3.169 \(\int \text{csch}^6(c+d x) (a+b \sinh ^3(c+d x))^3 \, dx\)
Optimal. Leaf size=131 \[ \frac{3 a^2 b \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{3 a^2 b \coth (c+d x) \text{csch}(c+d x)}{2 d}-\frac{a^3 \coth ^5(c+d x)}{5 d}+\frac{2 a^3 \coth ^3(c+d x)}{3 d}-\frac{a^3 \coth (c+d x)}{d}+3 a b^2 x+\frac{b^3 \cosh ^3(c+d x)}{3 d}-\frac{b^3 \cosh (c+d x)}{d} \]
[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)
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Rubi [A] time = 0.124965, antiderivative size = 131, normalized size of antiderivative = 1.,
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 =
{3220, 3768, 3770, 3767, 2633} \[ \frac{3 a^2 b \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{3 a^2 b \coth (c+d x) \text{csch}(c+d x)}{2 d}-\frac{a^3 \coth ^5(c+d x)}{5 d}+\frac{2 a^3 \coth ^3(c+d x)}{3 d}-\frac{a^3 \coth (c+d x)}{d}+3 a b^2 x+\frac{b^3 \cosh ^3(c+d x)}{3 d}-\frac{b^3 \cosh (c+d x)}{d} \]
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 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 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]
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 2633
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]
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 ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \coth (c+d x)\right )}{d}-\frac{b^3 \operatorname{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 = 1.84643, size = 225, normalized size = 1.72 \[ \frac{\frac{1}{2} a \left (8 \left (-32 a^2 \tanh \left (\frac{1}{2} (c+d x)\right )-24 a^2 \sinh ^6\left (\frac{1}{2} (c+d x)\right ) \text{csch}^5(c+d x)-38 a^2 \sinh ^4\left (\frac{1}{2} (c+d x)\right ) \text{csch}^3(c+d x)-45 a b \text{sech}^2\left (\frac{1}{2} (c+d x)\right )+180 b \left (2 b (c+d x)-a \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )-256 a^2 \coth \left (\frac{1}{2} (c+d x)\right )-3 a^2 \sinh (c+d x) \text{csch}^6\left (\frac{1}{2} (c+d x)\right )+19 a^2 \sinh (c+d x) \text{csch}^4\left (\frac{1}{2} (c+d x)\right )-360 a b \text{csch}^2\left (\frac{1}{2} (c+d x)\right )\right )-360 b^3 \cosh (c+d x)+40 b^3 \cosh (3 (c+d x))}{480 d} \]
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)
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Maple [A] time = 0.085, size = 99, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{8}{15}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{5}}+{\frac{4\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{15}} \right ){\rm coth} \left (dx+c\right )+3\,{a}^{2}b \left ( -1/2\,{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) +3\,a{b}^{2} \left ( dx+c \right ) +{b}^{3} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}
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)
[Out]
1/d*(a^3*(-8/15-1/5*csch(d*x+c)^4+4/15*csch(d*x+c)^2)*coth(d*x+c)+3*a^2*b*(-1/2*csch(d*x+c)*coth(d*x+c)+arctan
h(exp(d*x+c)))+3*a*b^2*(d*x+c)+b^3*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c))
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Maxima [B] time = 1.08746, size = 493, normalized size = 3.76 \begin{align*} 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{align*}
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)))
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Fricas [B] time = 2.36594, size = 12388, normalized size = 94.56 \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)^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*x + c)^5 - 70*a^2*b*cosh(d*x + c)^3 + 5*a^2*b*cosh(d*x + c))*si
nh(d*x + c)^4 + (286*a^2*b*cosh(d*x + c)^10 - 825*a^2*b*cosh(d*x + c)^8 + 840*a^2*b*cosh(d*x + c)^6 - 350*a^2*
b*cosh(d*x + c)^4 + 50*a^2*b*cosh(d*x + c)^2 - a^2*b)*sinh(d*x + c)^3 + (78*a^2*b*cosh(d*x + c)^11 - 275*a^2*b
*cosh(d*x + c)^9 + 360*a^2*b*cosh(d*x + c)^7 - 210*a^2*b*cosh(d*x + c)^5 + 50*a^2*b*cosh(d*x + c)^3 - 3*a^2*b*
cosh(d*x + c))*sinh(d*x + c)^2 + (13*a^2*b*cosh(d*x + c)^12 - 55*a^2*b*cosh(d*x + c)^10 + 90*a^2*b*cosh(d*x +
c)^8 - 70*a^2*b*cosh(d*x + c)^6 + 25*a^2*b*cosh(d*x + c)^4 - 3*a^2*b*cosh(d*x + c)^2)*sinh(d*x + c))*log(cosh(
d*x + c) + sinh(d*x + c) + 1) - 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*
sinh(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)*
sinh(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*cos
h(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 + 12
60*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*x + c)^5 - 70*a^2*b*cosh(d*x + c)^3 + 5*a^2*b*cosh(d*x + c))*s
inh(d*x + c)^4 + (286*a^2*b*cosh(d*x + c)^10 - 825*a^2*b*cosh(d*x + c)^8 + 840*a^2*b*cosh(d*x + c)^6 - 350*a^2
*b*cosh(d*x + c)^4 + 50*a^2*b*cosh(d*x + c)^2 - a^2*b)*sinh(d*x + c)^3 + (78*a^2*b*cosh(d*x + c)^11 - 275*a^2*
b*cosh(d*x + c)^9 + 360*a^2*b*cosh(d*x + c)^7 - 210*a^2*b*cosh(d*x + c)^5 + 50*a^2*b*cosh(d*x + c)^3 - 3*a^2*b
*cosh(d*x + c))*sinh(d*x + c)^2 + (13*a^2*b*cosh(d*x + c)^12 - 55*a^2*b*cosh(d*x + c)^10 + 90*a^2*b*cosh(d*x +
c)^8 - 70*a^2*b*cosh(d*x + c)^6 + 25*a^2*b*cosh(d*x + c)^4 - 3*a^2*b*cosh(d*x + c)^2)*sinh(d*x + c))*log(cosh
(d*x + c) + sinh(d*x + c) - 1) + 4*(20*b^3*cosh(d*x + c)^15 + 1170*a*b^2*d*x*cosh(d*x + c)^12 - 245*b^3*cosh(d
*x + c)^13 - 4950*a*b^2*d*x*cosh(d*x + c)^10 + 8100*a*b^2*d*x*cosh(d*x + c)^8 - 30*(36*a^2*b - 23*b^3)*cosh(d*
x + c)^11 + 225*(8*a^2*b - 3*b^3)*cosh(d*x + c)^9 - 140*(45*a*b^2*d*x + 16*a^3)*cosh(d*x + c)^6 - 135*(8*a^2*b
- 3*b^3)*cosh(d*x + c)^5 + 50*(45*a*b^2*d*x + 16*a^3)*cosh(d*x + c)^4 + 35*b^3*cosh(d*x + c) + 10*(36*a^2*b -
23*b^3)*cosh(d*x + c)^3 - 6*(45*a*b^2*d*x + 16*a^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)^13 + 13*
d*cosh(d*x + c)*sinh(d*x + c)^12 + d*sinh(d*x + c)^13 - 5*d*cosh(d*x + c)^11 + (78*d*cosh(d*x + c)^2 - 5*d)*si
nh(d*x + c)^11 + 11*(26*d*cosh(d*x + c)^3 - 5*d*cosh(d*x + c))*sinh(d*x + c)^10 + 10*d*cosh(d*x + c)^9 + 5*(14
3*d*cosh(d*x + c)^4 - 55*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^9 + 3*(429*d*cosh(d*x + c)^5 - 275*d*cosh(d*x
+ c)^3 + 30*d*cosh(d*x + c))*sinh(d*x + c)^8 - 10*d*cosh(d*x + c)^7 + 2*(858*d*cosh(d*x + c)^6 - 825*d*cosh(d*
x + c)^4 + 180*d*cosh(d*x + c)^2 - 5*d)*sinh(d*x + c)^7 + 2*(858*d*cosh(d*x + c)^7 - 1155*d*cosh(d*x + c)^5 +
420*d*cosh(d*x + c)^3 - 35*d*cosh(d*x + c))*sinh(d*x + c)^6 + 5*d*cosh(d*x + c)^5 + (1287*d*cosh(d*x + c)^8 -
2310*d*cosh(d*x + c)^6 + 1260*d*cosh(d*x + c)^4 - 210*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^5 + 5*(143*d*cosh
(d*x + c)^9 - 330*d*cosh(d*x + c)^7 + 252*d*cosh(d*x + c)^5 - 70*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d
*x + c)^4 - d*cosh(d*x + c)^3 + (286*d*cosh(d*x + c)^10 - 825*d*cosh(d*x + c)^8 + 840*d*cosh(d*x + c)^6 - 350*
d*cosh(d*x + c)^4 + 50*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^3 + (78*d*cosh(d*x + c)^11 - 275*d*cosh(d*x + c)^9
+ 360*d*cosh(d*x + c)^7 - 210*d*cosh(d*x + c)^5 + 50*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh(d*x + c)^2 +
(13*d*cosh(d*x + c)^12 - 55*d*cosh(d*x + c)^10 + 90*d*cosh(d*x + c)^8 - 70*d*cosh(d*x + c)^6 + 25*d*cosh(d*x
+ c)^4 - 3*d*cosh(d*x + c)^2)*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)**6*(a+b*sinh(d*x+c)**3)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.44204, size = 389, normalized size = 2.97 \begin{align*} \frac{3 \,{\left (d x + c\right )} a b^{2}}{d} + \frac{3 \, a^{2} b \log \left (e^{\left (d x + c\right )} + 1\right )}{2 \, d} - \frac{3 \, a^{2} b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{2 \, d} + \frac{b^{3} d^{2} e^{\left (3 \, d x + 3 \, c\right )} - 9 \, b^{3} d^{2} e^{\left (d x + c\right )}}{24 \, d^{3}} - \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 )}}{120 \, d{\left (e^{\left (d x + c\right )} + 1\right )}^{5}{\left (e^{\left (d x + c\right )} - 1\right )}^{5}} \end{align*}
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]
3*(d*x + c)*a*b^2/d + 3/2*a^2*b*log(e^(d*x + c) + 1)/d - 3/2*a^2*b*log(abs(e^(d*x + c) - 1))/d + 1/24*(b^3*d^2
*e^(3*d*x + 3*c) - 9*b^3*d^2*e^(d*x + c))/d^3 - 1/120*(475*b^3*e^(8*d*x + 8*c) + 1280*a^3*e^(7*d*x + 7*c) - 64
0*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)/(d*(e^(d*x + c) + 1)^5*(e^(d*x + c) - 1)^5)