3.167 \(\int \text{csch}^4(c+d x) (a+b \sinh ^3(c+d x))^3 \, dx\)

Optimal. Leaf size=129 \[ -\frac{3 a^2 b \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{a^3 \coth ^3(c+d x)}{3 d}+\frac{a^3 \coth (c+d x)}{d}+\frac{3 a b^2 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{3}{2} a b^2 x+\frac{b^3 \cosh ^5(c+d x)}{5 d}-\frac{2 b^3 \cosh ^3(c+d x)}{3 d}+\frac{b^3 \cosh (c+d x)}{d} \]

[Out]

(-3*a*b^2*x)/2 - (3*a^2*b*ArcTanh[Cosh[c + d*x]])/d + (b^3*Cosh[c + d*x])/d - (2*b^3*Cosh[c + d*x]^3)/(3*d) +
(b^3*Cosh[c + d*x]^5)/(5*d) + (a^3*Coth[c + d*x])/d - (a^3*Coth[c + d*x]^3)/(3*d) + (3*a*b^2*Cosh[c + d*x]*Sin
h[c + d*x])/(2*d)

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Rubi [A]  time = 0.120472, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3220, 3770, 3767, 2635, 8, 2633} \[ -\frac{3 a^2 b \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{a^3 \coth ^3(c+d x)}{3 d}+\frac{a^3 \coth (c+d x)}{d}+\frac{3 a b^2 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{3}{2} a b^2 x+\frac{b^3 \cosh ^5(c+d x)}{5 d}-\frac{2 b^3 \cosh ^3(c+d x)}{3 d}+\frac{b^3 \cosh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^3)^3,x]

[Out]

(-3*a*b^2*x)/2 - (3*a^2*b*ArcTanh[Cosh[c + d*x]])/d + (b^3*Cosh[c + d*x])/d - (2*b^3*Cosh[c + d*x]^3)/(3*d) +
(b^3*Cosh[c + d*x]^5)/(5*d) + (a^3*Coth[c + d*x])/d - (a^3*Coth[c + d*x]^3)/(3*d) + (3*a*b^2*Cosh[c + d*x]*Sin
h[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 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 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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx &=\int \left (3 a^2 b \text{csch}(c+d x)+a^3 \text{csch}^4(c+d x)+3 a b^2 \sinh ^2(c+d x)+b^3 \sinh ^5(c+d x)\right ) \, dx\\ &=a^3 \int \text{csch}^4(c+d x) \, dx+\left (3 a^2 b\right ) \int \text{csch}(c+d x) \, dx+\left (3 a b^2\right ) \int \sinh ^2(c+d x) \, dx+b^3 \int \sinh ^5(c+d x) \, dx\\ &=-\frac{3 a^2 b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{3 a b^2 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{1}{2} \left (3 a b^2\right ) \int 1 \, dx+\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (c+d x)\right )}{d}+\frac{b^3 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{3}{2} a b^2 x-\frac{3 a^2 b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b^3 \cosh (c+d x)}{d}-\frac{2 b^3 \cosh ^3(c+d x)}{3 d}+\frac{b^3 \cosh ^5(c+d x)}{5 d}+\frac{a^3 \coth (c+d x)}{d}-\frac{a^3 \coth ^3(c+d x)}{3 d}+\frac{3 a b^2 \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.444961, size = 169, normalized size = 1.31 \[ \frac{720 a^2 b \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+80 a^3 \tanh \left (\frac{1}{2} (c+d x)\right )+80 a^3 \coth \left (\frac{1}{2} (c+d x)\right )+80 a^3 \sinh ^4\left (\frac{1}{2} (c+d x)\right ) \text{csch}^3(c+d x)-5 a^3 \sinh (c+d x) \text{csch}^4\left (\frac{1}{2} (c+d x)\right )+180 a b^2 \sinh (2 (c+d x))-360 a b^2 c-360 a b^2 d x+150 b^3 \cosh (c+d x)-25 b^3 \cosh (3 (c+d x))+3 b^3 \cosh (5 (c+d x))}{240 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^3)^3,x]

[Out]

(-360*a*b^2*c - 360*a*b^2*d*x + 150*b^3*Cosh[c + d*x] - 25*b^3*Cosh[3*(c + d*x)] + 3*b^3*Cosh[5*(c + d*x)] + 8
0*a^3*Coth[(c + d*x)/2] + 720*a^2*b*Log[Tanh[(c + d*x)/2]] + 80*a^3*Csch[c + d*x]^3*Sinh[(c + d*x)/2]^4 - 5*a^
3*Csch[(c + d*x)/2]^4*Sinh[c + d*x] + 180*a*b^2*Sinh[2*(c + d*x)] + 80*a^3*Tanh[(c + d*x)/2])/(240*d)

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Maple [A]  time = 0.077, size = 101, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{2}{3}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{3}} \right ){\rm coth} \left (dx+c\right )-6\,{a}^{2}b{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +3\,a{b}^{2} \left ( 1/2\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) -1/2\,dx-c/2 \right ) +{b}^{3} \left ({\frac{8}{15}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15}} \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)^4*(a+b*sinh(d*x+c)^3)^3,x)

[Out]

1/d*(a^3*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)-6*a^2*b*arctanh(exp(d*x+c))+3*a*b^2*(1/2*cosh(d*x+c)*sinh(d*x+c)-
1/2*d*x-1/2*c)+b^3*(8/15+1/5*sinh(d*x+c)^4-4/15*sinh(d*x+c)^2)*cosh(d*x+c))

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Maxima [B]  time = 1.20844, size = 351, normalized size = 2.72 \begin{align*} -\frac{3}{8} \, a b^{2}{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + \frac{1}{480} \, b^{3}{\left (\frac{3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac{25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac{150 \, e^{\left (d x + c\right )}}{d} + \frac{150 \, e^{\left (-d x - c\right )}}{d} - \frac{25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac{3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} - 3 \, 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}\right )} + \frac{4}{3} \, a^{3}{\left (\frac{3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac{1}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4*(a+b*sinh(d*x+c)^3)^3,x, algorithm="maxima")

[Out]

-3/8*a*b^2*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) + 1/480*b^3*(3*e^(5*d*x + 5*c)/d - 25*e^(3*d*x + 3*c
)/d + 150*e^(d*x + c)/d + 150*e^(-d*x - c)/d - 25*e^(-3*d*x - 3*c)/d + 3*e^(-5*d*x - 5*c)/d) - 3*a^2*b*(log(e^
(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d) + 4/3*a^3*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*
x - 4*c) + e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)))

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Fricas [B]  time = 2.43673, size = 10166, normalized size = 78.81 \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)^4*(a+b*sinh(d*x+c)^3)^3,x, algorithm="fricas")

[Out]

1/480*(3*b^3*cosh(d*x + c)^16 + 48*b^3*cosh(d*x + c)*sinh(d*x + c)^15 + 3*b^3*sinh(d*x + c)^16 - 34*b^3*cosh(d
*x + c)^14 + 180*a*b^2*cosh(d*x + c)^13 + 234*b^3*cosh(d*x + c)^12 + 2*(180*b^3*cosh(d*x + c)^2 - 17*b^3)*sinh
(d*x + c)^14 + 4*(420*b^3*cosh(d*x + c)^3 - 119*b^3*cosh(d*x + c) + 45*a*b^2)*sinh(d*x + c)^13 - 378*b^3*cosh(
d*x + c)^10 + 26*(210*b^3*cosh(d*x + c)^4 - 119*b^3*cosh(d*x + c)^2 + 90*a*b^2*cosh(d*x + c) + 9*b^3)*sinh(d*x
+ c)^12 - 180*(4*a*b^2*d*x + 3*a*b^2)*cosh(d*x + c)^11 + 4*(3276*b^3*cosh(d*x + c)^5 - 3094*b^3*cosh(d*x + c)
^3 - 180*a*b^2*d*x + 3510*a*b^2*cosh(d*x + c)^2 + 702*b^3*cosh(d*x + c) - 135*a*b^2)*sinh(d*x + c)^11 + 2*(120
12*b^3*cosh(d*x + c)^6 - 17017*b^3*cosh(d*x + c)^4 + 25740*a*b^2*cosh(d*x + c)^3 + 7722*b^3*cosh(d*x + c)^2 -
189*b^3 - 990*(4*a*b^2*d*x + 3*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^10 + 360*(6*a*b^2*d*x + a*b^2)*cosh(d*x + c
)^9 + 4*(8580*b^3*cosh(d*x + c)^7 - 17017*b^3*cosh(d*x + c)^5 + 32175*a*b^2*cosh(d*x + c)^4 + 12870*b^3*cosh(d
*x + c)^3 + 540*a*b^2*d*x - 945*b^3*cosh(d*x + c) + 90*a*b^2 - 2475*(4*a*b^2*d*x + 3*a*b^2)*cosh(d*x + c)^2)*s
inh(d*x + c)^9 + 378*b^3*cosh(d*x + c)^6 + 6*(6435*b^3*cosh(d*x + c)^8 - 17017*b^3*cosh(d*x + c)^6 + 38610*a*b
^2*cosh(d*x + c)^5 + 19305*b^3*cosh(d*x + c)^4 - 2835*b^3*cosh(d*x + c)^2 - 4950*(4*a*b^2*d*x + 3*a*b^2)*cosh(
d*x + c)^3 + 540*(6*a*b^2*d*x + a*b^2)*cosh(d*x + c))*sinh(d*x + c)^8 - 120*(18*a*b^2*d*x + 16*a^3 - 3*a*b^2)*
cosh(d*x + c)^7 + 24*(1430*b^3*cosh(d*x + c)^9 - 4862*b^3*cosh(d*x + c)^7 + 12870*a*b^2*cosh(d*x + c)^6 + 7722
*b^3*cosh(d*x + c)^5 - 1890*b^3*cosh(d*x + c)^3 - 90*a*b^2*d*x - 2475*(4*a*b^2*d*x + 3*a*b^2)*cosh(d*x + c)^4
- 80*a^3 + 15*a*b^2 + 540*(6*a*b^2*d*x + a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^7 - 234*b^3*cosh(d*x + c)^4 + 6
*(4004*b^3*cosh(d*x + c)^10 - 17017*b^3*cosh(d*x + c)^8 + 51480*a*b^2*cosh(d*x + c)^7 + 36036*b^3*cosh(d*x + c
)^6 - 13230*b^3*cosh(d*x + c)^4 - 13860*(4*a*b^2*d*x + 3*a*b^2)*cosh(d*x + c)^5 + 5040*(6*a*b^2*d*x + a*b^2)*c
osh(d*x + c)^3 + 63*b^3 - 140*(18*a*b^2*d*x + 16*a^3 - 3*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^6 + 180*a*b^2*cos
h(d*x + c)^3 + 20*(36*a*b^2*d*x + 32*a^3 - 27*a*b^2)*cosh(d*x + c)^5 + 4*(3276*b^3*cosh(d*x + c)^11 - 17017*b^
3*cosh(d*x + c)^9 + 57915*a*b^2*cosh(d*x + c)^8 + 46332*b^3*cosh(d*x + c)^7 - 23814*b^3*cosh(d*x + c)^5 - 2079
0*(4*a*b^2*d*x + 3*a*b^2)*cosh(d*x + c)^6 + 180*a*b^2*d*x + 11340*(6*a*b^2*d*x + a*b^2)*cosh(d*x + c)^4 + 567*
b^3*cosh(d*x + c) + 160*a^3 - 135*a*b^2 - 630*(18*a*b^2*d*x + 16*a^3 - 3*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)
^5 + 34*b^3*cosh(d*x + c)^2 + 2*(2730*b^3*cosh(d*x + c)^12 - 17017*b^3*cosh(d*x + c)^10 + 64350*a*b^2*cosh(d*x
+ c)^9 + 57915*b^3*cosh(d*x + c)^8 - 39690*b^3*cosh(d*x + c)^6 - 29700*(4*a*b^2*d*x + 3*a*b^2)*cosh(d*x + c)^
7 + 22680*(6*a*b^2*d*x + a*b^2)*cosh(d*x + c)^5 + 2835*b^3*cosh(d*x + c)^2 - 2100*(18*a*b^2*d*x + 16*a^3 - 3*a
*b^2)*cosh(d*x + c)^3 - 117*b^3 + 50*(36*a*b^2*d*x + 32*a^3 - 27*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(42
0*b^3*cosh(d*x + c)^13 - 3094*b^3*cosh(d*x + c)^11 + 12870*a*b^2*cosh(d*x + c)^10 + 12870*b^3*cosh(d*x + c)^9
- 11340*b^3*cosh(d*x + c)^7 - 7425*(4*a*b^2*d*x + 3*a*b^2)*cosh(d*x + c)^8 + 7560*(6*a*b^2*d*x + a*b^2)*cosh(d
*x + c)^6 + 1890*b^3*cosh(d*x + c)^3 - 1050*(18*a*b^2*d*x + 16*a^3 - 3*a*b^2)*cosh(d*x + c)^4 - 234*b^3*cosh(d
*x + c) + 45*a*b^2 + 50*(36*a*b^2*d*x + 32*a^3 - 27*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 - 3*b^3 + 2*(180*b
^3*cosh(d*x + c)^14 - 1547*b^3*cosh(d*x + c)^12 + 7020*a*b^2*cosh(d*x + c)^11 + 7722*b^3*cosh(d*x + c)^10 - 85
05*b^3*cosh(d*x + c)^8 - 4950*(4*a*b^2*d*x + 3*a*b^2)*cosh(d*x + c)^9 + 6480*(6*a*b^2*d*x + a*b^2)*cosh(d*x +
c)^7 + 2835*b^3*cosh(d*x + c)^4 - 1260*(18*a*b^2*d*x + 16*a^3 - 3*a*b^2)*cosh(d*x + c)^5 - 702*b^3*cosh(d*x +
c)^2 + 270*a*b^2*cosh(d*x + c) + 100*(36*a*b^2*d*x + 32*a^3 - 27*a*b^2)*cosh(d*x + c)^3 + 17*b^3)*sinh(d*x + c
)^2 - 1440*(a^2*b*cosh(d*x + c)^11 + 11*a^2*b*cosh(d*x + c)*sinh(d*x + c)^10 + a^2*b*sinh(d*x + c)^11 - 3*a^2*
b*cosh(d*x + c)^9 + 3*a^2*b*cosh(d*x + c)^7 + (55*a^2*b*cosh(d*x + c)^2 - 3*a^2*b)*sinh(d*x + c)^9 + 3*(55*a^2
*b*cosh(d*x + c)^3 - 9*a^2*b*cosh(d*x + c))*sinh(d*x + c)^8 - a^2*b*cosh(d*x + c)^5 + 3*(110*a^2*b*cosh(d*x +
c)^4 - 36*a^2*b*cosh(d*x + c)^2 + a^2*b)*sinh(d*x + c)^7 + 21*(22*a^2*b*cosh(d*x + c)^5 - 12*a^2*b*cosh(d*x +
c)^3 + a^2*b*cosh(d*x + c))*sinh(d*x + c)^6 + (462*a^2*b*cosh(d*x + c)^6 - 378*a^2*b*cosh(d*x + c)^4 + 63*a^2*
b*cosh(d*x + c)^2 - a^2*b)*sinh(d*x + c)^5 + (330*a^2*b*cosh(d*x + c)^7 - 378*a^2*b*cosh(d*x + c)^5 + 105*a^2*
b*cosh(d*x + c)^3 - 5*a^2*b*cosh(d*x + c))*sinh(d*x + c)^4 + (165*a^2*b*cosh(d*x + c)^8 - 252*a^2*b*cosh(d*x +
c)^6 + 105*a^2*b*cosh(d*x + c)^4 - 10*a^2*b*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (55*a^2*b*cosh(d*x + c)^9 - 10
8*a^2*b*cosh(d*x + c)^7 + 63*a^2*b*cosh(d*x + c)^5 - 10*a^2*b*cosh(d*x + c)^3)*sinh(d*x + c)^2 + (11*a^2*b*cos
h(d*x + c)^10 - 27*a^2*b*cosh(d*x + c)^8 + 21*a^2*b*cosh(d*x + c)^6 - 5*a^2*b*cosh(d*x + c)^4)*sinh(d*x + c))*
log(cosh(d*x + c) + sinh(d*x + c) + 1) + 1440*(a^2*b*cosh(d*x + c)^11 + 11*a^2*b*cosh(d*x + c)*sinh(d*x + c)^1
0 + a^2*b*sinh(d*x + c)^11 - 3*a^2*b*cosh(d*x + c)^9 + 3*a^2*b*cosh(d*x + c)^7 + (55*a^2*b*cosh(d*x + c)^2 - 3
*a^2*b)*sinh(d*x + c)^9 + 3*(55*a^2*b*cosh(d*x + c)^3 - 9*a^2*b*cosh(d*x + c))*sinh(d*x + c)^8 - a^2*b*cosh(d*
x + c)^5 + 3*(110*a^2*b*cosh(d*x + c)^4 - 36*a^2*b*cosh(d*x + c)^2 + a^2*b)*sinh(d*x + c)^7 + 21*(22*a^2*b*cos
h(d*x + c)^5 - 12*a^2*b*cosh(d*x + c)^3 + a^2*b*cosh(d*x + c))*sinh(d*x + c)^6 + (462*a^2*b*cosh(d*x + c)^6 -
378*a^2*b*cosh(d*x + c)^4 + 63*a^2*b*cosh(d*x + c)^2 - a^2*b)*sinh(d*x + c)^5 + (330*a^2*b*cosh(d*x + c)^7 - 3
78*a^2*b*cosh(d*x + c)^5 + 105*a^2*b*cosh(d*x + c)^3 - 5*a^2*b*cosh(d*x + c))*sinh(d*x + c)^4 + (165*a^2*b*cos
h(d*x + c)^8 - 252*a^2*b*cosh(d*x + c)^6 + 105*a^2*b*cosh(d*x + c)^4 - 10*a^2*b*cosh(d*x + c)^2)*sinh(d*x + c)
^3 + (55*a^2*b*cosh(d*x + c)^9 - 108*a^2*b*cosh(d*x + c)^7 + 63*a^2*b*cosh(d*x + c)^5 - 10*a^2*b*cosh(d*x + c)
^3)*sinh(d*x + c)^2 + (11*a^2*b*cosh(d*x + c)^10 - 27*a^2*b*cosh(d*x + c)^8 + 21*a^2*b*cosh(d*x + c)^6 - 5*a^2
*b*cosh(d*x + c)^4)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 4*(12*b^3*cosh(d*x + c)^15 - 119*b
^3*cosh(d*x + c)^13 + 585*a*b^2*cosh(d*x + c)^12 + 702*b^3*cosh(d*x + c)^11 - 945*b^3*cosh(d*x + c)^9 - 495*(4
*a*b^2*d*x + 3*a*b^2)*cosh(d*x + c)^10 + 810*(6*a*b^2*d*x + a*b^2)*cosh(d*x + c)^8 + 567*b^3*cosh(d*x + c)^5 -
210*(18*a*b^2*d*x + 16*a^3 - 3*a*b^2)*cosh(d*x + c)^6 - 234*b^3*cosh(d*x + c)^3 + 135*a*b^2*cosh(d*x + c)^2 +
25*(36*a*b^2*d*x + 32*a^3 - 27*a*b^2)*cosh(d*x + c)^4 + 17*b^3*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)
^11 + 11*d*cosh(d*x + c)*sinh(d*x + c)^10 + d*sinh(d*x + c)^11 - 3*d*cosh(d*x + c)^9 + (55*d*cosh(d*x + c)^2 -
3*d)*sinh(d*x + c)^9 + 3*(55*d*cosh(d*x + c)^3 - 9*d*cosh(d*x + c))*sinh(d*x + c)^8 + 3*d*cosh(d*x + c)^7 + 3
*(110*d*cosh(d*x + c)^4 - 36*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^7 + 21*(22*d*cosh(d*x + c)^5 - 12*d*cosh(d*x
+ c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^6 - d*cosh(d*x + c)^5 + (462*d*cosh(d*x + c)^6 - 378*d*cosh(d*x + c)^
4 + 63*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^5 + (330*d*cosh(d*x + c)^7 - 378*d*cosh(d*x + c)^5 + 105*d*cosh(d*
x + c)^3 - 5*d*cosh(d*x + c))*sinh(d*x + c)^4 + (165*d*cosh(d*x + c)^8 - 252*d*cosh(d*x + c)^6 + 105*d*cosh(d*
x + c)^4 - 10*d*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (55*d*cosh(d*x + c)^9 - 108*d*cosh(d*x + c)^7 + 63*d*cosh(d
*x + c)^5 - 10*d*cosh(d*x + c)^3)*sinh(d*x + c)^2 + (11*d*cosh(d*x + c)^10 - 27*d*cosh(d*x + c)^8 + 21*d*cosh(
d*x + c)^6 - 5*d*cosh(d*x + c)^4)*sinh(d*x + c))

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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)**4*(a+b*sinh(d*x+c)**3)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.43691, size = 419, normalized size = 3.25 \begin{align*} -\frac{3 \,{\left (d x + c\right )} a b^{2}}{2 \, d} - \frac{3 \, a^{2} b \log \left (e^{\left (d x + c\right )} + 1\right )}{d} + \frac{3 \, a^{2} b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d} + \frac{3 \, b^{3} d^{4} e^{\left (5 \, d x + 5 \, c\right )} - 25 \, b^{3} d^{4} e^{\left (3 \, d x + 3 \, c\right )} + 180 \, a b^{2} d^{4} e^{\left (2 \, d x + 2 \, c\right )} + 150 \, b^{3} d^{4} e^{\left (d x + c\right )}}{480 \, d^{5}} + \frac{{\left (150 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 180 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} - 475 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 528 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 234 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 180 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 34 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b^{3} - 60 \,{\left (32 \, a^{3} - 9 \, a b^{2}\right )} e^{\left (7 \, d x + 7 \, c\right )} + 20 \,{\left (32 \, a^{3} - 27 \, a b^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{480 \, d{\left (e^{\left (d x + c\right )} + 1\right )}^{3}{\left (e^{\left (d x + c\right )} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4*(a+b*sinh(d*x+c)^3)^3,x, algorithm="giac")

[Out]

-3/2*(d*x + c)*a*b^2/d - 3*a^2*b*log(e^(d*x + c) + 1)/d + 3*a^2*b*log(abs(e^(d*x + c) - 1))/d + 1/480*(3*b^3*d
^4*e^(5*d*x + 5*c) - 25*b^3*d^4*e^(3*d*x + 3*c) + 180*a*b^2*d^4*e^(2*d*x + 2*c) + 150*b^3*d^4*e^(d*x + c))/d^5
+ 1/480*(150*b^3*e^(10*d*x + 10*c) - 180*a*b^2*e^(9*d*x + 9*c) - 475*b^3*e^(8*d*x + 8*c) + 528*b^3*e^(6*d*x +
6*c) - 234*b^3*e^(4*d*x + 4*c) + 180*a*b^2*e^(3*d*x + 3*c) + 34*b^3*e^(2*d*x + 2*c) - 3*b^3 - 60*(32*a^3 - 9*
a*b^2)*e^(7*d*x + 7*c) + 20*(32*a^3 - 27*a*b^2)*e^(5*d*x + 5*c))*e^(-5*d*x - 5*c)/(d*(e^(d*x + c) + 1)^3*(e^(d
*x + c) - 1)^3)