3.168 \(\int \text {csch}^5(c+d x) (a+b \sinh ^3(c+d x))^3 \, dx\)
Optimal. Leaf size=148 \[ -\frac {3 a^3 \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac {a^3 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}+\frac {3 a^3 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {3 a^2 b \coth (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {b^3 \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac {3 b^3 \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {3 b^3 x}{8} \]
[Out]
3/8*b^3*x-3/8*a^3*arctanh(cosh(d*x+c))/d+3*a*b^2*cosh(d*x+c)/d-3*a^2*b*coth(d*x+c)/d+3/8*a^3*coth(d*x+c)*csch(
d*x+c)/d-1/4*a^3*coth(d*x+c)*csch(d*x+c)^3/d-3/8*b^3*cosh(d*x+c)*sinh(d*x+c)/d+1/4*b^3*cosh(d*x+c)*sinh(d*x+c)
^3/d
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 148, normalized size of antiderivative = 1.00,
number of steps used = 11, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used
= {3220, 3767, 8, 3768, 3770, 2638, 2635} \[ -\frac {3 a^2 b \coth (c+d x)}{d}-\frac {3 a^3 \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac {a^3 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}+\frac {3 a^3 \coth (c+d x) \text {csch}(c+d x)}{8 d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {b^3 \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac {3 b^3 \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {3 b^3 x}{8} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^3)^3,x]
[Out]
(3*b^3*x)/8 - (3*a^3*ArcTanh[Cosh[c + d*x]])/(8*d) + (3*a*b^2*Cosh[c + d*x])/d - (3*a^2*b*Coth[c + d*x])/d + (
3*a^3*Coth[c + d*x]*Csch[c + d*x])/(8*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^3)/(4*d) - (3*b^3*Cosh[c + d*x]*Si
nh[c + d*x])/(8*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rule 2635
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]
Rule 2638
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rule 3220
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 3767
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 3768
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 3770
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}^5(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx &=i \int \left (-3 i a^2 b \text {csch}^2(c+d x)-i a^3 \text {csch}^5(c+d x)-3 i a b^2 \sinh (c+d x)-i b^3 \sinh ^4(c+d x)\right ) \, dx\\ &=a^3 \int \text {csch}^5(c+d x) \, dx+\left (3 a^2 b\right ) \int \text {csch}^2(c+d x) \, dx+\left (3 a b^2\right ) \int \sinh (c+d x) \, dx+b^3 \int \sinh ^4(c+d x) \, dx\\ &=\frac {3 a b^2 \cosh (c+d x)}{d}-\frac {a^3 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}+\frac {b^3 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac {1}{4} \left (3 a^3\right ) \int \text {csch}^3(c+d x) \, dx-\frac {1}{4} \left (3 b^3\right ) \int \sinh ^2(c+d x) \, dx-\frac {\left (3 i a^2 b\right ) \operatorname {Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{d}\\ &=\frac {3 a b^2 \cosh (c+d x)}{d}-\frac {3 a^2 b \coth (c+d x)}{d}+\frac {3 a^3 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a^3 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}-\frac {3 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^3 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}+\frac {1}{8} \left (3 a^3\right ) \int \text {csch}(c+d x) \, dx+\frac {1}{8} \left (3 b^3\right ) \int 1 \, dx\\ &=\frac {3 b^3 x}{8}-\frac {3 a^3 \tanh ^{-1}(\cosh (c+d x))}{8 d}+\frac {3 a b^2 \cosh (c+d x)}{d}-\frac {3 a^2 b \coth (c+d x)}{d}+\frac {3 a^3 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a^3 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}-\frac {3 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^3 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 6.14, size = 218, normalized size = 1.47 \[ -\frac {a^3 \text {csch}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {3 a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a^3 \text {sech}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {3 a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {3 a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {3 a^2 b \tanh \left (\frac {1}{2} (c+d x)\right )}{2 d}-\frac {3 a^2 b \coth \left (\frac {1}{2} (c+d x)\right )}{2 d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {3 b^3 (c+d x)}{8 d}-\frac {b^3 \sinh (2 (c+d x))}{4 d}+\frac {b^3 \sinh (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^3)^3,x]
[Out]
(3*b^3*(c + d*x))/(8*d) + (3*a*b^2*Cosh[c + d*x])/d - (3*a^2*b*Coth[(c + d*x)/2])/(2*d) + (3*a^3*Csch[(c + d*x
)/2]^2)/(32*d) - (a^3*Csch[(c + d*x)/2]^4)/(64*d) + (3*a^3*Log[Tanh[(c + d*x)/2]])/(8*d) + (3*a^3*Sech[(c + d*
x)/2]^2)/(32*d) + (a^3*Sech[(c + d*x)/2]^4)/(64*d) - (b^3*Sinh[2*(c + d*x)])/(4*d) + (b^3*Sinh[4*(c + d*x)])/(
32*d) - (3*a^2*b*Tanh[(c + d*x)/2])/(2*d)
________________________________________________________________________________________
fricas [B] time = 0.62, size = 4541, normalized size = 30.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^5*(a+b*sinh(d*x+c)^3)^3,x, algorithm="fricas")
[Out]
1/64*(b^3*cosh(d*x + c)^16 + 16*b^3*cosh(d*x + c)*sinh(d*x + c)^15 + b^3*sinh(d*x + c)^16 - 12*b^3*cosh(d*x +
c)^14 + 96*a*b^2*cosh(d*x + c)^13 + 12*(10*b^3*cosh(d*x + c)^2 - b^3)*sinh(d*x + c)^14 + 8*(70*b^3*cosh(d*x +
c)^3 - 21*b^3*cosh(d*x + c) + 12*a*b^2)*sinh(d*x + c)^13 + 2*(12*b^3*d*x + 19*b^3)*cosh(d*x + c)^12 + 2*(910*b
^3*cosh(d*x + c)^4 + 12*b^3*d*x - 546*b^3*cosh(d*x + c)^2 + 624*a*b^2*cosh(d*x + c) + 19*b^3)*sinh(d*x + c)^12
+ 48*(a^3 - 6*a*b^2)*cosh(d*x + c)^11 + 24*(182*b^3*cosh(d*x + c)^5 - 182*b^3*cosh(d*x + c)^3 + 312*a*b^2*cos
h(d*x + c)^2 + 2*a^3 - 12*a*b^2 + (12*b^3*d*x + 19*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 - 4*(24*b^3*d*x + 96*a
^2*b + 11*b^3)*cosh(d*x + c)^10 + 4*(2002*b^3*cosh(d*x + c)^6 - 3003*b^3*cosh(d*x + c)^4 + 6864*a*b^2*cosh(d*x
+ c)^3 - 24*b^3*d*x - 96*a^2*b - 11*b^3 + 33*(12*b^3*d*x + 19*b^3)*cosh(d*x + c)^2 + 132*(a^3 - 6*a*b^2)*cosh
(d*x + c))*sinh(d*x + c)^10 - 16*(11*a^3 - 12*a*b^2)*cosh(d*x + c)^9 + 8*(1430*b^3*cosh(d*x + c)^7 - 3003*b^3*
cosh(d*x + c)^5 + 8580*a*b^2*cosh(d*x + c)^4 + 55*(12*b^3*d*x + 19*b^3)*cosh(d*x + c)^3 - 22*a^3 + 24*a*b^2 +
330*(a^3 - 6*a*b^2)*cosh(d*x + c)^2 - 5*(24*b^3*d*x + 96*a^2*b + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 144*
(b^3*d*x + 8*a^2*b)*cosh(d*x + c)^8 + 18*(715*b^3*cosh(d*x + c)^8 - 2002*b^3*cosh(d*x + c)^6 + 6864*a*b^2*cosh
(d*x + c)^5 + 8*b^3*d*x + 55*(12*b^3*d*x + 19*b^3)*cosh(d*x + c)^4 + 440*(a^3 - 6*a*b^2)*cosh(d*x + c)^3 + 64*
a^2*b - 10*(24*b^3*d*x + 96*a^2*b + 11*b^3)*cosh(d*x + c)^2 - 8*(11*a^3 - 12*a*b^2)*cosh(d*x + c))*sinh(d*x +
c)^8 - 16*(11*a^3 - 12*a*b^2)*cosh(d*x + c)^7 + 16*(715*b^3*cosh(d*x + c)^9 - 2574*b^3*cosh(d*x + c)^7 + 10296
*a*b^2*cosh(d*x + c)^6 + 99*(12*b^3*d*x + 19*b^3)*cosh(d*x + c)^5 + 990*(a^3 - 6*a*b^2)*cosh(d*x + c)^4 - 30*(
24*b^3*d*x + 96*a^2*b + 11*b^3)*cosh(d*x + c)^3 - 11*a^3 + 12*a*b^2 - 36*(11*a^3 - 12*a*b^2)*cosh(d*x + c)^2 +
72*(b^3*d*x + 8*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^7 - 4*(24*b^3*d*x + 288*a^2*b - 11*b^3)*cosh(d*x + c)^6 +
4*(2002*b^3*cosh(d*x + c)^10 - 9009*b^3*cosh(d*x + c)^8 + 41184*a*b^2*cosh(d*x + c)^7 + 462*(12*b^3*d*x + 19*
b^3)*cosh(d*x + c)^6 + 5544*(a^3 - 6*a*b^2)*cosh(d*x + c)^5 - 24*b^3*d*x - 210*(24*b^3*d*x + 96*a^2*b + 11*b^3
)*cosh(d*x + c)^4 - 336*(11*a^3 - 12*a*b^2)*cosh(d*x + c)^3 - 288*a^2*b + 11*b^3 + 1008*(b^3*d*x + 8*a^2*b)*co
sh(d*x + c)^2 - 28*(11*a^3 - 12*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^6 + 96*a*b^2*cosh(d*x + c)^3 + 48*(a^3 - 6
*a*b^2)*cosh(d*x + c)^5 + 24*(182*b^3*cosh(d*x + c)^11 - 1001*b^3*cosh(d*x + c)^9 + 5148*a*b^2*cosh(d*x + c)^8
+ 66*(12*b^3*d*x + 19*b^3)*cosh(d*x + c)^7 + 924*(a^3 - 6*a*b^2)*cosh(d*x + c)^6 - 42*(24*b^3*d*x + 96*a^2*b
+ 11*b^3)*cosh(d*x + c)^5 - 84*(11*a^3 - 12*a*b^2)*cosh(d*x + c)^4 + 336*(b^3*d*x + 8*a^2*b)*cosh(d*x + c)^3 +
2*a^3 - 12*a*b^2 - 14*(11*a^3 - 12*a*b^2)*cosh(d*x + c)^2 - (24*b^3*d*x + 288*a^2*b - 11*b^3)*cosh(d*x + c))*
sinh(d*x + c)^5 + 12*b^3*cosh(d*x + c)^2 + 2*(12*b^3*d*x + 192*a^2*b - 19*b^3)*cosh(d*x + c)^4 + 2*(910*b^3*co
sh(d*x + c)^12 - 6006*b^3*cosh(d*x + c)^10 + 34320*a*b^2*cosh(d*x + c)^9 + 495*(12*b^3*d*x + 19*b^3)*cosh(d*x
+ c)^8 + 7920*(a^3 - 6*a*b^2)*cosh(d*x + c)^7 - 420*(24*b^3*d*x + 96*a^2*b + 11*b^3)*cosh(d*x + c)^6 - 1008*(1
1*a^3 - 12*a*b^2)*cosh(d*x + c)^5 + 12*b^3*d*x + 5040*(b^3*d*x + 8*a^2*b)*cosh(d*x + c)^4 - 280*(11*a^3 - 12*a
*b^2)*cosh(d*x + c)^3 + 192*a^2*b - 19*b^3 - 30*(24*b^3*d*x + 288*a^2*b - 11*b^3)*cosh(d*x + c)^2 + 120*(a^3 -
6*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 8*(70*b^3*cosh(d*x + c)^13 - 546*b^3*cosh(d*x + c)^11 + 3432*a*b^2*
cosh(d*x + c)^10 + 55*(12*b^3*d*x + 19*b^3)*cosh(d*x + c)^9 + 990*(a^3 - 6*a*b^2)*cosh(d*x + c)^8 - 60*(24*b^3
*d*x + 96*a^2*b + 11*b^3)*cosh(d*x + c)^7 - 168*(11*a^3 - 12*a*b^2)*cosh(d*x + c)^6 + 1008*(b^3*d*x + 8*a^2*b)
*cosh(d*x + c)^5 - 70*(11*a^3 - 12*a*b^2)*cosh(d*x + c)^4 - 10*(24*b^3*d*x + 288*a^2*b - 11*b^3)*cosh(d*x + c)
^3 + 12*a*b^2 + 60*(a^3 - 6*a*b^2)*cosh(d*x + c)^2 + (12*b^3*d*x + 192*a^2*b - 19*b^3)*cosh(d*x + c))*sinh(d*x
+ c)^3 - b^3 + 12*(10*b^3*cosh(d*x + c)^14 - 91*b^3*cosh(d*x + c)^12 + 624*a*b^2*cosh(d*x + c)^11 + 11*(12*b^
3*d*x + 19*b^3)*cosh(d*x + c)^10 + 220*(a^3 - 6*a*b^2)*cosh(d*x + c)^9 - 15*(24*b^3*d*x + 96*a^2*b + 11*b^3)*c
osh(d*x + c)^8 - 48*(11*a^3 - 12*a*b^2)*cosh(d*x + c)^7 + 336*(b^3*d*x + 8*a^2*b)*cosh(d*x + c)^6 - 28*(11*a^3
- 12*a*b^2)*cosh(d*x + c)^5 - 5*(24*b^3*d*x + 288*a^2*b - 11*b^3)*cosh(d*x + c)^4 + 24*a*b^2*cosh(d*x + c) +
40*(a^3 - 6*a*b^2)*cosh(d*x + c)^3 + b^3 + (12*b^3*d*x + 192*a^2*b - 19*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2
- 24*(a^3*cosh(d*x + c)^12 + 12*a^3*cosh(d*x + c)*sinh(d*x + c)^11 + a^3*sinh(d*x + c)^12 - 4*a^3*cosh(d*x + c
)^10 + 6*a^3*cosh(d*x + c)^8 + 2*(33*a^3*cosh(d*x + c)^2 - 2*a^3)*sinh(d*x + c)^10 + 20*(11*a^3*cosh(d*x + c)^
3 - 2*a^3*cosh(d*x + c))*sinh(d*x + c)^9 - 4*a^3*cosh(d*x + c)^6 + 3*(165*a^3*cosh(d*x + c)^4 - 60*a^3*cosh(d*
x + c)^2 + 2*a^3)*sinh(d*x + c)^8 + 24*(33*a^3*cosh(d*x + c)^5 - 20*a^3*cosh(d*x + c)^3 + 2*a^3*cosh(d*x + c))
*sinh(d*x + c)^7 + a^3*cosh(d*x + c)^4 + 4*(231*a^3*cosh(d*x + c)^6 - 210*a^3*cosh(d*x + c)^4 + 42*a^3*cosh(d*
x + c)^2 - a^3)*sinh(d*x + c)^6 + 24*(33*a^3*cosh(d*x + c)^7 - 42*a^3*cosh(d*x + c)^5 + 14*a^3*cosh(d*x + c)^3
- a^3*cosh(d*x + c))*sinh(d*x + c)^5 + (495*a^3*cosh(d*x + c)^8 - 840*a^3*cosh(d*x + c)^6 + 420*a^3*cosh(d*x
+ c)^4 - 60*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^4 + 4*(55*a^3*cosh(d*x + c)^9 - 120*a^3*cosh(d*x + c)^7 +
84*a^3*cosh(d*x + c)^5 - 20*a^3*cosh(d*x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(11*a^3*cosh(d*x + c
)^10 - 30*a^3*cosh(d*x + c)^8 + 28*a^3*cosh(d*x + c)^6 - 10*a^3*cosh(d*x + c)^4 + a^3*cosh(d*x + c)^2)*sinh(d*
x + c)^2 + 4*(3*a^3*cosh(d*x + c)^11 - 10*a^3*cosh(d*x + c)^9 + 12*a^3*cosh(d*x + c)^7 - 6*a^3*cosh(d*x + c)^5
+ a^3*cosh(d*x + c)^3)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 24*(a^3*cosh(d*x + c)^12 + 12*
a^3*cosh(d*x + c)*sinh(d*x + c)^11 + a^3*sinh(d*x + c)^12 - 4*a^3*cosh(d*x + c)^10 + 6*a^3*cosh(d*x + c)^8 + 2
*(33*a^3*cosh(d*x + c)^2 - 2*a^3)*sinh(d*x + c)^10 + 20*(11*a^3*cosh(d*x + c)^3 - 2*a^3*cosh(d*x + c))*sinh(d*
x + c)^9 - 4*a^3*cosh(d*x + c)^6 + 3*(165*a^3*cosh(d*x + c)^4 - 60*a^3*cosh(d*x + c)^2 + 2*a^3)*sinh(d*x + c)^
8 + 24*(33*a^3*cosh(d*x + c)^5 - 20*a^3*cosh(d*x + c)^3 + 2*a^3*cosh(d*x + c))*sinh(d*x + c)^7 + a^3*cosh(d*x
+ c)^4 + 4*(231*a^3*cosh(d*x + c)^6 - 210*a^3*cosh(d*x + c)^4 + 42*a^3*cosh(d*x + c)^2 - a^3)*sinh(d*x + c)^6
+ 24*(33*a^3*cosh(d*x + c)^7 - 42*a^3*cosh(d*x + c)^5 + 14*a^3*cosh(d*x + c)^3 - a^3*cosh(d*x + c))*sinh(d*x +
c)^5 + (495*a^3*cosh(d*x + c)^8 - 840*a^3*cosh(d*x + c)^6 + 420*a^3*cosh(d*x + c)^4 - 60*a^3*cosh(d*x + c)^2
+ a^3)*sinh(d*x + c)^4 + 4*(55*a^3*cosh(d*x + c)^9 - 120*a^3*cosh(d*x + c)^7 + 84*a^3*cosh(d*x + c)^5 - 20*a^3
*cosh(d*x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(11*a^3*cosh(d*x + c)^10 - 30*a^3*cosh(d*x + c)^8 +
28*a^3*cosh(d*x + c)^6 - 10*a^3*cosh(d*x + c)^4 + a^3*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(3*a^3*cosh(d*x + c
)^11 - 10*a^3*cosh(d*x + c)^9 + 12*a^3*cosh(d*x + c)^7 - 6*a^3*cosh(d*x + c)^5 + a^3*cosh(d*x + c)^3)*sinh(d*x
+ c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 8*(2*b^3*cosh(d*x + c)^15 - 21*b^3*cosh(d*x + c)^13 + 156*a*b^
2*cosh(d*x + c)^12 + 3*(12*b^3*d*x + 19*b^3)*cosh(d*x + c)^11 + 66*(a^3 - 6*a*b^2)*cosh(d*x + c)^10 - 5*(24*b^
3*d*x + 96*a^2*b + 11*b^3)*cosh(d*x + c)^9 - 18*(11*a^3 - 12*a*b^2)*cosh(d*x + c)^8 + 144*(b^3*d*x + 8*a^2*b)*
cosh(d*x + c)^7 - 14*(11*a^3 - 12*a*b^2)*cosh(d*x + c)^6 - 3*(24*b^3*d*x + 288*a^2*b - 11*b^3)*cosh(d*x + c)^5
+ 36*a*b^2*cosh(d*x + c)^2 + 30*(a^3 - 6*a*b^2)*cosh(d*x + c)^4 + 3*b^3*cosh(d*x + c) + (12*b^3*d*x + 192*a^2
*b - 19*b^3)*cosh(d*x + c)^3)*sinh(d*x + c))/(d*cosh(d*x + c)^12 + 12*d*cosh(d*x + c)*sinh(d*x + c)^11 + d*sin
h(d*x + c)^12 - 4*d*cosh(d*x + c)^10 + 2*(33*d*cosh(d*x + c)^2 - 2*d)*sinh(d*x + c)^10 + 20*(11*d*cosh(d*x + c
)^3 - 2*d*cosh(d*x + c))*sinh(d*x + c)^9 + 6*d*cosh(d*x + c)^8 + 3*(165*d*cosh(d*x + c)^4 - 60*d*cosh(d*x + c)
^2 + 2*d)*sinh(d*x + c)^8 + 24*(33*d*cosh(d*x + c)^5 - 20*d*cosh(d*x + c)^3 + 2*d*cosh(d*x + c))*sinh(d*x + c)
^7 - 4*d*cosh(d*x + c)^6 + 4*(231*d*cosh(d*x + c)^6 - 210*d*cosh(d*x + c)^4 + 42*d*cosh(d*x + c)^2 - d)*sinh(d
*x + c)^6 + 24*(33*d*cosh(d*x + c)^7 - 42*d*cosh(d*x + c)^5 + 14*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x
+ c)^5 + d*cosh(d*x + c)^4 + (495*d*cosh(d*x + c)^8 - 840*d*cosh(d*x + c)^6 + 420*d*cosh(d*x + c)^4 - 60*d*co
sh(d*x + c)^2 + d)*sinh(d*x + c)^4 + 4*(55*d*cosh(d*x + c)^9 - 120*d*cosh(d*x + c)^7 + 84*d*cosh(d*x + c)^5 -
20*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(11*d*cosh(d*x + c)^10 - 30*d*cosh(d*x + c)^8 + 28
*d*cosh(d*x + c)^6 - 10*d*cosh(d*x + c)^4 + d*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(3*d*cosh(d*x + c)^11 - 10*
d*cosh(d*x + c)^9 + 12*d*cosh(d*x + c)^7 - 6*d*cosh(d*x + c)^5 + d*cosh(d*x + c)^3)*sinh(d*x + c))
________________________________________________________________________________________
giac [B] time = 0.39, size = 329, normalized size = 2.22 \[ \frac {24 \, {\left (d x + c\right )} b^{3} + b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 96 \, a b^{2} e^{\left (d x + c\right )} - 24 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) + 24 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + \frac {{\left (96 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 12 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - b^{3} + 48 \, {\left (a^{3} + 2 \, a b^{2}\right )} e^{\left (11 \, d x + 11 \, c\right )} - 8 \, {\left (48 \, a^{2} b - b^{3}\right )} e^{\left (10 \, d x + 10 \, c\right )} - 16 \, {\left (11 \, a^{3} + 24 \, a b^{2}\right )} e^{\left (9 \, d x + 9 \, c\right )} + 3 \, {\left (384 \, a^{2} b - 11 \, b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )} - 16 \, {\left (11 \, a^{3} - 36 \, a b^{2}\right )} e^{\left (7 \, d x + 7 \, c\right )} - 4 \, {\left (288 \, a^{2} b - 13 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )} + 48 \, {\left (a^{3} - 8 \, a b^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )} + 2 \, {\left (192 \, a^{2} b - 19 \, b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )}^{4} {\left (e^{\left (d x + c\right )} - 1\right )}^{4}}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^5*(a+b*sinh(d*x+c)^3)^3,x, algorithm="giac")
[Out]
1/64*(24*(d*x + c)*b^3 + b^3*e^(4*d*x + 4*c) - 8*b^3*e^(2*d*x + 2*c) + 96*a*b^2*e^(d*x + c) - 24*a^3*log(e^(d*
x + c) + 1) + 24*a^3*log(abs(e^(d*x + c) - 1)) + (96*a*b^2*e^(3*d*x + 3*c) + 12*b^3*e^(2*d*x + 2*c) - b^3 + 48
*(a^3 + 2*a*b^2)*e^(11*d*x + 11*c) - 8*(48*a^2*b - b^3)*e^(10*d*x + 10*c) - 16*(11*a^3 + 24*a*b^2)*e^(9*d*x +
9*c) + 3*(384*a^2*b - 11*b^3)*e^(8*d*x + 8*c) - 16*(11*a^3 - 36*a*b^2)*e^(7*d*x + 7*c) - 4*(288*a^2*b - 13*b^3
)*e^(6*d*x + 6*c) + 48*(a^3 - 8*a*b^2)*e^(5*d*x + 5*c) + 2*(192*a^2*b - 19*b^3)*e^(4*d*x + 4*c))*e^(-4*d*x - 4
*c)/((e^(d*x + c) + 1)^4*(e^(d*x + c) - 1)^4))/d
________________________________________________________________________________________
maple [A] time = 0.19, size = 108, normalized size = 0.73 \[ \frac {a^{3} \left (\left (-\frac {\mathrm {csch}\left (d x +c \right )^{3}}{4}+\frac {3 \,\mathrm {csch}\left (d x +c \right )}{8}\right ) \coth \left (d x +c \right )-\frac {3 \arctanh \left ({\mathrm e}^{d x +c}\right )}{4}\right )-3 a^{2} b \coth \left (d x +c \right )+3 a \,b^{2} \cosh \left (d x +c \right )+b^{3} \left (\left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^5*(a+b*sinh(d*x+c)^3)^3,x)
[Out]
1/d*(a^3*((-1/4*csch(d*x+c)^3+3/8*csch(d*x+c))*coth(d*x+c)-3/4*arctanh(exp(d*x+c)))-3*a^2*b*coth(d*x+c)+3*a*b^
2*cosh(d*x+c)+b^3*((1/4*sinh(d*x+c)^3-3/8*sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c))
________________________________________________________________________________________
maxima [A] time = 0.34, size = 255, normalized size = 1.72 \[ \frac {1}{64} \, b^{3} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {3}{2} \, a b^{2} {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} - \frac {1}{8} \, a^{3} {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} - 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + \frac {6 \, a^{2} b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^5*(a+b*sinh(d*x+c)^3)^3,x, algorithm="maxima")
[Out]
1/64*b^3*(24*x + e^(4*d*x + 4*c)/d - 8*e^(2*d*x + 2*c)/d + 8*e^(-2*d*x - 2*c)/d - e^(-4*d*x - 4*c)/d) + 3/2*a*
b^2*(e^(d*x + c)/d + e^(-d*x - c)/d) - 1/8*a^3*(3*log(e^(-d*x - c) + 1)/d - 3*log(e^(-d*x - c) - 1)/d + 2*(3*e
^(-d*x - c) - 11*e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) + 3*e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) - 6*e^(-4
*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) - 1))) + 6*a^2*b/(d*(e^(-2*d*x - 2*c) - 1))
________________________________________________________________________________________
mupad [B] time = 0.87, size = 451, normalized size = 3.05 \[ \frac {3\,b^3\,x}{8}-\frac {\frac {3\,a^2\,b}{2\,d}+\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}}{d}-\frac {3\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{d}+\frac {3\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{2\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}-\frac {\frac {4\,a^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{d}-\frac {3\,a^2\,b}{2\,d}+\frac {9\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{2\,d}-\frac {9\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{2\,d}+\frac {3\,a^2\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{2\,d}}{6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {\frac {3\,a^2\,b}{d}-\frac {3\,a^3\,{\mathrm {e}}^{c+d\,x}}{4\,d}}{{\mathrm {e}}^{2\,c+2\,d\,x}-1}-\frac {3\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^6}}\right )\,\sqrt {a^6}}{4\,\sqrt {-d^2}}+\frac {b^3\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}-\frac {b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {b^3\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,d}+\frac {b^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,d}+\frac {3\,a\,b^2\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {3\,a\,b^2\,{\mathrm {e}}^{c+d\,x}}{2\,d}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{2\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*sinh(c + d*x)^3)^3/sinh(c + d*x)^5,x)
[Out]
(3*b^3*x)/8 - ((3*a^2*b)/(2*d) + (2*a^3*exp(c + d*x))/d - (3*a^2*b*exp(2*c + 2*d*x))/d + (3*a^2*b*exp(4*c + 4*
d*x))/(2*d))/(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) - ((4*a^3*exp(3*c + 3*d*x))/d -
(3*a^2*b)/(2*d) + (9*a^2*b*exp(2*c + 2*d*x))/(2*d) - (9*a^2*b*exp(4*c + 4*d*x))/(2*d) + (3*a^2*b*exp(6*c + 6*d
*x))/(2*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*a^2*b)
/d - (3*a^3*exp(c + d*x))/(4*d))/(exp(2*c + 2*d*x) - 1) - (3*atan((a^3*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d*(a^6)^
(1/2)))*(a^6)^(1/2))/(4*(-d^2)^(1/2)) + (b^3*exp(- 2*c - 2*d*x))/(8*d) - (b^3*exp(2*c + 2*d*x))/(8*d) - (b^3*e
xp(- 4*c - 4*d*x))/(64*d) + (b^3*exp(4*c + 4*d*x))/(64*d) + (3*a*b^2*exp(- c - d*x))/(2*d) + (3*a*b^2*exp(c +
d*x))/(2*d) - (a^3*exp(c + d*x))/(2*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))
________________________________________________________________________________________
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)**5*(a+b*sinh(d*x+c)**3)**3,x)
[Out]
Timed out
________________________________________________________________________________________