3.2.69 \(\int \text {csch}^6(c+d x) (a+b \sinh ^3(c+d x))^3 \, dx\) [169]
Optimal. Leaf size=131 \[ 3 a b^2 x+\frac {3 a^2 b \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {b^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}-\frac {a^3 \coth (c+d x)}{d}+\frac {2 a^3 \coth ^3(c+d x)}{3 d}-\frac {a^3 \coth ^5(c+d x)}{5 d}-\frac {3 a^2 b \coth (c+d x) \text {csch}(c+d x)}{2 d} \]
[Out]
3*a*b^2*x+3/2*a^2*b*arctanh(cosh(d*x+c))/d-b^3*cosh(d*x+c)/d+1/3*b^3*cosh(d*x+c)^3/d-a^3*coth(d*x+c)/d+2/3*a^3
*coth(d*x+c)^3/d-1/5*a^3*coth(d*x+c)^5/d-3/2*a^2*b*coth(d*x+c)*csch(d*x+c)/d
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3299, 3853,
3855, 3852, 2713} \begin {gather*} -\frac {a^3 \coth ^5(c+d x)}{5 d}+\frac {2 a^3 \coth ^3(c+d x)}{3 d}-\frac {a^3 \coth (c+d x)}{d}+\frac {3 a^2 b \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {3 a^2 b \coth (c+d x) \text {csch}(c+d x)}{2 d}+3 a b^2 x+\frac {b^3 \cosh ^3(c+d x)}{3 d}-\frac {b^3 \cosh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^3)^3,x]
[Out]
3*a*b^2*x + (3*a^2*b*ArcTanh[Cosh[c + d*x]])/(2*d) - (b^3*Cosh[c + d*x])/d + (b^3*Cosh[c + d*x]^3)/(3*d) - (a^
3*Coth[c + d*x])/d + (2*a^3*Coth[c + d*x]^3)/(3*d) - (a^3*Coth[c + d*x]^5)/(5*d) - (3*a^2*b*Coth[c + d*x]*Csch
[c + d*x])/(2*d)
Rule 2713
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Rule 3299
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
Rule 3852
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Rule 3853
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]
Rule 3855
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin {align*} \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx &=-\int \left (-3 a b^2-3 a^2 b \text {csch}^3(c+d x)-a^3 \text {csch}^6(c+d x)-b^3 \sinh ^3(c+d x)\right ) \, dx\\ &=3 a b^2 x+a^3 \int \text {csch}^6(c+d x) \, dx+\left (3 a^2 b\right ) \int \text {csch}^3(c+d x) \, dx+b^3 \int \sinh ^3(c+d x) \, dx\\ &=3 a b^2 x-\frac {3 a^2 b \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {1}{2} \left (3 a^2 b\right ) \int \text {csch}(c+d x) \, dx-\frac {\left (i a^3\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \coth (c+d x)\right )}{d}-\frac {b^3 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=3 a b^2 x+\frac {3 a^2 b \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {b^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}-\frac {a^3 \coth (c+d x)}{d}+\frac {2 a^3 \coth ^3(c+d x)}{3 d}-\frac {a^3 \coth ^5(c+d x)}{5 d}-\frac {3 a^2 b \coth (c+d x) \text {csch}(c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.27, size = 225, normalized size = 1.72 \begin {gather*} \frac {-360 b^3 \cosh (c+d x)+40 b^3 \cosh (3 (c+d x))+\frac {1}{2} a \left (-256 a^2 \coth \left (\frac {1}{2} (c+d x)\right )-360 a b \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+19 a^2 \text {csch}^4\left (\frac {1}{2} (c+d x)\right ) \sinh (c+d x)-3 a^2 \text {csch}^6\left (\frac {1}{2} (c+d x)\right ) \sinh (c+d x)+8 \left (180 b \left (2 b (c+d x)-a \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )-45 a b \text {sech}^2\left (\frac {1}{2} (c+d x)\right )-38 a^2 \text {csch}^3(c+d x) \sinh ^4\left (\frac {1}{2} (c+d x)\right )-24 a^2 \text {csch}^5(c+d x) \sinh ^6\left (\frac {1}{2} (c+d x)\right )-32 a^2 \tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{480 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^3)^3,x]
[Out]
(-360*b^3*Cosh[c + d*x] + 40*b^3*Cosh[3*(c + d*x)] + (a*(-256*a^2*Coth[(c + d*x)/2] - 360*a*b*Csch[(c + d*x)/2
]^2 + 19*a^2*Csch[(c + d*x)/2]^4*Sinh[c + d*x] - 3*a^2*Csch[(c + d*x)/2]^6*Sinh[c + d*x] + 8*(180*b*(2*b*(c +
d*x) - a*Log[Tanh[(c + d*x)/2]]) - 45*a*b*Sech[(c + d*x)/2]^2 - 38*a^2*Csch[c + d*x]^3*Sinh[(c + d*x)/2]^4 - 2
4*a^2*Csch[c + d*x]^5*Sinh[(c + d*x)/2]^6 - 32*a^2*Tanh[(c + d*x)/2])))/2)/(480*d)
________________________________________________________________________________________
Maple [A]
time = 2.08, size = 204, normalized size = 1.56
| | |
| method |
result |
size |
| | |
| risch |
\(3 a \,b^{2} x +\frac {b^{3} {\mathrm e}^{3 d x +3 c}}{24 d}-\frac {3 b^{3} {\mathrm e}^{d x +c}}{8 d}-\frac {3 b^{3} {\mathrm e}^{-d x -c}}{8 d}+\frac {b^{3} {\mathrm e}^{-3 d x -3 c}}{24 d}-\frac {a^{2} \left (45 b \,{\mathrm e}^{9 d x +9 c}-90 b \,{\mathrm e}^{7 d x +7 c}+160 a \,{\mathrm e}^{4 d x +4 c}+90 b \,{\mathrm e}^{3 d x +3 c}-80 a \,{\mathrm e}^{2 d x +2 c}-45 b \,{\mathrm e}^{d x +c}+16 a \right )}{15 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{5}}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}\) |
\(204\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^6*(a+b*sinh(d*x+c)^3)^3,x,method=_RETURNVERBOSE)
[Out]
3*a*b^2*x+1/24*b^3/d*exp(3*d*x+3*c)-3/8*b^3/d*exp(d*x+c)-3/8*b^3/d*exp(-d*x-c)+1/24*b^3/d*exp(-3*d*x-3*c)-1/15
*a^2*(45*b*exp(9*d*x+9*c)-90*b*exp(7*d*x+7*c)+160*a*exp(4*d*x+4*c)+90*b*exp(3*d*x+3*c)-80*a*exp(2*d*x+2*c)-45*
b*exp(d*x+c)+16*a)/d/(exp(2*d*x+2*c)-1)^5-3/2*a^2*b/d*ln(exp(d*x+c)-1)+3/2*a^2*b/d*ln(exp(d*x+c)+1)
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 365 vs.
\(2 (121) = 242\).
time = 0.28, size = 365, normalized size = 2.79 \begin {gather*} 3 \, a b^{2} x + \frac {1}{24} \, b^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - \frac {16}{15} \, a^{3} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^6*(a+b*sinh(d*x+c)^3)^3,x, algorithm="maxima")
[Out]
3*a*b^2*x + 1/24*b^3*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d) + 3/2*a^2*b
*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*
c) - e^(-4*d*x - 4*c) - 1))) - 16/15*a^3*(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)))
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4629 vs.
\(2 (121) = 242\).
time = 0.48, size = 4629, normalized size = 35.34 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^6*(a+b*sinh(d*x+c)^3)^3,x, algorithm="fricas")
[Out]
1/120*(5*b^3*cosh(d*x + c)^16 + 80*b^3*cosh(d*x + c)*sinh(d*x + c)^15 + 5*b^3*sinh(d*x + c)^16 + 360*a*b^2*d*x
*cosh(d*x + c)^13 - 70*b^3*cosh(d*x + c)^14 - 1800*a*b^2*d*x*cosh(d*x + c)^11 + 10*(60*b^3*cosh(d*x + c)^2 - 7
*b^3)*sinh(d*x + c)^14 + 3600*a*b^2*d*x*cosh(d*x + c)^9 + 20*(140*b^3*cosh(d*x + c)^3 + 18*a*b^2*d*x - 49*b^3*
cosh(d*x + c))*sinh(d*x + c)^13 - 10*(36*a^2*b - 23*b^3)*cosh(d*x + c)^12 + 10*(910*b^3*cosh(d*x + c)^4 + 468*
a*b^2*d*x*cosh(d*x + c) - 637*b^3*cosh(d*x + c)^2 - 36*a^2*b + 23*b^3)*sinh(d*x + c)^12 + 40*(546*b^3*cosh(d*x
+ c)^5 + 702*a*b^2*d*x*cosh(d*x + c)^2 - 637*b^3*cosh(d*x + c)^3 - 45*a*b^2*d*x - 3*(36*a^2*b - 23*b^3)*cosh(
d*x + c))*sinh(d*x + c)^11 + 90*(8*a^2*b - 3*b^3)*cosh(d*x + c)^10 + 10*(4004*b^3*cosh(d*x + c)^6 + 10296*a*b^
2*d*x*cosh(d*x + c)^3 - 7007*b^3*cosh(d*x + c)^4 - 1980*a*b^2*d*x*cosh(d*x + c) + 72*a^2*b - 27*b^3 - 66*(36*a
^2*b - 23*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 20*(2860*b^3*cosh(d*x + c)^7 + 12870*a*b^2*d*x*cosh(d*x + c
)^4 - 7007*b^3*cosh(d*x + c)^5 - 4950*a*b^2*d*x*cosh(d*x + c)^2 + 180*a*b^2*d*x - 110*(36*a^2*b - 23*b^3)*cosh
(d*x + c)^3 + 45*(8*a^2*b - 3*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 30*(2145*b^3*cosh(d*x + c)^8 + 15444*a*b^2
*d*x*cosh(d*x + c)^5 - 7007*b^3*cosh(d*x + c)^6 - 9900*a*b^2*d*x*cosh(d*x + c)^3 + 1080*a*b^2*d*x*cosh(d*x + c
) - 165*(36*a^2*b - 23*b^3)*cosh(d*x + c)^4 + 135*(8*a^2*b - 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 - 80*(45*
a*b^2*d*x + 16*a^3)*cosh(d*x + c)^7 + 80*(715*b^3*cosh(d*x + c)^9 + 7722*a*b^2*d*x*cosh(d*x + c)^6 - 3003*b^3*
cosh(d*x + c)^7 - 7425*a*b^2*d*x*cosh(d*x + c)^4 + 1620*a*b^2*d*x*cosh(d*x + c)^2 - 99*(36*a^2*b - 23*b^3)*cos
h(d*x + c)^5 - 45*a*b^2*d*x + 135*(8*a^2*b - 3*b^3)*cosh(d*x + c)^3 - 16*a^3)*sinh(d*x + c)^7 - 90*(8*a^2*b -
3*b^3)*cosh(d*x + c)^6 + 10*(4004*b^3*cosh(d*x + c)^10 + 61776*a*b^2*d*x*cosh(d*x + c)^7 - 21021*b^3*cosh(d*x
+ c)^8 - 83160*a*b^2*d*x*cosh(d*x + c)^5 + 30240*a*b^2*d*x*cosh(d*x + c)^3 - 924*(36*a^2*b - 23*b^3)*cosh(d*x
+ c)^6 + 1890*(8*a^2*b - 3*b^3)*cosh(d*x + c)^4 - 72*a^2*b + 27*b^3 - 56*(45*a*b^2*d*x + 16*a^3)*cosh(d*x + c)
)*sinh(d*x + c)^6 + 40*(45*a*b^2*d*x + 16*a^3)*cosh(d*x + c)^5 + 20*(1092*b^3*cosh(d*x + c)^11 + 23166*a*b^2*d
*x*cosh(d*x + c)^8 - 7007*b^3*cosh(d*x + c)^9 - 41580*a*b^2*d*x*cosh(d*x + c)^6 + 22680*a*b^2*d*x*cosh(d*x + c
)^4 - 396*(36*a^2*b - 23*b^3)*cosh(d*x + c)^7 + 1134*(8*a^2*b - 3*b^3)*cosh(d*x + c)^5 + 90*a*b^2*d*x + 32*a^3
- 84*(45*a*b^2*d*x + 16*a^3)*cosh(d*x + c)^2 - 27*(8*a^2*b - 3*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 70*b^3*c
osh(d*x + c)^2 + 10*(36*a^2*b - 23*b^3)*cosh(d*x + c)^4 + 10*(910*b^3*cosh(d*x + c)^12 + 25740*a*b^2*d*x*cosh(
d*x + c)^9 - 7007*b^3*cosh(d*x + c)^10 - 59400*a*b^2*d*x*cosh(d*x + c)^7 + 45360*a*b^2*d*x*cosh(d*x + c)^5 - 4
95*(36*a^2*b - 23*b^3)*cosh(d*x + c)^8 + 1890*(8*a^2*b - 3*b^3)*cosh(d*x + c)^6 - 280*(45*a*b^2*d*x + 16*a^3)*
cosh(d*x + c)^3 + 36*a^2*b - 23*b^3 - 135*(8*a^2*b - 3*b^3)*cosh(d*x + c)^2 + 20*(45*a*b^2*d*x + 16*a^3)*cosh(
d*x + c))*sinh(d*x + c)^4 - 8*(45*a*b^2*d*x + 16*a^3)*cosh(d*x + c)^3 + 8*(350*b^3*cosh(d*x + c)^13 + 12870*a*
b^2*d*x*cosh(d*x + c)^10 - 3185*b^3*cosh(d*x + c)^11 - 37125*a*b^2*d*x*cosh(d*x + c)^8 + 37800*a*b^2*d*x*cosh(
d*x + c)^6 - 275*(36*a^2*b - 23*b^3)*cosh(d*x + c)^9 + 1350*(8*a^2*b - 3*b^3)*cosh(d*x + c)^7 - 45*a*b^2*d*x -
350*(45*a*b^2*d*x + 16*a^3)*cosh(d*x + c)^4 - 225*(8*a^2*b - 3*b^3)*cosh(d*x + c)^3 - 16*a^3 + 50*(45*a*b^2*d
*x + 16*a^3)*cosh(d*x + c)^2 + 5*(36*a^2*b - 23*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 5*b^3 + 2*(300*b^3*cosh(
d*x + c)^14 + 14040*a*b^2*d*x*cosh(d*x + c)^11 - 3185*b^3*cosh(d*x + c)^12 - 49500*a*b^2*d*x*cosh(d*x + c)^9 +
64800*a*b^2*d*x*cosh(d*x + c)^7 - 330*(36*a^2*b - 23*b^3)*cosh(d*x + c)^10 + 2025*(8*a^2*b - 3*b^3)*cosh(d*x
+ c)^8 - 840*(45*a*b^2*d*x + 16*a^3)*cosh(d*x + c)^5 - 675*(8*a^2*b - 3*b^3)*cosh(d*x + c)^4 + 200*(45*a*b^2*d
*x + 16*a^3)*cosh(d*x + c)^3 + 35*b^3 + 30*(36*a^2*b - 23*b^3)*cosh(d*x + c)^2 - 12*(45*a*b^2*d*x + 16*a^3)*co
sh(d*x + c))*sinh(d*x + c)^2 + 180*(a^2*b*cosh(d*x + c)^13 + 13*a^2*b*cosh(d*x + c)*sinh(d*x + c)^12 + a^2*b*s
inh(d*x + c)^13 - 5*a^2*b*cosh(d*x + c)^11 + 10*a^2*b*cosh(d*x + c)^9 + (78*a^2*b*cosh(d*x + c)^2 - 5*a^2*b)*s
inh(d*x + c)^11 + 11*(26*a^2*b*cosh(d*x + c)^3 - 5*a^2*b*cosh(d*x + c))*sinh(d*x + c)^10 - 10*a^2*b*cosh(d*x +
c)^7 + 5*(143*a^2*b*cosh(d*x + c)^4 - 55*a^2*b*cosh(d*x + c)^2 + 2*a^2*b)*sinh(d*x + c)^9 + 3*(429*a^2*b*cosh
(d*x + c)^5 - 275*a^2*b*cosh(d*x + c)^3 + 30*a^2*b*cosh(d*x + c))*sinh(d*x + c)^8 + 5*a^2*b*cosh(d*x + c)^5 +
2*(858*a^2*b*cosh(d*x + c)^6 - 825*a^2*b*cosh(d*x + c)^4 + 180*a^2*b*cosh(d*x + c)^2 - 5*a^2*b)*sinh(d*x + c)^
7 + 2*(858*a^2*b*cosh(d*x + c)^7 - 1155*a^2*b*cosh(d*x + c)^5 + 420*a^2*b*cosh(d*x + c)^3 - 35*a^2*b*cosh(d*x
+ c))*sinh(d*x + c)^6 - a^2*b*cosh(d*x + c)^3 + (1287*a^2*b*cosh(d*x + c)^8 - 2310*a^2*b*cosh(d*x + c)^6 + 126
0*a^2*b*cosh(d*x + c)^4 - 210*a^2*b*cosh(d*x + c)^2 + 5*a^2*b)*sinh(d*x + c)^5 + 5*(143*a^2*b*cosh(d*x + c)^9
- 330*a^2*b*cosh(d*x + c)^7 + 252*a^2*b*cosh(d*...
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)**6*(a+b*sinh(d*x+c)**3)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 270 vs.
\(2 (121) = 242\).
time = 0.51, size = 270, normalized size = 2.06 \begin {gather*} \frac {360 \, {\left (d x + c\right )} a b^{2} + 5 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 45 \, b^{3} e^{\left (d x + c\right )} + 180 \, a^{2} b \log \left (e^{\left (d x + c\right )} + 1\right ) - 180 \, a^{2} b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {{\left (475 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 1280 \, a^{3} e^{\left (7 \, d x + 7 \, c\right )} - 640 \, a^{3} e^{\left (5 \, d x + 5 \, c\right )} + 128 \, a^{3} e^{\left (3 \, d x + 3 \, c\right )} - 70 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{3} + 45 \, {\left (8 \, a^{2} b + b^{3}\right )} e^{\left (12 \, d x + 12 \, c\right )} - 10 \, {\left (72 \, a^{2} b + 23 \, b^{3}\right )} e^{\left (10 \, d x + 10 \, c\right )} + 20 \, {\left (36 \, a^{2} b - 25 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )} - 5 \, {\left (72 \, a^{2} b - 55 \, b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )}^{5} {\left (e^{\left (d x + c\right )} - 1\right )}^{5}}}{120 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^6*(a+b*sinh(d*x+c)^3)^3,x, algorithm="giac")
[Out]
1/120*(360*(d*x + c)*a*b^2 + 5*b^3*e^(3*d*x + 3*c) - 45*b^3*e^(d*x + c) + 180*a^2*b*log(e^(d*x + c) + 1) - 180
*a^2*b*log(abs(e^(d*x + c) - 1)) - (475*b^3*e^(8*d*x + 8*c) + 1280*a^3*e^(7*d*x + 7*c) - 640*a^3*e^(5*d*x + 5*
c) + 128*a^3*e^(3*d*x + 3*c) - 70*b^3*e^(2*d*x + 2*c) + 5*b^3 + 45*(8*a^2*b + b^3)*e^(12*d*x + 12*c) - 10*(72*
a^2*b + 23*b^3)*e^(10*d*x + 10*c) + 20*(36*a^2*b - 25*b^3)*e^(6*d*x + 6*c) - 5*(72*a^2*b - 55*b^3)*e^(4*d*x +
4*c))*e^(-3*d*x - 3*c)/((e^(d*x + c) + 1)^5*(e^(d*x + c) - 1)^5))/d
________________________________________________________________________________________
Mupad [B]
time = 0.81, size = 432, normalized size = 3.30 \begin {gather*} \frac {b^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}-\frac {3\,b^3\,{\mathrm {e}}^{c+d\,x}}{8\,d}-\frac {3\,b^3\,{\mathrm {e}}^{-c-d\,x}}{8\,d}-\frac {\frac {32\,a^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,d}+\frac {36\,a^2\,b\,{\mathrm {e}}^{3\,c+3\,d\,x}}{5\,d}-\frac {36\,a^2\,b\,{\mathrm {e}}^{5\,c+5\,d\,x}}{5\,d}+\frac {12\,a^2\,b\,{\mathrm {e}}^{7\,c+7\,d\,x}}{5\,d}-\frac {12\,a^2\,b\,{\mathrm {e}}^{c+d\,x}}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1}+\frac {b^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {64\,a^3}{15\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {16\,a^3}{5\,d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {3\,\mathrm {atan}\left (\frac {a^2\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^4\,b^2}}\right )\,\sqrt {a^4\,b^2}}{\sqrt {-d^2}}+3\,a\,b^2\,x-\frac {3\,a^2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {18\,a^2\,b\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*sinh(c + d*x)^3)^3/sinh(c + d*x)^6,x)
[Out]
(b^3*exp(- 3*c - 3*d*x))/(24*d) - (3*b^3*exp(c + d*x))/(8*d) - (3*b^3*exp(- c - d*x))/(8*d) - ((32*a^3*exp(4*c
+ 4*d*x))/(5*d) + (36*a^2*b*exp(3*c + 3*d*x))/(5*d) - (36*a^2*b*exp(5*c + 5*d*x))/(5*d) + (12*a^2*b*exp(7*c +
7*d*x))/(5*d) - (12*a^2*b*exp(c + d*x))/(5*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) + (b^3*exp(3*c + 3*d*x))/(24*d) - (64*a^3)/(15*d*(3*exp(2*c +
2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (16*a^3)/(5*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)) + (3*atan((a^2*b*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d*(a^4*b^2)^(1/
2)))*(a^4*b^2)^(1/2))/(-d^2)^(1/2) + 3*a*b^2*x - (3*a^2*b*exp(c + d*x))/(d*(exp(2*c + 2*d*x) - 1)) - (18*a^2*b
*exp(c + d*x))/(5*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))
________________________________________________________________________________________