*(c + d*x)] - 676936260*b^3*Cosh[7*(c + d*x)] + 65345280*a^2*b*Cosh[9*(c + d*x)] + 31
8558240*a*b^2*Cosh[9*(c + d*x)] + 202502300*b^3*Cosh[9*(c + d*x)] - 43439760*a*b^2*Cosh[11*(c + d*x)] - 473382
00*b^3*Cosh[11*(c + d*x)] + 2827440*a*b^2*Cosh[13*(c + d*x)] + 8011080*b^3*Cosh[13*(c + d*x)] - 867867*b^3*Cos
h[15*(c + d*x)] + 45045*b^3*Cosh[17*(c + d*x)])/(50185175040*d)
________________________________________________________________________________________
Maple [A] time = 0.096, size = 258, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ({\frac{32768}{109395}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{16}}{17}}-{\frac{16\, \left ( \sinh \left ( dx+c \right ) \right ) ^{14}}{255}}+{\frac{224\, \left ( \sinh \left ( dx+c \right ) \right ) ^{12}}{3315}}-{\frac{896\, \left ( \sinh \left ( dx+c \right ) \right ) ^{10}}{12155}}+{\frac{1792\, \left ( \sinh \left ( dx+c \right ) \right ) ^{8}}{21879}}-{\frac{2048\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{21879}}+{\frac{4096\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{36465}}-{\frac{16384\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{109395}} \right ) \cosh \left ( dx+c \right ) +3\,a{b}^{2} \left ({\frac{1024}{3003}}+1/13\, \left ( \sinh \left ( dx+c \right ) \right ) ^{12}-{\frac{12\, \left ( \sinh \left ( dx+c \right ) \right ) ^{10}}{143}}+{\frac{40\, \left ( \sinh \left ( dx+c \right ) \right ) ^{8}}{429}}-{\frac{320\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{3003}}+{\frac{128\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{1001}}-{\frac{512\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3003}} \right ) \cosh \left ( dx+c \right ) +3\,{a}^{2}b \left ({\frac{128}{315}}+1/9\, \left ( \sinh \left ( dx+c \right ) \right ) ^{8}-{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{63}}+{\frac{16\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{105}}-{\frac{64\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{315}} \right ) \cosh \left ( dx+c \right ) +{a}^{3} \left ({\frac{8}{15}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15}} \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)^5*(a+b*sinh(d*x+c)^4)^3,x)
[Out]
1/d*(b^3*(32768/109395+1/17*sinh(d*x+c)^16-16/255*sinh(d*x+c)^14+224/3315*sinh(d*x+c)^12-896/12155*sinh(d*x+c)
^10+1792/21879*sinh(d*x+c)^8-2048/21879*sinh(d*x+c)^6+4096/36465*sinh(d*x+c)^4-16384/109395*sinh(d*x+c)^2)*cos
h(d*x+c)+3*a*b^2*(1024/3003+1/13*sinh(d*x+c)^12-12/143*sinh(d*x+c)^10+40/429*sinh(d*x+c)^8-320/3003*sinh(d*x+c
)^6+128/1001*sinh(d*x+c)^4-512/3003*sinh(d*x+c)^2)*cosh(d*x+c)+3*a^2*b*(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)+a^3*(8/15+1/5*sinh(d*x+c)^4-4/15*sinh(d*x+c)^2)*
cosh(d*x+c))
________________________________________________________________________________________
Maxima [B] time = 1.07614, size = 810, normalized size = 3.68 \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)^5*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")
[Out]
-1/14338621440*b^3*((123981*e^(-2*d*x - 2*c) - 1144440*e^(-4*d*x - 4*c) + 6762600*e^(-6*d*x - 6*c) - 28928900*
e^(-8*d*x - 8*c) + 96705180*e^(-10*d*x - 10*c) - 270774504*e^(-12*d*x - 12*c) + 709171320*e^(-14*d*x - 14*c) -
2659392450*e^(-16*d*x - 16*c) - 6435)*e^(17*d*x + 17*c)/d - (2659392450*e^(-d*x - c) - 709171320*e^(-3*d*x -
3*c) + 270774504*e^(-5*d*x - 5*c) - 96705180*e^(-7*d*x - 7*c) + 28928900*e^(-9*d*x - 9*c) - 6762600*e^(-11*d*x
- 11*c) + 1144440*e^(-13*d*x - 13*c) - 123981*e^(-15*d*x - 15*c) + 6435*e^(-17*d*x - 17*c))/d) - 1/8200192*a*
b^2*((3549*e^(-2*d*x - 2*c) - 26026*e^(-4*d*x - 4*c) + 122694*e^(-6*d*x - 6*c) - 429429*e^(-8*d*x - 8*c) + 128
8287*e^(-10*d*x - 10*c) - 5153148*e^(-12*d*x - 12*c) - 231)*e^(13*d*x + 13*c)/d - (5153148*e^(-d*x - c) - 1288
287*e^(-3*d*x - 3*c) + 429429*e^(-5*d*x - 5*c) - 122694*e^(-7*d*x - 7*c) + 26026*e^(-9*d*x - 9*c) - 3549*e^(-1
1*d*x - 11*c) + 231*e^(-13*d*x - 13*c))/d) - 1/53760*a^2*b*((405*e^(-2*d*x - 2*c) - 2268*e^(-4*d*x - 4*c) + 88
20*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) + 1/480*a^3*(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)
________________________________________________________________________________________
Fricas [B] time = 1.78734, size = 2898, normalized size = 13.17 \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)^5*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")
[Out]
1/50185175040*(45045*b^3*cosh(d*x + c)^17 + 765765*b^3*cosh(d*x + c)*sinh(d*x + c)^16 - 867867*b^3*cosh(d*x +
c)^15 + 765765*(40*b^3*cosh(d*x + c)^3 - 17*b^3*cosh(d*x + c))*sinh(d*x + c)^14 + 471240*(6*a*b^2 + 17*b^3)*co
sh(d*x + c)^13 + 255255*(1092*b^3*cosh(d*x + c)^5 - 1547*b^3*cosh(d*x + c)^3 + 24*(6*a*b^2 + 17*b^3)*cosh(d*x
+ c))*sinh(d*x + c)^12 - 556920*(78*a*b^2 + 85*b^3)*cosh(d*x + c)^11 + 153153*(5720*b^3*cosh(d*x + c)^7 - 1701
7*b^3*cosh(d*x + c)^5 + 880*(6*a*b^2 + 17*b^3)*cosh(d*x + c)^3 - 40*(78*a*b^2 + 85*b^3)*cosh(d*x + c))*sinh(d*
x + c)^10 + 340340*(192*a^2*b + 936*a*b^2 + 595*b^3)*cosh(d*x + c)^9 + 765765*(1430*b^3*cosh(d*x + c)^9 - 7293
*b^3*cosh(d*x + c)^7 + 792*(6*a*b^2 + 17*b^3)*cosh(d*x + c)^5 - 120*(78*a*b^2 + 85*b^3)*cosh(d*x + c)^3 + 4*(1
92*a^2*b + 936*a*b^2 + 595*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 - 437580*(1728*a^2*b + 3432*a*b^2 + 1547*b^3)*c
osh(d*x + c)^7 + 255255*(2184*b^3*cosh(d*x + c)^11 - 17017*b^3*cosh(d*x + c)^9 + 3168*(6*a*b^2 + 17*b^3)*cosh(
d*x + c)^7 - 1008*(78*a*b^2 + 85*b^3)*cosh(d*x + c)^5 + 112*(192*a^2*b + 936*a*b^2 + 595*b^3)*cosh(d*x + c)^3
- 12*(1728*a^2*b + 3432*a*b^2 + 1547*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 1225224*(512*a^3 + 3456*a^2*b + 429
0*a*b^2 + 1547*b^3)*cosh(d*x + c)^5 + 765765*(140*b^3*cosh(d*x + c)^13 - 1547*b^3*cosh(d*x + c)^11 + 440*(6*a*
b^2 + 17*b^3)*cosh(d*x + c)^9 - 240*(78*a*b^2 + 85*b^3)*cosh(d*x + c)^7 + 56*(192*a^2*b + 936*a*b^2 + 595*b^3)
*cosh(d*x + c)^5 - 20*(1728*a^2*b + 3432*a*b^2 + 1547*b^3)*cosh(d*x + c)^3 + 8*(512*a^3 + 3456*a^2*b + 4290*a*
b^2 + 1547*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 2042040*(2560*a^3 + 8064*a^2*b + 7722*a*b^2 + 2431*b^3)*cosh(
d*x + c)^3 + 765765*(8*b^3*cosh(d*x + c)^15 - 119*b^3*cosh(d*x + c)^13 + 48*(6*a*b^2 + 17*b^3)*cosh(d*x + c)^1
1 - 40*(78*a*b^2 + 85*b^3)*cosh(d*x + c)^9 + 16*(192*a^2*b + 936*a*b^2 + 595*b^3)*cosh(d*x + c)^7 - 12*(1728*a
^2*b + 3432*a*b^2 + 1547*b^3)*cosh(d*x + c)^5 + 16*(512*a^3 + 3456*a^2*b + 4290*a*b^2 + 1547*b^3)*cosh(d*x + c
)^3 - 8*(2560*a^3 + 8064*a^2*b + 7722*a*b^2 + 2431*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 1531530*(20480*a^3 +
48384*a^2*b + 41184*a*b^2 + 12155*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)**5*(a+b*sinh(d*x+c)**4)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.7945, size = 934, normalized size = 4.25 \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)^5*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")
[Out]
1/100370350080*(45045*b^3*e^(17*d*x + 17*c) - 867867*b^3*e^(15*d*x + 15*c) + 2827440*a*b^2*e^(13*d*x + 13*c) +
8011080*b^3*e^(13*d*x + 13*c) - 43439760*a*b^2*e^(11*d*x + 11*c) - 47338200*b^3*e^(11*d*x + 11*c) + 65345280*
a^2*b*e^(9*d*x + 9*c) + 318558240*a*b^2*e^(9*d*x + 9*c) + 202502300*b^3*e^(9*d*x + 9*c) - 756138240*a^2*b*e^(7
*d*x + 7*c) - 1501774560*a*b^2*e^(7*d*x + 7*c) - 676936260*b^3*e^(7*d*x + 7*c) + 627314688*a^3*e^(5*d*x + 5*c)
+ 4234374144*a^2*b*e^(5*d*x + 5*c) + 5256210960*a*b^2*e^(5*d*x + 5*c) + 1895421528*b^3*e^(5*d*x + 5*c) - 5227
622400*a^3*e^(3*d*x + 3*c) - 16467010560*a^2*b*e^(3*d*x + 3*c) - 15768632880*a*b^2*e^(3*d*x + 3*c) - 496419924
0*b^3*e^(3*d*x + 3*c) + 31365734400*a^3*e^(d*x + c) + 74101547520*a^2*b*e^(d*x + c) + 63074531520*a*b^2*e^(d*x
+ c) + 18615747150*b^3*e^(d*x + c) + (31365734400*a^3*e^(16*d*x + 16*c) + 74101547520*a^2*b*e^(16*d*x + 16*c)
+ 63074531520*a*b^2*e^(16*d*x + 16*c) + 18615747150*b^3*e^(16*d*x + 16*c) - 5227622400*a^3*e^(14*d*x + 14*c)
- 16467010560*a^2*b*e^(14*d*x + 14*c) - 15768632880*a*b^2*e^(14*d*x + 14*c) - 4964199240*b^3*e^(14*d*x + 14*c)
+ 627314688*a^3*e^(12*d*x + 12*c) + 4234374144*a^2*b*e^(12*d*x + 12*c) + 5256210960*a*b^2*e^(12*d*x + 12*c) +
1895421528*b^3*e^(12*d*x + 12*c) - 756138240*a^2*b*e^(10*d*x + 10*c) - 1501774560*a*b^2*e^(10*d*x + 10*c) - 6
76936260*b^3*e^(10*d*x + 10*c) + 65345280*a^2*b*e^(8*d*x + 8*c) + 318558240*a*b^2*e^(8*d*x + 8*c) + 202502300*
b^3*e^(8*d*x + 8*c) - 43439760*a*b^2*e^(6*d*x + 6*c) - 47338200*b^3*e^(6*d*x + 6*c) + 2827440*a*b^2*e^(4*d*x +
4*c) + 8011080*b^3*e^(4*d*x + 4*c) - 867867*b^3*e^(2*d*x + 2*c) + 45045*b^3)*e^(-17*d*x - 17*c))/d