3.225 \(\int \text{csch}^{14}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)
Optimal. Leaf size=144 \[ -\frac{a^2 (5 a+b) \coth ^9(c+d x)}{3 d}+\frac{4 a^2 (5 a+3 b) \coth ^7(c+d x)}{7 d}-\frac{a^3 \coth ^{13}(c+d x)}{13 d}+\frac{6 a^3 \coth ^{11}(c+d x)}{11 d}-\frac{3 a (a+b) (5 a+b) \coth ^5(c+d x)}{5 d}+\frac{2 a (a+b)^2 \coth ^3(c+d x)}{d}-\frac{(a+b)^3 \coth (c+d x)}{d} \]
[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)
________________________________________________________________________________________
Rubi [A] time = 0.130776, antiderivative size = 144, 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 =
{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 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]
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]
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*}
Mathematica [B] time = 3.20677, size = 350, normalized size = 2.43 \[ -\frac{\text{csch}^{13}(c+d x) \left (8580 \left (1152 a^2 b+1024 a^3+840 a b^2+231 b^3\right ) \cosh (c+d x)-6435 \left (2944 a^2 b+1024 a^3+2408 a b^2+693 b^3\right ) \cosh (3 (c+d x))+13087360 a^2 b \cosh (5 (c+d x))-5234944 a^2 b \cosh (7 (c+d x))+1427712 a^2 b \cosh (9 (c+d x))-237952 a^2 b \cosh (11 (c+d x))+18304 a^2 b \cosh (13 (c+d x))+3660800 a^3 \cosh (5 (c+d x))-1464320 a^3 \cosh (7 (c+d x))+399360 a^3 \cosh (9 (c+d x))-66560 a^3 \cosh (11 (c+d x))+5120 a^3 \cosh (13 (c+d x))+13093080 a b^2 \cosh (5 (c+d x))-6390384 a b^2 \cosh (7 (c+d x))+1873872 a b^2 \cosh (9 (c+d x))-312312 a b^2 \cosh (11 (c+d x))+24024 a b^2 \cosh (13 (c+d x))+4129125 b^3 \cosh (5 (c+d x))-2312310 b^3 \cosh (7 (c+d x))+810810 b^3 \cosh (9 (c+d x))-165165 b^3 \cosh (11 (c+d x))+15015 b^3 \cosh (13 (c+d x))\right )}{61501440 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^14*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
-((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)] + 1
427712*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)] - 66560*a^3*Co
sh[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*Cosh[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)/(61501440*d)
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Maple [A] time = 0.098, size = 177, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{1024}{3003}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{12}}{13}}+{\frac{12\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{10}}{143}}-{\frac{40\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{8}}{429}}+{\frac{320\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}}{3003}}-{\frac{128\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{1001}}+{\frac{512\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{3003}} \right ){\rm coth} \left (dx+c\right )+3\,{a}^{2}b \left ( -{\frac{128}{315}}-1/9\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{8}+{\frac{8\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}}{63}}-{\frac{16\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{105}}+{\frac{64\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{315}} \right ){\rm coth} \left (dx+c\right )+3\,a{b}^{2} \left ( -{\frac{8}{15}}-1/5\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}+{\frac{4\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{15}} \right ){\rm coth} \left (dx+c\right )-{b}^{3}{\rm coth} \left (dx+c\right ) \right ) } \end{align*}
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))
________________________________________________________________________________________
Maxima [B] time = 1.30355, size = 2587, normalized size = 17.97 \begin{align*} \text{result too large to display} \end{align*}
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))
________________________________________________________________________________________
Fricas [B] time = 1.80262, size = 6778, normalized size = 47.07 \begin{align*} \text{result too large to display} \end{align*}
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)
________________________________________________________________________________________
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(csch(d*x+c)**14*(a+b*sinh(d*x+c)**4)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.66735, size = 760, normalized size = 5.28 \begin{align*} \text{result too large to display} \end{align*}
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)