3.225 \(\int \text {csch}^{14}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)
Optimal. Leaf size=144 \[ -\frac {a^3 \coth ^{13}(c+d x)}{13 d}+\frac {6 a^3 \coth ^{11}(c+d x)}{11 d}-\frac {a^2 (5 a+b) \coth ^9(c+d x)}{3 d}+\frac {4 a^2 (5 a+3 b) \coth ^7(c+d x)}{7 d}-\frac {3 a (a+b) (5 a+b) \coth ^5(c+d x)}{5 d}+\frac {2 a (a+b)^2 \coth ^3(c+d x)}{d}-\frac {(a+b)^3 \coth (c+d x)}{d} \]
[Out]
-(a+b)^3*coth(d*x+c)/d+2*a*(a+b)^2*coth(d*x+c)^3/d-3/5*a*(a+b)*(5*a+b)*coth(d*x+c)^5/d+4/7*a^2*(5*a+3*b)*coth(
d*x+c)^7/d-1/3*a^2*(5*a+b)*coth(d*x+c)^9/d+6/11*a^3*coth(d*x+c)^11/d-1/13*a^3*coth(d*x+c)^13/d
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Rubi [A] time = 0.13, antiderivative size = 144, 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 =
{3217, 1108} \[ -\frac {a^2 (5 a+b) \coth ^9(c+d x)}{3 d}+\frac {4 a^2 (5 a+3 b) \coth ^7(c+d x)}{7 d}-\frac {a^3 \coth ^{13}(c+d x)}{13 d}+\frac {6 a^3 \coth ^{11}(c+d x)}{11 d}-\frac {3 a (a+b) (5 a+b) \coth ^5(c+d x)}{5 d}+\frac {2 a (a+b)^2 \coth ^3(c+d x)}{d}-\frac {(a+b)^3 \coth (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^14*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
-(((a + b)^3*Coth[c + d*x])/d) + (2*a*(a + b)^2*Coth[c + d*x]^3)/d - (3*a*(a + b)*(5*a + b)*Coth[c + d*x]^5)/(
5*d) + (4*a^2*(5*a + 3*b)*Coth[c + d*x]^7)/(7*d) - (a^2*(5*a + b)*Coth[c + d*x]^9)/(3*d) + (6*a^3*Coth[c + d*x
]^11)/(11*d) - (a^3*Coth[c + d*x]^13)/(13*d)
Rule 1108
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && !IntegerQ[(m + 1)/2]
Rule 3217
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p)/(1 + ff^2
*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
Rubi steps
\begin {align*} \int \text {csch}^{14}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a-2 a x^2+(a+b) x^4\right )^3}{x^{14}} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^3}{x^{14}}-\frac {6 a^3}{x^{12}}+\frac {3 a^2 (5 a+b)}{x^{10}}-\frac {4 a^2 (5 a+3 b)}{x^8}+\frac {3 a (a+b) (5 a+b)}{x^6}-\frac {6 a (a+b)^2}{x^4}+\frac {(a+b)^3}{x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {(a+b)^3 \coth (c+d x)}{d}+\frac {2 a (a+b)^2 \coth ^3(c+d x)}{d}-\frac {3 a (a+b) (5 a+b) \coth ^5(c+d x)}{5 d}+\frac {4 a^2 (5 a+3 b) \coth ^7(c+d x)}{7 d}-\frac {a^2 (5 a+b) \coth ^9(c+d x)}{3 d}+\frac {6 a^3 \coth ^{11}(c+d x)}{11 d}-\frac {a^3 \coth ^{13}(c+d x)}{13 d}\\ \end {align*}
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Mathematica [B] time = 3.24, size = 350, normalized size = 2.43 \[ -\frac {\text {csch}^{13}(c+d x) \left (3660800 a^3 \cosh (5 (c+d x))-1464320 a^3 \cosh (7 (c+d x))+399360 a^3 \cosh (9 (c+d x))-66560 a^3 \cosh (11 (c+d x))+5120 a^3 \cosh (13 (c+d x))+13087360 a^2 b \cosh (5 (c+d x))-5234944 a^2 b \cosh (7 (c+d x))+1427712 a^2 b \cosh (9 (c+d x))-237952 a^2 b \cosh (11 (c+d x))+18304 a^2 b \cosh (13 (c+d x))+8580 \left (1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right ) \cosh (c+d x)-6435 \left (1024 a^3+2944 a^2 b+2408 a b^2+693 b^3\right ) \cosh (3 (c+d x))+13093080 a b^2 \cosh (5 (c+d x))-6390384 a b^2 \cosh (7 (c+d x))+1873872 a b^2 \cosh (9 (c+d x))-312312 a b^2 \cosh (11 (c+d x))+24024 a b^2 \cosh (13 (c+d x))+4129125 b^3 \cosh (5 (c+d x))-2312310 b^3 \cosh (7 (c+d x))+810810 b^3 \cosh (9 (c+d x))-165165 b^3 \cosh (11 (c+d x))+15015 b^3 \cosh (13 (c+d x))\right )}{61501440 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^14*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
-1/61501440*((8580*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*Cosh[c + d*x] - 6435*(1024*a^3 + 2944*a^2*b +
2408*a*b^2 + 693*b^3)*Cosh[3*(c + d*x)] + 3660800*a^3*Cosh[5*(c + d*x)] + 13087360*a^2*b*Cosh[5*(c + d*x)] +
13093080*a*b^2*Cosh[5*(c + d*x)] + 4129125*b^3*Cosh[5*(c + d*x)] - 1464320*a^3*Cosh[7*(c + d*x)] - 5234944*a^2
*b*Cosh[7*(c + d*x)] - 6390384*a*b^2*Cosh[7*(c + d*x)] - 2312310*b^3*Cosh[7*(c + d*x)] + 399360*a^3*Cosh[9*(c
+ d*x)] + 1427712*a^2*b*Cosh[9*(c + d*x)] + 1873872*a*b^2*Cosh[9*(c + d*x)] + 810810*b^3*Cosh[9*(c + d*x)] - 6
6560*a^3*Cosh[11*(c + d*x)] - 237952*a^2*b*Cosh[11*(c + d*x)] - 312312*a*b^2*Cosh[11*(c + d*x)] - 165165*b^3*C
osh[11*(c + d*x)] + 5120*a^3*Cosh[13*(c + d*x)] + 18304*a^2*b*Cosh[13*(c + d*x)] + 24024*a*b^2*Cosh[13*(c + d*
x)] + 15015*b^3*Cosh[13*(c + d*x)])*Csch[c + d*x]^13)/d
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fricas [B] time = 0.65, size = 2323, normalized size = 16.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^14*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")
[Out]
-4/15015*((2560*a^3 + 9152*a^2*b + 12012*a*b^2 + 15015*b^3)*cosh(d*x + c)^12 - 48*(640*a^3 + 2288*a^2*b + 3003
*a*b^2)*cosh(d*x + c)*sinh(d*x + c)^11 + (2560*a^3 + 9152*a^2*b + 12012*a*b^2 + 15015*b^3)*sinh(d*x + c)^12 -
52*(640*a^3 + 2288*a^2*b + 3003*a*b^2 + 3465*b^3)*cosh(d*x + c)^10 - 2*(16640*a^3 + 59488*a^2*b + 78078*a*b^2
+ 90090*b^3 - 33*(2560*a^3 + 9152*a^2*b + 12012*a*b^2 + 15015*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 - 40*(22*
(640*a^3 + 2288*a^2*b + 3003*a*b^2)*cosh(d*x + c)^3 - 13*(640*a^3 + 2288*a^2*b + 3003*a*b^2)*cosh(d*x + c))*si
nh(d*x + c)^9 + 78*(2560*a^3 + 9152*a^2*b + 13552*a*b^2 + 12705*b^3)*cosh(d*x + c)^8 + 3*(165*(2560*a^3 + 9152
*a^2*b + 12012*a*b^2 + 15015*b^3)*cosh(d*x + c)^4 + 66560*a^3 + 237952*a^2*b + 352352*a*b^2 + 330330*b^3 - 780
*(640*a^3 + 2288*a^2*b + 3003*a*b^2 + 3465*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 - 96*(33*(640*a^3 + 2288*a^2*
b + 3003*a*b^2)*cosh(d*x + c)^5 - 65*(640*a^3 + 2288*a^2*b + 3003*a*b^2)*cosh(d*x + c)^3 + 52*(320*a^3 + 1144*
a^2*b + 1309*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 572*(1280*a^3 + 4576*a^2*b + 7581*a*b^2 + 5775*b^3)*cosh(
d*x + c)^6 + 4*(231*(2560*a^3 + 9152*a^2*b + 12012*a*b^2 + 15015*b^3)*cosh(d*x + c)^6 - 2730*(640*a^3 + 2288*a
^2*b + 3003*a*b^2 + 3465*b^3)*cosh(d*x + c)^4 - 183040*a^3 - 654368*a^2*b - 1084083*a*b^2 - 825825*b^3 + 546*(
2560*a^3 + 9152*a^2*b + 13552*a*b^2 + 12705*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 - 24*(132*(640*a^3 + 2288*a^
2*b + 3003*a*b^2)*cosh(d*x + c)^7 - 546*(640*a^3 + 2288*a^2*b + 3003*a*b^2)*cosh(d*x + c)^5 + 1456*(320*a^3 +
1144*a^2*b + 1309*a*b^2)*cosh(d*x + c)^3 - 143*(1280*a^3 + 4576*a^2*b + 4011*a*b^2)*cosh(d*x + c))*sinh(d*x +
c)^5 + 143*(12800*a^3 + 53824*a^2*b + 79884*a*b^2 + 51975*b^3)*cosh(d*x + c)^4 + (495*(2560*a^3 + 9152*a^2*b +
12012*a*b^2 + 15015*b^3)*cosh(d*x + c)^8 - 10920*(640*a^3 + 2288*a^2*b + 3003*a*b^2 + 3465*b^3)*cosh(d*x + c)
^6 + 5460*(2560*a^3 + 9152*a^2*b + 13552*a*b^2 + 12705*b^3)*cosh(d*x + c)^4 + 1830400*a^3 + 7696832*a^2*b + 11
423412*a*b^2 + 7432425*b^3 - 8580*(1280*a^3 + 4576*a^2*b + 7581*a*b^2 + 5775*b^3)*cosh(d*x + c)^2)*sinh(d*x +
c)^4 - 16*(55*(640*a^3 + 2288*a^2*b + 3003*a*b^2)*cosh(d*x + c)^9 - 390*(640*a^3 + 2288*a^2*b + 3003*a*b^2)*co
sh(d*x + c)^7 + 2184*(320*a^3 + 1144*a^2*b + 1309*a*b^2)*cosh(d*x + c)^5 - 715*(1280*a^3 + 4576*a^2*b + 4011*a
*b^2)*cosh(d*x + c)^3 + 143*(3200*a^3 + 9424*a^2*b + 6489*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4392960*a^3
+ 10323456*a^2*b + 12108096*a*b^2 + 6936930*b^3 - 3432*(960*a^3 + 4664*a^2*b + 5859*a*b^2 + 3465*b^3)*cosh(d*x
+ c)^2 + 6*(11*(2560*a^3 + 9152*a^2*b + 12012*a*b^2 + 15015*b^3)*cosh(d*x + c)^10 - 390*(640*a^3 + 2288*a^2*b
+ 3003*a*b^2 + 3465*b^3)*cosh(d*x + c)^8 + 364*(2560*a^3 + 9152*a^2*b + 13552*a*b^2 + 12705*b^3)*cosh(d*x + c
)^6 - 1430*(1280*a^3 + 4576*a^2*b + 7581*a*b^2 + 5775*b^3)*cosh(d*x + c)^4 - 549120*a^3 - 2667808*a^2*b - 3351
348*a*b^2 - 1981980*b^3 + 143*(12800*a^3 + 53824*a^2*b + 79884*a*b^2 + 51975*b^3)*cosh(d*x + c)^2)*sinh(d*x +
c)^2 - 8*(6*(640*a^3 + 2288*a^2*b + 3003*a*b^2)*cosh(d*x + c)^11 - 65*(640*a^3 + 2288*a^2*b + 3003*a*b^2)*cosh
(d*x + c)^9 + 624*(320*a^3 + 1144*a^2*b + 1309*a*b^2)*cosh(d*x + c)^7 - 429*(1280*a^3 + 4576*a^2*b + 4011*a*b^
2)*cosh(d*x + c)^5 + 286*(3200*a^3 + 9424*a^2*b + 6489*a*b^2)*cosh(d*x + c)^3 - 858*(960*a^3 + 1528*a^2*b + 90
3*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^14 + 14*d*cosh(d*x + c)*sinh(d*x + c)^13 + d*sinh(d*x
+ c)^14 - 14*d*cosh(d*x + c)^12 + 7*(13*d*cosh(d*x + c)^2 - 2*d)*sinh(d*x + c)^12 + 4*(91*d*cosh(d*x + c)^3 -
36*d*cosh(d*x + c))*sinh(d*x + c)^11 + 91*d*cosh(d*x + c)^10 + 7*(143*d*cosh(d*x + c)^4 - 132*d*cosh(d*x + c)^
2 + 13*d)*sinh(d*x + c)^10 + 2*(1001*d*cosh(d*x + c)^5 - 1320*d*cosh(d*x + c)^3 + 325*d*cosh(d*x + c))*sinh(d*
x + c)^9 - 364*d*cosh(d*x + c)^8 + 7*(429*d*cosh(d*x + c)^6 - 990*d*cosh(d*x + c)^4 + 585*d*cosh(d*x + c)^2 -
52*d)*sinh(d*x + c)^8 + 8*(429*d*cosh(d*x + c)^7 - 1188*d*cosh(d*x + c)^5 + 975*d*cosh(d*x + c)^3 - 208*d*cosh
(d*x + c))*sinh(d*x + c)^7 + 1001*d*cosh(d*x + c)^6 + 7*(429*d*cosh(d*x + c)^8 - 1848*d*cosh(d*x + c)^6 + 2730
*d*cosh(d*x + c)^4 - 1456*d*cosh(d*x + c)^2 + 143*d)*sinh(d*x + c)^6 + 2*(1001*d*cosh(d*x + c)^9 - 4752*d*cosh
(d*x + c)^7 + 8190*d*cosh(d*x + c)^5 - 5824*d*cosh(d*x + c)^3 + 1287*d*cosh(d*x + c))*sinh(d*x + c)^5 - 2002*d
*cosh(d*x + c)^4 + 7*(143*d*cosh(d*x + c)^10 - 990*d*cosh(d*x + c)^8 + 2730*d*cosh(d*x + c)^6 - 3640*d*cosh(d*
x + c)^4 + 2145*d*cosh(d*x + c)^2 - 286*d)*sinh(d*x + c)^4 + 4*(91*d*cosh(d*x + c)^11 - 660*d*cosh(d*x + c)^9
+ 1950*d*cosh(d*x + c)^7 - 2912*d*cosh(d*x + c)^5 + 2145*d*cosh(d*x + c)^3 - 572*d*cosh(d*x + c))*sinh(d*x + c
)^3 + 3003*d*cosh(d*x + c)^2 + 7*(13*d*cosh(d*x + c)^12 - 132*d*cosh(d*x + c)^10 + 585*d*cosh(d*x + c)^8 - 145
6*d*cosh(d*x + c)^6 + 2145*d*cosh(d*x + c)^4 - 1716*d*cosh(d*x + c)^2 + 429*d)*sinh(d*x + c)^2 + 2*(7*d*cosh(d
*x + c)^13 - 72*d*cosh(d*x + c)^11 + 325*d*cosh(d*x + c)^9 - 832*d*cosh(d*x + c)^7 + 1287*d*cosh(d*x + c)^5 -
1144*d*cosh(d*x + c)^3 + 429*d*cosh(d*x + c))*sinh(d*x + c) - 1716*d)
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giac [B] time = 0.52, size = 563, normalized size = 3.91 \[ -\frac {2 \, {\left (15015 \, b^{3} e^{\left (24 \, d x + 24 \, c\right )} - 180180 \, b^{3} e^{\left (22 \, d x + 22 \, c\right )} + 240240 \, a b^{2} e^{\left (20 \, d x + 20 \, c\right )} + 990990 \, b^{3} e^{\left (20 \, d x + 20 \, c\right )} - 2042040 \, a b^{2} e^{\left (18 \, d x + 18 \, c\right )} - 3303300 \, b^{3} e^{\left (18 \, d x + 18 \, c\right )} + 2306304 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} + 7711704 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} + 7432425 \, b^{3} e^{\left (16 \, d x + 16 \, c\right )} - 10762752 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} - 17008992 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} - 11891880 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 8785920 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 20646912 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 24216192 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 13873860 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} - 6589440 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} - 21250944 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 23207184 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 11891880 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 3660800 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 13087360 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 15135120 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 7432425 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 1464320 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 5234944 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 6630624 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 3303300 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 399360 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 1427712 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 1873872 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 990990 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 66560 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 237952 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 312312 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 180180 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5120 \, a^{3} + 18304 \, a^{2} b + 24024 \, a b^{2} + 15015 \, b^{3}\right )}}{15015 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^14*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")
[Out]
-2/15015*(15015*b^3*e^(24*d*x + 24*c) - 180180*b^3*e^(22*d*x + 22*c) + 240240*a*b^2*e^(20*d*x + 20*c) + 990990
*b^3*e^(20*d*x + 20*c) - 2042040*a*b^2*e^(18*d*x + 18*c) - 3303300*b^3*e^(18*d*x + 18*c) + 2306304*a^2*b*e^(16
*d*x + 16*c) + 7711704*a*b^2*e^(16*d*x + 16*c) + 7432425*b^3*e^(16*d*x + 16*c) - 10762752*a^2*b*e^(14*d*x + 14
*c) - 17008992*a*b^2*e^(14*d*x + 14*c) - 11891880*b^3*e^(14*d*x + 14*c) + 8785920*a^3*e^(12*d*x + 12*c) + 2064
6912*a^2*b*e^(12*d*x + 12*c) + 24216192*a*b^2*e^(12*d*x + 12*c) + 13873860*b^3*e^(12*d*x + 12*c) - 6589440*a^3
*e^(10*d*x + 10*c) - 21250944*a^2*b*e^(10*d*x + 10*c) - 23207184*a*b^2*e^(10*d*x + 10*c) - 11891880*b^3*e^(10*
d*x + 10*c) + 3660800*a^3*e^(8*d*x + 8*c) + 13087360*a^2*b*e^(8*d*x + 8*c) + 15135120*a*b^2*e^(8*d*x + 8*c) +
7432425*b^3*e^(8*d*x + 8*c) - 1464320*a^3*e^(6*d*x + 6*c) - 5234944*a^2*b*e^(6*d*x + 6*c) - 6630624*a*b^2*e^(6
*d*x + 6*c) - 3303300*b^3*e^(6*d*x + 6*c) + 399360*a^3*e^(4*d*x + 4*c) + 1427712*a^2*b*e^(4*d*x + 4*c) + 18738
72*a*b^2*e^(4*d*x + 4*c) + 990990*b^3*e^(4*d*x + 4*c) - 66560*a^3*e^(2*d*x + 2*c) - 237952*a^2*b*e^(2*d*x + 2*
c) - 312312*a*b^2*e^(2*d*x + 2*c) - 180180*b^3*e^(2*d*x + 2*c) + 5120*a^3 + 18304*a^2*b + 24024*a*b^2 + 15015*
b^3)/(d*(e^(2*d*x + 2*c) - 1)^13)
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maple [A] time = 0.13, size = 177, normalized size = 1.23 \[ \frac {a^{3} \left (-\frac {1024}{3003}-\frac {\mathrm {csch}\left (d x +c \right )^{12}}{13}+\frac {12 \mathrm {csch}\left (d x +c \right )^{10}}{143}-\frac {40 \mathrm {csch}\left (d x +c \right )^{8}}{429}+\frac {320 \mathrm {csch}\left (d x +c \right )^{6}}{3003}-\frac {128 \mathrm {csch}\left (d x +c \right )^{4}}{1001}+\frac {512 \mathrm {csch}\left (d x +c \right )^{2}}{3003}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (-\frac {128}{315}-\frac {\mathrm {csch}\left (d x +c \right )^{8}}{9}+\frac {8 \mathrm {csch}\left (d x +c \right )^{6}}{63}-\frac {16 \mathrm {csch}\left (d x +c \right )^{4}}{105}+\frac {64 \mathrm {csch}\left (d x +c \right )^{2}}{315}\right ) \coth \left (d x +c \right )+3 a \,b^{2} \left (-\frac {8}{15}-\frac {\mathrm {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )-b^{3} \coth \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^14*(a+b*sinh(d*x+c)^4)^3,x)
[Out]
1/d*(a^3*(-1024/3003-1/13*csch(d*x+c)^12+12/143*csch(d*x+c)^10-40/429*csch(d*x+c)^8+320/3003*csch(d*x+c)^6-128
/1001*csch(d*x+c)^4+512/3003*csch(d*x+c)^2)*coth(d*x+c)+3*a^2*b*(-128/315-1/9*csch(d*x+c)^8+8/63*csch(d*x+c)^6
-16/105*csch(d*x+c)^4+64/315*csch(d*x+c)^2)*coth(d*x+c)+3*a*b^2*(-8/15-1/5*csch(d*x+c)^4+4/15*csch(d*x+c)^2)*c
oth(d*x+c)-b^3*coth(d*x+c))
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maxima [B] time = 0.37, size = 1916, normalized size = 13.31 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^14*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")
[Out]
-2048/3003*a^3*(13*e^(-2*d*x - 2*c)/(d*(13*e^(-2*d*x - 2*c) - 78*e^(-4*d*x - 4*c) + 286*e^(-6*d*x - 6*c) - 715
*e^(-8*d*x - 8*c) + 1287*e^(-10*d*x - 10*c) - 1716*e^(-12*d*x - 12*c) + 1716*e^(-14*d*x - 14*c) - 1287*e^(-16*
d*x - 16*c) + 715*e^(-18*d*x - 18*c) - 286*e^(-20*d*x - 20*c) + 78*e^(-22*d*x - 22*c) - 13*e^(-24*d*x - 24*c)
+ e^(-26*d*x - 26*c) - 1)) - 78*e^(-4*d*x - 4*c)/(d*(13*e^(-2*d*x - 2*c) - 78*e^(-4*d*x - 4*c) + 286*e^(-6*d*x
- 6*c) - 715*e^(-8*d*x - 8*c) + 1287*e^(-10*d*x - 10*c) - 1716*e^(-12*d*x - 12*c) + 1716*e^(-14*d*x - 14*c) -
1287*e^(-16*d*x - 16*c) + 715*e^(-18*d*x - 18*c) - 286*e^(-20*d*x - 20*c) + 78*e^(-22*d*x - 22*c) - 13*e^(-24
*d*x - 24*c) + e^(-26*d*x - 26*c) - 1)) + 286*e^(-6*d*x - 6*c)/(d*(13*e^(-2*d*x - 2*c) - 78*e^(-4*d*x - 4*c) +
286*e^(-6*d*x - 6*c) - 715*e^(-8*d*x - 8*c) + 1287*e^(-10*d*x - 10*c) - 1716*e^(-12*d*x - 12*c) + 1716*e^(-14
*d*x - 14*c) - 1287*e^(-16*d*x - 16*c) + 715*e^(-18*d*x - 18*c) - 286*e^(-20*d*x - 20*c) + 78*e^(-22*d*x - 22*
c) - 13*e^(-24*d*x - 24*c) + e^(-26*d*x - 26*c) - 1)) - 715*e^(-8*d*x - 8*c)/(d*(13*e^(-2*d*x - 2*c) - 78*e^(-
4*d*x - 4*c) + 286*e^(-6*d*x - 6*c) - 715*e^(-8*d*x - 8*c) + 1287*e^(-10*d*x - 10*c) - 1716*e^(-12*d*x - 12*c)
+ 1716*e^(-14*d*x - 14*c) - 1287*e^(-16*d*x - 16*c) + 715*e^(-18*d*x - 18*c) - 286*e^(-20*d*x - 20*c) + 78*e^
(-22*d*x - 22*c) - 13*e^(-24*d*x - 24*c) + e^(-26*d*x - 26*c) - 1)) + 1287*e^(-10*d*x - 10*c)/(d*(13*e^(-2*d*x
- 2*c) - 78*e^(-4*d*x - 4*c) + 286*e^(-6*d*x - 6*c) - 715*e^(-8*d*x - 8*c) + 1287*e^(-10*d*x - 10*c) - 1716*e
^(-12*d*x - 12*c) + 1716*e^(-14*d*x - 14*c) - 1287*e^(-16*d*x - 16*c) + 715*e^(-18*d*x - 18*c) - 286*e^(-20*d*
x - 20*c) + 78*e^(-22*d*x - 22*c) - 13*e^(-24*d*x - 24*c) + e^(-26*d*x - 26*c) - 1)) - 1716*e^(-12*d*x - 12*c)
/(d*(13*e^(-2*d*x - 2*c) - 78*e^(-4*d*x - 4*c) + 286*e^(-6*d*x - 6*c) - 715*e^(-8*d*x - 8*c) + 1287*e^(-10*d*x
- 10*c) - 1716*e^(-12*d*x - 12*c) + 1716*e^(-14*d*x - 14*c) - 1287*e^(-16*d*x - 16*c) + 715*e^(-18*d*x - 18*c
) - 286*e^(-20*d*x - 20*c) + 78*e^(-22*d*x - 22*c) - 13*e^(-24*d*x - 24*c) + e^(-26*d*x - 26*c) - 1)) - 1/(d*(
13*e^(-2*d*x - 2*c) - 78*e^(-4*d*x - 4*c) + 286*e^(-6*d*x - 6*c) - 715*e^(-8*d*x - 8*c) + 1287*e^(-10*d*x - 10
*c) - 1716*e^(-12*d*x - 12*c) + 1716*e^(-14*d*x - 14*c) - 1287*e^(-16*d*x - 16*c) + 715*e^(-18*d*x - 18*c) - 2
86*e^(-20*d*x - 20*c) + 78*e^(-22*d*x - 22*c) - 13*e^(-24*d*x - 24*c) + e^(-26*d*x - 26*c) - 1))) - 256/105*a^
2*b*(9*e^(-2*d*x - 2*c)/(d*(9*e^(-2*d*x - 2*c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x - 8
*c) + 126*e^(-10*d*x - 10*c) - 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) - 9*e^(-16*d*x - 16*c) + e^(-18*d
*x - 18*c) - 1)) - 36*e^(-4*d*x - 4*c)/(d*(9*e^(-2*d*x - 2*c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) - 12
6*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) - 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) - 9*e^(-16*d*x - 1
6*c) + e^(-18*d*x - 18*c) - 1)) + 84*e^(-6*d*x - 6*c)/(d*(9*e^(-2*d*x - 2*c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*
d*x - 6*c) - 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) - 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) - 9
*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) - 1)) - 126*e^(-8*d*x - 8*c)/(d*(9*e^(-2*d*x - 2*c) - 36*e^(-4*d*x -
4*c) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) - 84*e^(-12*d*x - 12*c) + 36*e^(-14
*d*x - 14*c) - 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) - 1)) - 1/(d*(9*e^(-2*d*x - 2*c) - 36*e^(-4*d*x - 4*c
) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) - 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*
x - 14*c) - 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) - 1))) - 16/5*a*b^2*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x
- 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 10*e^(-4*
d*x - 4*c)/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*
x - 10*c) - 1)) - 1/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) +
e^(-10*d*x - 10*c) - 1))) + 2*b^3/(d*(e^(-2*d*x - 2*c) - 1))
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mupad [B] time = 1.14, size = 3138, normalized size = 21.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*sinh(c + d*x)^4)^3/sinh(c + d*x)^14,x)
[Out]
((6*b^3*exp(4*c + 4*d*x))/(13*d) - (2*b^3*exp(6*c + 6*d*x))/(13*d) + (2*b^2*(96*a + 55*b))/(715*d) - (6*b^2*ex
p(2*c + 2*d*x)*(8*a + 11*b))/(143*d))/(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c
+ 8*d*x) + 1) - ((2*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(3003*d) - (12*b^3*exp(10*c + 10*d*x))/(13*
d) + (2*b^3*exp(12*c + 12*d*x))/(13*d) - (4*b*exp(2*c + 2*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(143*d) + (2*b*ex
p(4*c + 4*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(143*d) + (30*b^2*exp(8*c + 8*d*x)*(8*a + 11*b))/(143*d) - (8*b^
2*exp(6*c + 6*d*x)*(96*a + 55*b))/(143*d))/(7*exp(2*c + 2*d*x) - 21*exp(4*c + 4*d*x) + 35*exp(6*c + 6*d*x) - 3
5*exp(8*c + 8*d*x) + 21*exp(10*c + 10*d*x) - 7*exp(12*c + 12*d*x) + exp(14*c + 14*d*x) - 1) - ((8*exp(6*c + 6*
d*x)*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(143*d) - (18*b^3*exp(16*c + 16*d*x))/(13*d) + (2*b^3*exp(
18*c + 18*d*x))/(13*d) - (2*b^2*(96*a + 55*b))/(715*d) - (24*b*exp(4*c + 4*d*x)*(112*a*b + 128*a^2 + 33*b^2))/
(143*d) - (84*b*exp(8*c + 8*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(143*d) + (6*b*exp(2*c + 2*d*x)*(448*a*b + 256*
a^2 + 165*b^2))/(715*d) + (84*b*exp(10*c + 10*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(715*d) + (72*b^2*exp(14*c +
14*d*x)*(8*a + 11*b))/(143*d) - (168*b^2*exp(12*c + 12*d*x)*(96*a + 55*b))/(715*d))/(45*exp(4*c + 4*d*x) - 10
*exp(2*c + 2*d*x) - 120*exp(6*c + 6*d*x) + 210*exp(8*c + 8*d*x) - 252*exp(10*c + 10*d*x) + 210*exp(12*c + 12*d
*x) - 120*exp(14*c + 14*d*x) + 45*exp(16*c + 16*d*x) - 10*exp(18*c + 18*d*x) + exp(20*c + 20*d*x) + 1) - ((2*b
*(448*a*b + 256*a^2 + 165*b^2))/(2145*d) - (8*b^3*exp(6*c + 6*d*x))/(13*d) + (2*b^3*exp(8*c + 8*d*x))/(13*d) +
(12*b^2*exp(4*c + 4*d*x)*(8*a + 11*b))/(143*d) - (8*b^2*exp(2*c + 2*d*x)*(96*a + 55*b))/(715*d))/(5*exp(2*c +
2*d*x) - 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) - 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) - 1) + ((2*b*(11
2*a*b + 128*a^2 + 33*b^2))/(429*d) - (2*exp(2*c + 2*d*x)*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(429*d
) + (14*b^3*exp(12*c + 12*d*x))/(13*d) - (2*b^3*exp(14*c + 14*d*x))/(13*d) + (14*b*exp(4*c + 4*d*x)*(112*a*b +
128*a^2 + 33*b^2))/(143*d) - (14*b*exp(6*c + 6*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(429*d) - (42*b^2*exp(10*c
+ 10*d*x)*(8*a + 11*b))/(143*d) + (14*b^2*exp(8*c + 8*d*x)*(96*a + 55*b))/(143*d))/(28*exp(4*c + 4*d*x) - 8*e
xp(2*c + 2*d*x) - 56*exp(6*c + 6*d*x) + 70*exp(8*c + 8*d*x) - 56*exp(10*c + 10*d*x) + 28*exp(12*c + 12*d*x) -
8*exp(14*c + 14*d*x) + exp(16*c + 16*d*x) + 1) - ((2*b^3)/(13*d) + (8*exp(12*c + 12*d*x)*(840*a*b^2 + 1152*a^2
*b + 1024*a^3 + 231*b^3))/(13*d) - (24*b^3*exp(2*c + 2*d*x))/(13*d) - (24*b^3*exp(22*c + 22*d*x))/(13*d) + (2*
b^3*exp(24*c + 24*d*x))/(13*d) - (48*b*exp(10*c + 10*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(13*d) - (48*b*exp(14*
c + 14*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(13*d) + (6*b*exp(8*c + 8*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(13*d)
+ (6*b*exp(16*c + 16*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(13*d) + (12*b^2*exp(4*c + 4*d*x)*(8*a + 11*b))/(13*
d) + (12*b^2*exp(20*c + 20*d*x)*(8*a + 11*b))/(13*d) - (8*b^2*exp(6*c + 6*d*x)*(96*a + 55*b))/(13*d) - (8*b^2*
exp(18*c + 18*d*x)*(96*a + 55*b))/(13*d))/(13*exp(2*c + 2*d*x) - 78*exp(4*c + 4*d*x) + 286*exp(6*c + 6*d*x) -
715*exp(8*c + 8*d*x) + 1287*exp(10*c + 10*d*x) - 1716*exp(12*c + 12*d*x) + 1716*exp(14*c + 14*d*x) - 1287*exp(
16*c + 16*d*x) + 715*exp(18*c + 18*d*x) - 286*exp(20*c + 20*d*x) + 78*exp(22*c + 22*d*x) - 13*exp(24*c + 24*d*
x) + exp(26*c + 26*d*x) - 1) - ((2*b^3*exp(4*c + 4*d*x))/(13*d) - (4*b^3*exp(2*c + 2*d*x))/(13*d) + (2*b^2*(8*
a + 11*b))/(143*d))/(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) + ((2*b*(112*a*b + 128*a^
2 + 33*b^2))/(429*d) + (10*b^3*exp(8*c + 8*d*x))/(13*d) - (2*b^3*exp(10*c + 10*d*x))/(13*d) - (2*b*exp(2*c + 2
*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(429*d) - (20*b^2*exp(6*c + 6*d*x)*(8*a + 11*b))/(143*d) + (4*b^2*exp(4*c
+ 4*d*x)*(96*a + 55*b))/(143*d))/(15*exp(4*c + 4*d*x) - 6*exp(2*c + 2*d*x) - 20*exp(6*c + 6*d*x) + 15*exp(8*c
+ 8*d*x) - 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1) - ((2*b*(448*a*b + 256*a^2 + 165*b^2))/(2145*d) + (
8*exp(4*c + 4*d*x)*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(429*d) - (16*b^3*exp(14*c + 14*d*x))/(13*d)
+ (2*b^3*exp(16*c + 16*d*x))/(13*d) - (16*b*exp(2*c + 2*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(429*d) - (112*b*e
xp(6*c + 6*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(429*d) + (28*b*exp(8*c + 8*d*x)*(448*a*b + 256*a^2 + 165*b^2))/
(429*d) + (56*b^2*exp(12*c + 12*d*x)*(8*a + 11*b))/(143*d) - (112*b^2*exp(10*c + 10*d*x)*(96*a + 55*b))/(715*d
))/(9*exp(2*c + 2*d*x) - 36*exp(4*c + 4*d*x) + 84*exp(6*c + 6*d*x) - 126*exp(8*c + 8*d*x) + 126*exp(10*c + 10*
d*x) - 84*exp(12*c + 12*d*x) + 36*exp(14*c + 14*d*x) - 9*exp(16*c + 16*d*x) + exp(18*c + 18*d*x) - 1) + ((2*b^
3)/(13*d) - (4*exp(10*c + 10*d*x)*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(13*d) + (22*b^3*exp(20*c + 2
0*d*x))/(13*d) - (2*b^3*exp(22*c + 22*d*x))/(13*d) + (20*b*exp(8*c + 8*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(13*
d) + (28*b*exp(12*c + 12*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(13*d) - (2*b*exp(6*c + 6*d*x)*(448*a*b + 256*a^2
+ 165*b^2))/(13*d) - (4*b*exp(14*c + 14*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(13*d) - (2*b^2*exp(2*c + 2*d*x)*(
8*a + 11*b))/(13*d) - (10*b^2*exp(18*c + 18*d*x)*(8*a + 11*b))/(13*d) + (2*b^2*exp(4*c + 4*d*x)*(96*a + 55*b))
/(13*d) + (6*b^2*exp(16*c + 16*d*x)*(96*a + 55*b))/(13*d))/(66*exp(4*c + 4*d*x) - 12*exp(2*c + 2*d*x) - 220*ex
p(6*c + 6*d*x) + 495*exp(8*c + 8*d*x) - 792*exp(10*c + 10*d*x) + 924*exp(12*c + 12*d*x) - 792*exp(14*c + 14*d*
x) + 495*exp(16*c + 16*d*x) - 220*exp(18*c + 18*d*x) + 66*exp(20*c + 20*d*x) - 12*exp(22*c + 22*d*x) + exp(24*
c + 24*d*x) + 1) - ((20*exp(8*c + 8*d*x)*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(143*d) - (20*b^3*exp(
18*c + 18*d*x))/(13*d) + (2*b^3*exp(20*c + 20*d*x))/(13*d) + (2*b^2*(8*a + 11*b))/(143*d) - (80*b*exp(6*c + 6*
d*x)*(112*a*b + 128*a^2 + 33*b^2))/(143*d) - (168*b*exp(10*c + 10*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(143*d) +
(6*b*exp(4*c + 4*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(143*d) + (28*b*exp(12*c + 12*d*x)*(448*a*b + 256*a^2 +
165*b^2))/(143*d) + (90*b^2*exp(16*c + 16*d*x)*(8*a + 11*b))/(143*d) - (4*b^2*exp(2*c + 2*d*x)*(96*a + 55*b))/
(143*d) - (48*b^2*exp(14*c + 14*d*x)*(96*a + 55*b))/(143*d))/(11*exp(2*c + 2*d*x) - 55*exp(4*c + 4*d*x) + 165*
exp(6*c + 6*d*x) - 330*exp(8*c + 8*d*x) + 462*exp(10*c + 10*d*x) - 462*exp(12*c + 12*d*x) + 330*exp(14*c + 14*
d*x) - 165*exp(16*c + 16*d*x) + 55*exp(18*c + 18*d*x) - 11*exp(20*c + 20*d*x) + exp(22*c + 22*d*x) - 1) - (4*b
^3)/(13*d*(exp(2*c + 2*d*x) - 1))
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)**14*(a+b*sinh(d*x+c)**4)**3,x)
[Out]
Timed out
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