3.228 \(\int \text{csch}^{20}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)
Optimal. Leaf size=248 \[ -\frac{3 a \left (42 a^2+21 a b+b^2\right ) \coth ^{11}(c+d x)}{11 d}+\frac{a \left (42 a^2+35 a b+5 b^2\right ) \coth ^9(c+d x)}{3 d}-\frac{\left (105 a^2 b+84 a^3+30 a b^2+b^3\right ) \coth ^7(c+d x)}{7 d}+\frac{3 (a+b) \left (12 a^2+9 a b+b^2\right ) \coth ^5(c+d x)}{5 d}-\frac{a^2 (12 a+b) \coth ^{15}(c+d x)}{5 d}+\frac{21 a^2 (4 a+b) \coth ^{13}(c+d x)}{13 d}-\frac{a^3 \coth ^{19}(c+d x)}{19 d}+\frac{9 a^3 \coth ^{17}(c+d x)}{17 d}-\frac{(a+b)^2 (3 a+b) \coth ^3(c+d x)}{d}+\frac{(a+b)^3 \coth (c+d x)}{d} \]
[Out]
((a + b)^3*Coth[c + d*x])/d - ((a + b)^2*(3*a + b)*Coth[c + d*x]^3)/d + (3*(a + b)*(12*a^2 + 9*a*b + b^2)*Coth
[c + d*x]^5)/(5*d) - ((84*a^3 + 105*a^2*b + 30*a*b^2 + b^3)*Coth[c + d*x]^7)/(7*d) + (a*(42*a^2 + 35*a*b + 5*b
^2)*Coth[c + d*x]^9)/(3*d) - (3*a*(42*a^2 + 21*a*b + b^2)*Coth[c + d*x]^11)/(11*d) + (21*a^2*(4*a + b)*Coth[c
+ d*x]^13)/(13*d) - (a^2*(12*a + b)*Coth[c + d*x]^15)/(5*d) + (9*a^3*Coth[c + d*x]^17)/(17*d) - (a^3*Coth[c +
d*x]^19)/(19*d)
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Rubi [A] time = 0.222522, antiderivative size = 248, 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, 1261} \[ -\frac{3 a \left (42 a^2+21 a b+b^2\right ) \coth ^{11}(c+d x)}{11 d}+\frac{a \left (42 a^2+35 a b+5 b^2\right ) \coth ^9(c+d x)}{3 d}-\frac{\left (105 a^2 b+84 a^3+30 a b^2+b^3\right ) \coth ^7(c+d x)}{7 d}+\frac{3 (a+b) \left (12 a^2+9 a b+b^2\right ) \coth ^5(c+d x)}{5 d}-\frac{a^2 (12 a+b) \coth ^{15}(c+d x)}{5 d}+\frac{21 a^2 (4 a+b) \coth ^{13}(c+d x)}{13 d}-\frac{a^3 \coth ^{19}(c+d x)}{19 d}+\frac{9 a^3 \coth ^{17}(c+d x)}{17 d}-\frac{(a+b)^2 (3 a+b) \coth ^3(c+d x)}{d}+\frac{(a+b)^3 \coth (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^20*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
((a + b)^3*Coth[c + d*x])/d - ((a + b)^2*(3*a + b)*Coth[c + d*x]^3)/d + (3*(a + b)*(12*a^2 + 9*a*b + b^2)*Coth
[c + d*x]^5)/(5*d) - ((84*a^3 + 105*a^2*b + 30*a*b^2 + b^3)*Coth[c + d*x]^7)/(7*d) + (a*(42*a^2 + 35*a*b + 5*b
^2)*Coth[c + d*x]^9)/(3*d) - (3*a*(42*a^2 + 21*a*b + b^2)*Coth[c + d*x]^11)/(11*d) + (21*a^2*(4*a + b)*Coth[c
+ d*x]^13)/(13*d) - (a^2*(12*a + b)*Coth[c + d*x]^15)/(5*d) + (9*a^3*Coth[c + d*x]^17)/(17*d) - (a^3*Coth[c +
d*x]^19)/(19*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 1261
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Rubi steps
\begin{align*} \int \text{csch}^{20}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3 \left (a-2 a x^2+(a+b) x^4\right )^3}{x^{20}} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^3}{x^{20}}-\frac{9 a^3}{x^{18}}+\frac{3 a^2 (12 a+b)}{x^{16}}-\frac{21 a^2 (4 a+b)}{x^{14}}+\frac{3 a \left (42 a^2+21 a b+b^2\right )}{x^{12}}-\frac{3 a \left (42 a^2+35 a b+5 b^2\right )}{x^{10}}+\frac{84 a^3+105 a^2 b+30 a b^2+b^3}{x^8}+\frac{3 (a+b) \left (-12 a^2-9 a b-b^2\right )}{x^6}+\frac{3 (a+b)^2 (3 a+b)}{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{(a+b)^2 (3 a+b) \coth ^3(c+d x)}{d}+\frac{3 (a+b) \left (12 a^2+9 a b+b^2\right ) \coth ^5(c+d x)}{5 d}-\frac{\left (84 a^3+105 a^2 b+30 a b^2+b^3\right ) \coth ^7(c+d x)}{7 d}+\frac{a \left (42 a^2+35 a b+5 b^2\right ) \coth ^9(c+d x)}{3 d}-\frac{3 a \left (42 a^2+21 a b+b^2\right ) \coth ^{11}(c+d x)}{11 d}+\frac{21 a^2 (4 a+b) \coth ^{13}(c+d x)}{13 d}-\frac{a^2 (12 a+b) \coth ^{15}(c+d x)}{5 d}+\frac{9 a^3 \coth ^{17}(c+d x)}{17 d}-\frac{a^3 \coth ^{19}(c+d x)}{19 d}\\ \end{align*}
Mathematica [B] time = 6.16765, size = 548, normalized size = 2.21 \[ \frac{\text{csch}^{19}(c+d x) \left (-8939234304 a^2 b \cosh (c+d x)+18149354496 a^2 b \cosh (3 (c+d x))-14582690304 a^2 b \cosh (5 (c+d x))+7852217856 a^2 b \cosh (7 (c+d x))-3365236224 a^2 b \cosh (9 (c+d x))+1121745408 a^2 b \cosh (11 (c+d x))-280436352 a^2 b \cosh (13 (c+d x))+49488768 a^2 b \cosh (15 (c+d x))-5498752 a^2 b \cosh (17 (c+d x))+289408 a^2 b \cosh (19 (c+d x))-7945986048 a^3 \cosh (c+d x)+6501261312 a^3 \cosh (3 (c+d x))-4334174208 a^3 \cosh (5 (c+d x))+2333786112 a^3 \cosh (7 (c+d x))-1000194048 a^3 \cosh (9 (c+d x))+333398016 a^3 \cosh (11 (c+d x))-83349504 a^3 \cosh (13 (c+d x))+14708736 a^3 \cosh (15 (c+d x))-1634304 a^3 \cosh (17 (c+d x))+86016 a^3 \cosh (19 (c+d x))-6518191680 a b^2 \cosh (c+d x)+14814072000 a b^2 \cosh (3 (c+d x))-14221509120 a b^2 \cosh (5 (c+d x))+8803791360 a b^2 \cosh (7 (c+d x))-3906077760 a b^2 \cosh (9 (c+d x))+1302025920 a b^2 \cosh (11 (c+d x))-325506480 a b^2 \cosh (13 (c+d x))+57442320 a b^2 \cosh (15 (c+d x))-6382480 a b^2 \cosh (17 (c+d x))+335920 a b^2 \cosh (19 (c+d x))-1792502712 b^3 \cosh (c+d x)+4260103848 b^3 \cosh (3 (c+d x))-4440518082 b^3 \cosh (5 (c+d x))+3047642598 b^3 \cosh (7 (c+d x))-1489040982 b^3 \cosh (9 (c+d x))+527386002 b^3 \cosh (11 (c+d x))-134271423 b^3 \cosh (13 (c+d x))+23694957 b^3 \cosh (15 (c+d x))-2632773 b^3 \cosh (17 (c+d x))+138567 b^3 \cosh (19 (c+d x))\right )}{79459860480 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^20*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
((-7945986048*a^3*Cosh[c + d*x] - 8939234304*a^2*b*Cosh[c + d*x] - 6518191680*a*b^2*Cosh[c + d*x] - 1792502712
*b^3*Cosh[c + d*x] + 6501261312*a^3*Cosh[3*(c + d*x)] + 18149354496*a^2*b*Cosh[3*(c + d*x)] + 14814072000*a*b^
2*Cosh[3*(c + d*x)] + 4260103848*b^3*Cosh[3*(c + d*x)] - 4334174208*a^3*Cosh[5*(c + d*x)] - 14582690304*a^2*b*
Cosh[5*(c + d*x)] - 14221509120*a*b^2*Cosh[5*(c + d*x)] - 4440518082*b^3*Cosh[5*(c + d*x)] + 2333786112*a^3*Co
sh[7*(c + d*x)] + 7852217856*a^2*b*Cosh[7*(c + d*x)] + 8803791360*a*b^2*Cosh[7*(c + d*x)] + 3047642598*b^3*Cos
h[7*(c + d*x)] - 1000194048*a^3*Cosh[9*(c + d*x)] - 3365236224*a^2*b*Cosh[9*(c + d*x)] - 3906077760*a*b^2*Cosh
[9*(c + d*x)] - 1489040982*b^3*Cosh[9*(c + d*x)] + 333398016*a^3*Cosh[11*(c + d*x)] + 1121745408*a^2*b*Cosh[11
*(c + d*x)] + 1302025920*a*b^2*Cosh[11*(c + d*x)] + 527386002*b^3*Cosh[11*(c + d*x)] - 83349504*a^3*Cosh[13*(c
+ d*x)] - 280436352*a^2*b*Cosh[13*(c + d*x)] - 325506480*a*b^2*Cosh[13*(c + d*x)] - 134271423*b^3*Cosh[13*(c
+ d*x)] + 14708736*a^3*Cosh[15*(c + d*x)] + 49488768*a^2*b*Cosh[15*(c + d*x)] + 57442320*a*b^2*Cosh[15*(c + d*
x)] + 23694957*b^3*Cosh[15*(c + d*x)] - 1634304*a^3*Cosh[17*(c + d*x)] - 5498752*a^2*b*Cosh[17*(c + d*x)] - 63
82480*a*b^2*Cosh[17*(c + d*x)] - 2632773*b^3*Cosh[17*(c + d*x)] + 86016*a^3*Cosh[19*(c + d*x)] + 289408*a^2*b*
Cosh[19*(c + d*x)] + 335920*a*b^2*Cosh[19*(c + d*x)] + 138567*b^3*Cosh[19*(c + d*x)])*Csch[c + d*x]^19)/(79459
860480*d)
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Maple [A] time = 0.09, size = 298, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{65536}{230945}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{18}}{19}}+{\frac{18\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{16}}{323}}-{\frac{96\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{14}}{1615}}+{\frac{1344\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{12}}{20995}}-{\frac{16128\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{10}}{230945}}+{\frac{3584\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{8}}{46189}}-{\frac{4096\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}}{46189}}+{\frac{24576\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{230945}}-{\frac{32768\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{230945}} \right ){\rm coth} \left (dx+c\right )+3\,{a}^{2}b \left ({\frac{2048}{6435}}-1/15\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{14}+{\frac{14\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{12}}{195}}-{\frac{56\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{10}}{715}}+{\frac{112\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{8}}{1287}}-{\frac{128\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}}{1287}}+{\frac{256\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{2145}}-{\frac{1024\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{6435}} \right ){\rm coth} \left (dx+c\right )+3\,a{b}^{2} \left ({\frac{256}{693}}-1/11\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{10}+{\frac{10\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{8}}{99}}-{\frac{80\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}}{693}}+{\frac{32\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{231}}-{\frac{128\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{693}} \right ){\rm coth} \left (dx+c\right )+{b}^{3} \left ({\frac{16}{35}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}}{7}}+{\frac{6\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{35}}-{\frac{8\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{35}} \right ){\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)^20*(a+b*sinh(d*x+c)^4)^3,x)
[Out]
1/d*(a^3*(65536/230945-1/19*csch(d*x+c)^18+18/323*csch(d*x+c)^16-96/1615*csch(d*x+c)^14+1344/20995*csch(d*x+c)
^12-16128/230945*csch(d*x+c)^10+3584/46189*csch(d*x+c)^8-4096/46189*csch(d*x+c)^6+24576/230945*csch(d*x+c)^4-3
2768/230945*csch(d*x+c)^2)*coth(d*x+c)+3*a^2*b*(2048/6435-1/15*csch(d*x+c)^14+14/195*csch(d*x+c)^12-56/715*csc
h(d*x+c)^10+112/1287*csch(d*x+c)^8-128/1287*csch(d*x+c)^6+256/2145*csch(d*x+c)^4-1024/6435*csch(d*x+c)^2)*coth
(d*x+c)+3*a*b^2*(256/693-1/11*csch(d*x+c)^10+10/99*csch(d*x+c)^8-80/693*csch(d*x+c)^6+32/231*csch(d*x+c)^4-128
/693*csch(d*x+c)^2)*coth(d*x+c)+b^3*(16/35-1/7*csch(d*x+c)^6+6/35*csch(d*x+c)^4-8/35*csch(d*x+c)^2)*coth(d*x+c
))
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Maxima [B] time = 1.29003, size = 6592, normalized size = 26.58 \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)^20*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")
[Out]
131072/230945*a^3*(19*e^(-2*d*x - 2*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) -
3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 7558
2*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 50388*
e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-3
2*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) - 171*e^(-4*d*x - 4*
c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-1
0*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(-16*d*x - 16*c) + 92378*e^(-18*
d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*
x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*
c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) + 969*e^(-6*d*x - 6*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(
-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 1
2*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*
c) + 75582*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c)
+ 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*
d*x - 38*c) - 1)) - 3876*e^(-8*d*x - 8*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c
) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 7
5582*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 503
88*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^
(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) + 11628*e^(-10*d*
x - 10*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 1162
8*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(-16*d*x - 16*c) + 92378*
e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^
(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 32*c) + 171*e^(-34*d*
x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) - 27132*e^(-12*d*x - 12*c)/(d*(19*e^(-2*d*x - 2*c
) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(
-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-2
0*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*
d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c
) + e^(-38*d*x - 38*c) - 1)) + 50388*e^(-14*d*x - 14*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e
^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d
*x - 14*c) - 75582*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x
- 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x -
30*c) - 969*e^(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) - 7
5582*e^(-16*d*x - 16*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x
- 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(-16*d*x -
16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24
*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 32*c) +
171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) + 92378*e^(-18*d*x - 18*c)/(d*(19*e
^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*
c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c)
- 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) -
11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(
-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) - 1/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x
- 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c
) - 75582*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c)
- 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 9
69*e^(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1))) + 4096/2145
*a^2*b*(15*e^(-2*d*x - 2*c)/(d*(15*e^(-2*d*x - 2*c) - 105*e^(-4*d*x - 4*c) + 455*e^(-6*d*x - 6*c) - 1365*e^(-8
*d*x - 8*c) + 3003*e^(-10*d*x - 10*c) - 5005*e^(-12*d*x - 12*c) + 6435*e^(-14*d*x - 14*c) - 6435*e^(-16*d*x -
16*c) + 5005*e^(-18*d*x - 18*c) - 3003*e^(-20*d*x - 20*c) + 1365*e^(-22*d*x - 22*c) - 455*e^(-24*d*x - 24*c) +
105*e^(-26*d*x - 26*c) - 15*e^(-28*d*x - 28*c) + e^(-30*d*x - 30*c) - 1)) - 105*e^(-4*d*x - 4*c)/(d*(15*e^(-2
*d*x - 2*c) - 105*e^(-4*d*x - 4*c) + 455*e^(-6*d*x - 6*c) - 1365*e^(-8*d*x - 8*c) + 3003*e^(-10*d*x - 10*c) -
5005*e^(-12*d*x - 12*c) + 6435*e^(-14*d*x - 14*c) - 6435*e^(-16*d*x - 16*c) + 5005*e^(-18*d*x - 18*c) - 3003*e
^(-20*d*x - 20*c) + 1365*e^(-22*d*x - 22*c) - 455*e^(-24*d*x - 24*c) + 105*e^(-26*d*x - 26*c) - 15*e^(-28*d*x
- 28*c) + e^(-30*d*x - 30*c) - 1)) + 455*e^(-6*d*x - 6*c)/(d*(15*e^(-2*d*x - 2*c) - 105*e^(-4*d*x - 4*c) + 455
*e^(-6*d*x - 6*c) - 1365*e^(-8*d*x - 8*c) + 3003*e^(-10*d*x - 10*c) - 5005*e^(-12*d*x - 12*c) + 6435*e^(-14*d*
x - 14*c) - 6435*e^(-16*d*x - 16*c) + 5005*e^(-18*d*x - 18*c) - 3003*e^(-20*d*x - 20*c) + 1365*e^(-22*d*x - 22
*c) - 455*e^(-24*d*x - 24*c) + 105*e^(-26*d*x - 26*c) - 15*e^(-28*d*x - 28*c) + e^(-30*d*x - 30*c) - 1)) - 136
5*e^(-8*d*x - 8*c)/(d*(15*e^(-2*d*x - 2*c) - 105*e^(-4*d*x - 4*c) + 455*e^(-6*d*x - 6*c) - 1365*e^(-8*d*x - 8*
c) + 3003*e^(-10*d*x - 10*c) - 5005*e^(-12*d*x - 12*c) + 6435*e^(-14*d*x - 14*c) - 6435*e^(-16*d*x - 16*c) + 5
005*e^(-18*d*x - 18*c) - 3003*e^(-20*d*x - 20*c) + 1365*e^(-22*d*x - 22*c) - 455*e^(-24*d*x - 24*c) + 105*e^(-
26*d*x - 26*c) - 15*e^(-28*d*x - 28*c) + e^(-30*d*x - 30*c) - 1)) + 3003*e^(-10*d*x - 10*c)/(d*(15*e^(-2*d*x -
2*c) - 105*e^(-4*d*x - 4*c) + 455*e^(-6*d*x - 6*c) - 1365*e^(-8*d*x - 8*c) + 3003*e^(-10*d*x - 10*c) - 5005*e
^(-12*d*x - 12*c) + 6435*e^(-14*d*x - 14*c) - 6435*e^(-16*d*x - 16*c) + 5005*e^(-18*d*x - 18*c) - 3003*e^(-20*
d*x - 20*c) + 1365*e^(-22*d*x - 22*c) - 455*e^(-24*d*x - 24*c) + 105*e^(-26*d*x - 26*c) - 15*e^(-28*d*x - 28*c
) + e^(-30*d*x - 30*c) - 1)) - 5005*e^(-12*d*x - 12*c)/(d*(15*e^(-2*d*x - 2*c) - 105*e^(-4*d*x - 4*c) + 455*e^
(-6*d*x - 6*c) - 1365*e^(-8*d*x - 8*c) + 3003*e^(-10*d*x - 10*c) - 5005*e^(-12*d*x - 12*c) + 6435*e^(-14*d*x -
14*c) - 6435*e^(-16*d*x - 16*c) + 5005*e^(-18*d*x - 18*c) - 3003*e^(-20*d*x - 20*c) + 1365*e^(-22*d*x - 22*c)
- 455*e^(-24*d*x - 24*c) + 105*e^(-26*d*x - 26*c) - 15*e^(-28*d*x - 28*c) + e^(-30*d*x - 30*c) - 1)) + 6435*e
^(-14*d*x - 14*c)/(d*(15*e^(-2*d*x - 2*c) - 105*e^(-4*d*x - 4*c) + 455*e^(-6*d*x - 6*c) - 1365*e^(-8*d*x - 8*c
) + 3003*e^(-10*d*x - 10*c) - 5005*e^(-12*d*x - 12*c) + 6435*e^(-14*d*x - 14*c) - 6435*e^(-16*d*x - 16*c) + 50
05*e^(-18*d*x - 18*c) - 3003*e^(-20*d*x - 20*c) + 1365*e^(-22*d*x - 22*c) - 455*e^(-24*d*x - 24*c) + 105*e^(-2
6*d*x - 26*c) - 15*e^(-28*d*x - 28*c) + e^(-30*d*x - 30*c) - 1)) - 1/(d*(15*e^(-2*d*x - 2*c) - 105*e^(-4*d*x -
4*c) + 455*e^(-6*d*x - 6*c) - 1365*e^(-8*d*x - 8*c) + 3003*e^(-10*d*x - 10*c) - 5005*e^(-12*d*x - 12*c) + 643
5*e^(-14*d*x - 14*c) - 6435*e^(-16*d*x - 16*c) + 5005*e^(-18*d*x - 18*c) - 3003*e^(-20*d*x - 20*c) + 1365*e^(-
22*d*x - 22*c) - 455*e^(-24*d*x - 24*c) + 105*e^(-26*d*x - 26*c) - 15*e^(-28*d*x - 28*c) + e^(-30*d*x - 30*c)
- 1))) + 512/231*a*b^2*(11*e^(-2*d*x - 2*c)/(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165*e^(-6*d*x - 6*
c) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*x - 10*c) - 462*e^(-12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(
-16*d*x - 16*c) + 55*e^(-18*d*x - 18*c) - 11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1)) - 55*e^(-4*d*x - 4*
c)/(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165*e^(-6*d*x - 6*c) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*
x - 10*c) - 462*e^(-12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(-16*d*x - 16*c) + 55*e^(-18*d*x - 18*c) -
11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1)) + 165*e^(-6*d*x - 6*c)/(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*
x - 4*c) + 165*e^(-6*d*x - 6*c) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*x - 10*c) - 462*e^(-12*d*x - 12*c) + 330
*e^(-14*d*x - 14*c) - 165*e^(-16*d*x - 16*c) + 55*e^(-18*d*x - 18*c) - 11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22
*c) - 1)) - 330*e^(-8*d*x - 8*c)/(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165*e^(-6*d*x - 6*c) - 330*e^
(-8*d*x - 8*c) + 462*e^(-10*d*x - 10*c) - 462*e^(-12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(-16*d*x - 1
6*c) + 55*e^(-18*d*x - 18*c) - 11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1)) + 462*e^(-10*d*x - 10*c)/(d*(1
1*e^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165*e^(-6*d*x - 6*c) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*x - 10*c
) - 462*e^(-12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(-16*d*x - 16*c) + 55*e^(-18*d*x - 18*c) - 11*e^(-
20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1)) - 1/(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165*e^(-6*d*x -
6*c) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*x - 10*c) - 462*e^(-12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e
^(-16*d*x - 16*c) + 55*e^(-18*d*x - 18*c) - 11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1))) + 32/35*b^3*(7*e
^(-2*d*x - 2*c)/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*
e^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1)) - 21*e^(-4*d*x - 4*c)/(d*(7*e^(-2*d*x - 2
*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x -
12*c) + e^(-14*d*x - 14*c) - 1)) + 35*e^(-6*d*x - 6*c)/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-
6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1)) -
1/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x -
10*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1)))
________________________________________________________________________________________
Fricas [B] time = 2.26097, size = 13925, normalized size = 56.15 \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)^20*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")
[Out]
64/4849845*((43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 2355639*b^3)*cosh(d*x + c)^15 + 15*(43008*a^3 + 144704*
a^2*b + 167960*a*b^2 - 2355639*b^3)*cosh(d*x + c)*sinh(d*x + c)^14 - 2*(21504*a^3 + 72352*a^2*b + 83980*a*b^2
+ 1247103*b^3)*sinh(d*x + c)^15 - 19*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 1538823*b^3)*cosh(d*x + c)^13
+ 2*(408576*a^3 + 1374688*a^2*b + 1595620*a*b^2 + 15935205*b^3 - 105*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 +
1247103*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^13 + 13*(35*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 2355639*b^3
)*cosh(d*x + c)^3 - 19*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 1538823*b^3)*cosh(d*x + c))*sinh(d*x + c)^12
+ 57*(129024*a^3 + 434112*a^2*b - 857480*a*b^2 - 2914769*b^3)*cosh(d*x + c)^11 - 6*(455*(21504*a^3 + 72352*a^
2*b + 83980*a*b^2 + 1247103*b^3)*cosh(d*x + c)^4 + 1225728*a^3 + 4124064*a^2*b + 17719780*a*b^2 + 31639465*b^3
- 494*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 838695*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^11 + 11*(273*(43008
*a^3 + 144704*a^2*b + 167960*a*b^2 - 2355639*b^3)*cosh(d*x + c)^5 - 494*(43008*a^3 + 144704*a^2*b + 167960*a*b
^2 - 1538823*b^3)*cosh(d*x + c)^3 + 57*(129024*a^3 + 434112*a^2*b - 857480*a*b^2 - 2914769*b^3)*cosh(d*x + c))
*sinh(d*x + c)^10 - 969*(43008*a^3 + 144704*a^2*b - 529880*a*b^2 - 586443*b^3)*cosh(d*x + c)^9 - 2*(5005*(2150
4*a^3 + 72352*a^2*b + 83980*a*b^2 + 1247103*b^3)*cosh(d*x + c)^6 - 13585*(21504*a^3 + 72352*a^2*b + 83980*a*b^
2 + 838695*b^3)*cosh(d*x + c)^4 - 20837376*a^3 - 70109088*a^2*b - 419480100*a*b^2 - 351267345*b^3 + 3135*(6451
2*a^3 + 217056*a^2*b + 932620*a*b^2 + 1665235*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 9*(715*(43008*a^3 + 1447
04*a^2*b + 167960*a*b^2 - 2355639*b^3)*cosh(d*x + c)^7 - 2717*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 15388
23*b^3)*cosh(d*x + c)^5 + 1045*(129024*a^3 + 434112*a^2*b - 857480*a*b^2 - 2914769*b^3)*cosh(d*x + c)^3 - 969*
(43008*a^3 + 144704*a^2*b - 529880*a*b^2 - 586443*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 + 6783*(24576*a^3 - 5459
2*a^2*b - 293800*a*b^2 - 189761*b^3)*cosh(d*x + c)^7 - 6*(2145*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 124710
3*b^3)*cosh(d*x + c)^8 - 10868*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 838695*b^3)*cosh(d*x + c)^6 + 6270*(64
512*a^3 + 217056*a^2*b + 932620*a*b^2 + 1665235*b^3)*cosh(d*x + c)^4 + 27783168*a^3 + 248673824*a^2*b + 549145
220*a*b^2 + 303230785*b^3 - 11628*(21504*a^3 + 72352*a^2*b + 432900*a*b^2 + 362505*b^3)*cosh(d*x + c)^2)*sinh(
d*x + c)^7 + (5005*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 2355639*b^3)*cosh(d*x + c)^9 - 32604*(43008*a^3
+ 144704*a^2*b + 167960*a*b^2 - 1538823*b^3)*cosh(d*x + c)^7 + 26334*(129024*a^3 + 434112*a^2*b - 857480*a*b^2
- 2914769*b^3)*cosh(d*x + c)^5 - 81396*(43008*a^3 + 144704*a^2*b - 529880*a*b^2 - 586443*b^3)*cosh(d*x + c)^3
+ 47481*(24576*a^3 - 54592*a^2*b - 293800*a*b^2 - 189761*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 323*(1548288*a
^3 - 8564416*a^2*b - 12926680*a*b^2 - 6120543*b^3)*cosh(d*x + c)^5 - 2*(3003*(21504*a^3 + 72352*a^2*b + 83980*
a*b^2 + 1247103*b^3)*cosh(d*x + c)^10 - 24453*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 838695*b^3)*cosh(d*x +
c)^8 + 26334*(64512*a^3 + 217056*a^2*b + 932620*a*b^2 + 1665235*b^3)*cosh(d*x + c)^6 - 122094*(21504*a^3 + 723
52*a^2*b + 432900*a*b^2 + 362505*b^3)*cosh(d*x + c)^4 - 250048512*a^3 - 3065771296*a^2*b - 4040697700*a*b^2 -
1763542209*b^3 + 20349*(86016*a^3 + 769888*a^2*b + 1700140*a*b^2 + 938795*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^
5 + (1365*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 2355639*b^3)*cosh(d*x + c)^11 - 13585*(43008*a^3 + 144704
*a^2*b + 167960*a*b^2 - 1538823*b^3)*cosh(d*x + c)^9 + 18810*(129024*a^3 + 434112*a^2*b - 857480*a*b^2 - 29147
69*b^3)*cosh(d*x + c)^7 - 122094*(43008*a^3 + 144704*a^2*b - 529880*a*b^2 - 586443*b^3)*cosh(d*x + c)^5 + 2374
05*(24576*a^3 - 54592*a^2*b - 293800*a*b^2 - 189761*b^3)*cosh(d*x + c)^3 - 1615*(1548288*a^3 - 8564416*a^2*b -
12926680*a*b^2 - 6120543*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 323*(8687616*a^3 + 15456448*a^2*b + 15194920*a
*b^2 + 6026163*b^3)*cosh(d*x + c)^3 - 2*(455*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 1247103*b^3)*cosh(d*x +
c)^12 - 5434*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 838695*b^3)*cosh(d*x + c)^10 + 9405*(64512*a^3 + 217056*
a^2*b + 932620*a*b^2 + 1665235*b^3)*cosh(d*x + c)^8 - 81396*(21504*a^3 + 72352*a^2*b + 432900*a*b^2 + 362505*b
^3)*cosh(d*x + c)^6 + 33915*(86016*a^3 + 769888*a^2*b + 1700140*a*b^2 + 938795*b^3)*cosh(d*x + c)^4 + 25699430
40*a^3 + 6422325280*a^2*b + 6933472780*a*b^2 + 2675035935*b^3 - 3230*(774144*a^3 + 9491552*a^2*b + 12509900*a*
b^2 + 5459883*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (105*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 2355639*
b^3)*cosh(d*x + c)^13 - 1482*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 1538823*b^3)*cosh(d*x + c)^11 + 3135*(
129024*a^3 + 434112*a^2*b - 857480*a*b^2 - 2914769*b^3)*cosh(d*x + c)^9 - 34884*(43008*a^3 + 144704*a^2*b - 52
9880*a*b^2 - 586443*b^3)*cosh(d*x + c)^7 + 142443*(24576*a^3 - 54592*a^2*b - 293800*a*b^2 - 189761*b^3)*cosh(d
*x + c)^5 - 3230*(1548288*a^3 - 8564416*a^2*b - 12926680*a*b^2 - 6120543*b^3)*cosh(d*x + c)^3 - 969*(8687616*a
^3 + 15456448*a^2*b + 15194920*a*b^2 + 6026163*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 12597*(86016*a^3 + 215488
*a^2*b + 179720*a*b^2 + 65703*b^3)*cosh(d*x + c) - 2*(15*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 1247103*b^3)
*cosh(d*x + c)^14 - 247*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 838695*b^3)*cosh(d*x + c)^12 + 627*(64512*a^3
+ 217056*a^2*b + 932620*a*b^2 + 1665235*b^3)*cosh(d*x + c)^10 - 8721*(21504*a^3 + 72352*a^2*b + 432900*a*b^2
+ 362505*b^3)*cosh(d*x + c)^8 + 6783*(86016*a^3 + 769888*a^2*b + 1700140*a*b^2 + 938795*b^3)*cosh(d*x + c)^6 -
1615*(774144*a^3 + 9491552*a^2*b + 12509900*a*b^2 + 5459883*b^3)*cosh(d*x + c)^4 - 2708858880*a^3 - 864859632
0*a^2*b - 8918927940*a*b^2 - 3269488365*b^3 + 4845*(1591296*a^3 + 3976672*a^2*b + 4293172*a*b^2 + 1656369*b^3)
*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)^23 + 23*d*cosh(d*x + c)*sinh(d*x + c)^22 + d*sinh(d*x + c)^2
3 - 19*d*cosh(d*x + c)^21 + (253*d*cosh(d*x + c)^2 - 19*d)*sinh(d*x + c)^21 + 7*(253*d*cosh(d*x + c)^3 - 57*d*
cosh(d*x + c))*sinh(d*x + c)^20 + 171*d*cosh(d*x + c)^19 + (8855*d*cosh(d*x + c)^4 - 3990*d*cosh(d*x + c)^2 +
171*d)*sinh(d*x + c)^19 + 19*(1771*d*cosh(d*x + c)^5 - 1330*d*cosh(d*x + c)^3 + 171*d*cosh(d*x + c))*sinh(d*x
+ c)^18 - 969*d*cosh(d*x + c)^17 + 57*(1771*d*cosh(d*x + c)^6 - 1995*d*cosh(d*x + c)^4 + 513*d*cosh(d*x + c)^2
- 17*d)*sinh(d*x + c)^17 + 969*(253*d*cosh(d*x + c)^7 - 399*d*cosh(d*x + c)^5 + 171*d*cosh(d*x + c)^3 - 17*d*
cosh(d*x + c))*sinh(d*x + c)^16 + 3875*d*cosh(d*x + c)^15 + (490314*d*cosh(d*x + c)^8 - 1031016*d*cosh(d*x + c
)^6 + 662796*d*cosh(d*x + c)^4 - 131784*d*cosh(d*x + c)^2 + 3877*d)*sinh(d*x + c)^15 + (817190*d*cosh(d*x + c)
^9 - 2209320*d*cosh(d*x + c)^7 + 1988388*d*cosh(d*x + c)^5 - 658920*d*cosh(d*x + c)^3 + 58125*d*cosh(d*x + c))
*sinh(d*x + c)^14 - 11609*d*cosh(d*x + c)^13 + (1144066*d*cosh(d*x + c)^10 - 3866310*d*cosh(d*x + c)^8 + 46395
72*d*cosh(d*x + c)^6 - 2306220*d*cosh(d*x + c)^4 + 407085*d*cosh(d*x + c)^2 - 11647*d)*sinh(d*x + c)^13 + 13*(
104006*d*cosh(d*x + c)^11 - 429590*d*cosh(d*x + c)^9 + 662796*d*cosh(d*x + c)^7 - 461244*d*cosh(d*x + c)^5 + 1
35625*d*cosh(d*x + c)^3 - 11609*d*cosh(d*x + c))*sinh(d*x + c)^12 + 26961*d*cosh(d*x + c)^11 + (1352078*d*cosh
(d*x + c)^12 - 6701604*d*cosh(d*x + c)^10 + 12924522*d*cosh(d*x + c)^8 - 11992344*d*cosh(d*x + c)^6 + 5292105*
d*cosh(d*x + c)^4 - 908466*d*cosh(d*x + c)^2 + 27303*d)*sinh(d*x + c)^11 + (1144066*d*cosh(d*x + c)^13 - 67016
04*d*cosh(d*x + c)^11 + 15796638*d*cosh(d*x + c)^9 - 18845112*d*cosh(d*x + c)^7 + 11636625*d*cosh(d*x + c)^5 -
3320174*d*cosh(d*x + c)^3 + 296571*d*cosh(d*x + c))*sinh(d*x + c)^10 - 49419*d*cosh(d*x + c)^9 + (817190*d*co
sh(d*x + c)^14 - 5584670*d*cosh(d*x + c)^12 + 15796638*d*cosh(d*x + c)^10 - 23556390*d*cosh(d*x + c)^8 + 19404
385*d*cosh(d*x + c)^6 - 8327605*d*cosh(d*x + c)^4 + 1501665*d*cosh(d*x + c)^2 - 51357*d)*sinh(d*x + c)^9 + 3*(
163438*d*cosh(d*x + c)^15 - 1288770*d*cosh(d*x + c)^13 + 4308174*d*cosh(d*x + c)^11 - 7852130*d*cosh(d*x + c)^
9 + 8311875*d*cosh(d*x + c)^7 - 4980261*d*cosh(d*x + c)^5 + 1482855*d*cosh(d*x + c)^3 - 148257*d*cosh(d*x + c)
)*sinh(d*x + c)^8 + 71706*d*cosh(d*x + c)^7 + 3*(81719*d*cosh(d*x + c)^16 - 736440*d*cosh(d*x + c)^14 + 287211
6*d*cosh(d*x + c)^12 - 6281704*d*cosh(d*x + c)^10 + 8316165*d*cosh(d*x + c)^8 - 6662084*d*cosh(d*x + c)^6 + 30
03330*d*cosh(d*x + c)^4 - 616284*d*cosh(d*x + c)^2 + 26486*d)*sinh(d*x + c)^7 + (100947*d*cosh(d*x + c)^17 - 1
031016*d*cosh(d*x + c)^15 + 4639572*d*cosh(d*x + c)^13 - 11992344*d*cosh(d*x + c)^11 + 19394375*d*cosh(d*x + c
)^9 - 19921044*d*cosh(d*x + c)^7 + 12455982*d*cosh(d*x + c)^5 - 4151196*d*cosh(d*x + c)^3 + 501942*d*cosh(d*x
+ c))*sinh(d*x + c)^6 - 80750*d*cosh(d*x + c)^5 + (33649*d*cosh(d*x + c)^18 - 386631*d*cosh(d*x + c)^16 + 1988
388*d*cosh(d*x + c)^14 - 5996172*d*cosh(d*x + c)^12 + 11642631*d*cosh(d*x + c)^10 - 14989689*d*cosh(d*x + c)^8
+ 12613986*d*cosh(d*x + c)^6 - 6470982*d*cosh(d*x + c)^4 + 1668618*d*cosh(d*x + c)^2 - 104006*d)*sinh(d*x + c
)^5 + (8855*d*cosh(d*x + c)^19 - 113715*d*cosh(d*x + c)^17 + 662796*d*cosh(d*x + c)^15 - 2306220*d*cosh(d*x +
c)^13 + 5289375*d*cosh(d*x + c)^11 - 8300435*d*cosh(d*x + c)^9 + 8897130*d*cosh(d*x + c)^7 - 6226794*d*cosh(d*
x + c)^5 + 2509710*d*cosh(d*x + c)^3 - 403750*d*cosh(d*x + c))*sinh(d*x + c)^4 + 65246*d*cosh(d*x + c)^3 + (17
71*d*cosh(d*x + c)^20 - 25270*d*cosh(d*x + c)^18 + 165699*d*cosh(d*x + c)^16 - 658920*d*cosh(d*x + c)^14 + 176
4035*d*cosh(d*x + c)^12 - 3331042*d*cosh(d*x + c)^10 + 4504995*d*cosh(d*x + c)^8 - 4313988*d*cosh(d*x + c)^6 +
2781030*d*cosh(d*x + c)^4 - 1040060*d*cosh(d*x + c)^2 + 119510*d)*sinh(d*x + c)^3 + (253*d*cosh(d*x + c)^21 -
3990*d*cosh(d*x + c)^19 + 29241*d*cosh(d*x + c)^17 - 131784*d*cosh(d*x + c)^15 + 406875*d*cosh(d*x + c)^13 -
905502*d*cosh(d*x + c)^11 + 1482855*d*cosh(d*x + c)^9 - 1779084*d*cosh(d*x + c)^7 + 1505826*d*cosh(d*x + c)^5
- 807500*d*cosh(d*x + c)^3 + 195738*d*cosh(d*x + c))*sinh(d*x + c)^2 - 25194*d*cosh(d*x + c) + (23*d*cosh(d*x
+ c)^22 - 399*d*cosh(d*x + c)^20 + 3249*d*cosh(d*x + c)^18 - 16473*d*cosh(d*x + c)^16 + 58155*d*cosh(d*x + c)^
14 - 151411*d*cosh(d*x + c)^12 + 300333*d*cosh(d*x + c)^10 - 462213*d*cosh(d*x + c)^8 + 556206*d*cosh(d*x + c)
^6 - 520030*d*cosh(d*x + c)^4 + 358530*d*cosh(d*x + c)^2 - 125970*d)*sinh(d*x + c))
________________________________________________________________________________________
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)**20*(a+b*sinh(d*x+c)**4)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.66426, size = 995, normalized size = 4.01 \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)^20*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")
[Out]
-32/4849845*(4849845*b^3*e^(30*d*x + 30*c) - 61108047*b^3*e^(28*d*x + 28*c) + 155195040*a*b^2*e^(26*d*x + 26*c
) + 355978623*b^3*e^(26*d*x + 26*c) - 1352413920*a*b^2*e^(24*d*x + 24*c) - 1270797957*b^3*e^(24*d*x + 24*c) +
1862340480*a^2*b*e^(22*d*x + 22*c) + 5287716720*a*b^2*e^(22*d*x + 22*c) + 3106533573*b^3*e^(22*d*x + 22*c) - 8
897848960*a^2*b*e^(20*d*x + 20*c) - 12256713040*a*b^2*e^(20*d*x + 20*c) - 5504019807*b^3*e^(20*d*x + 20*c) + 7
945986048*a^3*e^(18*d*x + 18*c) + 17837083264*a^2*b*e^(18*d*x + 18*c) + 18774904720*a*b^2*e^(18*d*x + 18*c) +
7296522519*b^3*e^(18*d*x + 18*c) - 6501261312*a^3*e^(16*d*x + 16*c) - 20011694976*a^2*b*e^(16*d*x + 16*c) - 20
101788720*a*b^2*e^(16*d*x + 16*c) - 7366637421*b^3*e^(16*d*x + 16*c) + 4334174208*a^3*e^(14*d*x + 14*c) + 1458
2690304*a^2*b*e^(14*d*x + 14*c) + 15573923040*a*b^2*e^(14*d*x + 14*c) + 5711316039*b^3*e^(14*d*x + 14*c) - 233
3786112*a^3*e^(12*d*x + 12*c) - 7852217856*a^2*b*e^(12*d*x + 12*c) - 8958986400*a*b^2*e^(12*d*x + 12*c) - 3403
621221*b^3*e^(12*d*x + 12*c) + 1000194048*a^3*e^(10*d*x + 10*c) + 3365236224*a^2*b*e^(10*d*x + 10*c) + 3906077
760*a*b^2*e^(10*d*x + 10*c) + 1550149029*b^3*e^(10*d*x + 10*c) - 333398016*a^3*e^(8*d*x + 8*c) - 1121745408*a^
2*b*e^(8*d*x + 8*c) - 1302025920*a*b^2*e^(8*d*x + 8*c) - 532235847*b^3*e^(8*d*x + 8*c) + 83349504*a^3*e^(6*d*x
+ 6*c) + 280436352*a^2*b*e^(6*d*x + 6*c) + 325506480*a*b^2*e^(6*d*x + 6*c) + 134271423*b^3*e^(6*d*x + 6*c) -
14708736*a^3*e^(4*d*x + 4*c) - 49488768*a^2*b*e^(4*d*x + 4*c) - 57442320*a*b^2*e^(4*d*x + 4*c) - 23694957*b^3*
e^(4*d*x + 4*c) + 1634304*a^3*e^(2*d*x + 2*c) + 5498752*a^2*b*e^(2*d*x + 2*c) + 6382480*a*b^2*e^(2*d*x + 2*c)
+ 2632773*b^3*e^(2*d*x + 2*c) - 86016*a^3 - 289408*a^2*b - 335920*a*b^2 - 138567*b^3)/(d*(e^(2*d*x + 2*c) - 1)
^19)