3.64 \(\int \text {csch}^4(c+d x) (a+b \tanh ^3(c+d x))^2 \, dx\)

Optimal. Leaf size=97 \[ -\frac {a^2 \coth ^3(c+d x)}{3 d}+\frac {a^2 \coth (c+d x)}{d}-\frac {a b \tanh ^2(c+d x)}{d}+\frac {2 a b \log (\tanh (c+d x))}{d}-\frac {b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^2 \tanh ^3(c+d x)}{3 d} \]

[Out]

a^2*coth(d*x+c)/d-1/3*a^2*coth(d*x+c)^3/d+2*a*b*ln(tanh(d*x+c))/d-a*b*tanh(d*x+c)^2/d+1/3*b^2*tanh(d*x+c)^3/d-
1/5*b^2*tanh(d*x+c)^5/d

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Rubi [A]  time = 0.10, antiderivative size = 97, normalized size of antiderivative = 1.00, 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 = {3663, 1802} \[ -\frac {a^2 \coth ^3(c+d x)}{3 d}+\frac {a^2 \coth (c+d x)}{d}-\frac {a b \tanh ^2(c+d x)}{d}+\frac {2 a b \log (\tanh (c+d x))}{d}-\frac {b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^2 \tanh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^2,x]

[Out]

(a^2*Coth[c + d*x])/d - (a^2*Coth[c + d*x]^3)/(3*d) + (2*a*b*Log[Tanh[c + d*x]])/d - (a*b*Tanh[c + d*x]^2)/d +
(b^2*Tanh[c + d*x]^3)/(3*d) - (b^2*Tanh[c + d*x]^5)/(5*d)

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 3663

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff^(m + 1))/f, Subst[Int[(x^m*(a + b*(ff*x)^n)^p)/(c^2 + ff^2*x^2
)^(m/2 + 1), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2]

Rubi steps

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

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Mathematica [A]  time = 0.22, size = 147, normalized size = 1.52 \[ \frac {2 a^2 \coth (c+d x)}{3 d}-\frac {a^2 \coth (c+d x) \text {csch}^2(c+d x)}{3 d}+\frac {a b \text {sech}^2(c+d x)}{d}+\frac {2 a b \log (\sinh (c+d x))}{d}-\frac {2 a b \log (\cosh (c+d x))}{d}+\frac {2 b^2 \tanh (c+d x)}{15 d}-\frac {b^2 \tanh (c+d x) \text {sech}^4(c+d x)}{5 d}+\frac {b^2 \tanh (c+d x) \text {sech}^2(c+d x)}{15 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^2,x]

[Out]

(2*a^2*Coth[c + d*x])/(3*d) - (a^2*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (2*a*b*Log[Cosh[c + d*x]])/d + (2*a*
b*Log[Sinh[c + d*x]])/d + (a*b*Sech[c + d*x]^2)/d + (2*b^2*Tanh[c + d*x])/(15*d) + (b^2*Sech[c + d*x]^2*Tanh[c
+ d*x])/(15*d) - (b^2*Sech[c + d*x]^4*Tanh[c + d*x])/(5*d)

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fricas [B]  time = 0.62, size = 4125, normalized size = 42.53 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

2/15*(30*a*b*cosh(d*x + c)^14 + 420*a*b*cosh(d*x + c)*sinh(d*x + c)^13 + 30*a*b*sinh(d*x + c)^14 - 30*(a^2 + b
^2)*cosh(d*x + c)^12 + 30*(91*a*b*cosh(d*x + c)^2 - a^2 - b^2)*sinh(d*x + c)^12 + 120*(91*a*b*cosh(d*x + c)^3
- 3*(a^2 + b^2)*cosh(d*x + c))*sinh(d*x + c)^11 - 10*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^10 + 10*(3003*a*b
*cosh(d*x + c)^4 - 198*(a^2 + b^2)*cosh(d*x + c)^2 - 14*a^2 - 9*a*b + 10*b^2)*sinh(d*x + c)^10 + 20*(3003*a*b*
cosh(d*x + c)^5 - 330*(a^2 + b^2)*cosh(d*x + c)^3 - 5*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c))*sinh(d*x + c)^9
- 10*(25*a^2 + 13*b^2)*cosh(d*x + c)^8 + 10*(9009*a*b*cosh(d*x + c)^6 - 1485*(a^2 + b^2)*cosh(d*x + c)^4 - 45
*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^2 - 25*a^2 - 13*b^2)*sinh(d*x + c)^8 + 80*(1287*a*b*cosh(d*x + c)^7 -
297*(a^2 + b^2)*cosh(d*x + c)^5 - 15*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^3 - (25*a^2 + 13*b^2)*cosh(d*x +
c))*sinh(d*x + c)^7 - 2*(100*a^2 - 45*a*b - 44*b^2)*cosh(d*x + c)^6 + 2*(45045*a*b*cosh(d*x + c)^8 - 13860*(a
^2 + b^2)*cosh(d*x + c)^6 - 1050*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^4 - 140*(25*a^2 + 13*b^2)*cosh(d*x +
c)^2 - 100*a^2 + 45*a*b + 44*b^2)*sinh(d*x + c)^6 + 4*(15015*a*b*cosh(d*x + c)^9 - 5940*(a^2 + b^2)*cosh(d*x +
c)^7 - 630*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^5 - 140*(25*a^2 + 13*b^2)*cosh(d*x + c)^3 - 3*(100*a^2 - 4
5*a*b - 44*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(25*a^2 + 17*b^2)*cosh(d*x + c)^4 + 2*(15015*a*b*cosh(d*x +
c)^10 - 7425*(a^2 + b^2)*cosh(d*x + c)^8 - 1050*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^6 - 350*(25*a^2 + 13*
b^2)*cosh(d*x + c)^4 - 15*(100*a^2 - 45*a*b - 44*b^2)*cosh(d*x + c)^2 - 25*a^2 - 17*b^2)*sinh(d*x + c)^4 + 8*(
1365*a*b*cosh(d*x + c)^11 - 825*(a^2 + b^2)*cosh(d*x + c)^9 - 150*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^7 -
70*(25*a^2 + 13*b^2)*cosh(d*x + c)^5 - 5*(100*a^2 - 45*a*b - 44*b^2)*cosh(d*x + c)^3 - (25*a^2 + 17*b^2)*cosh(
d*x + c))*sinh(d*x + c)^3 + 2*(10*a^2 - 15*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(1365*a*b*cosh(d*x + c)^12 - 990*(
a^2 + b^2)*cosh(d*x + c)^10 - 225*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^8 - 140*(25*a^2 + 13*b^2)*cosh(d*x +
c)^6 - 15*(100*a^2 - 45*a*b - 44*b^2)*cosh(d*x + c)^4 - 6*(25*a^2 + 17*b^2)*cosh(d*x + c)^2 + 10*a^2 - 15*a*b
+ 2*b^2)*sinh(d*x + c)^2 + 10*a^2 + 2*b^2 - 15*(a*b*cosh(d*x + c)^16 + 16*a*b*cosh(d*x + c)*sinh(d*x + c)^15
+ a*b*sinh(d*x + c)^16 + 2*a*b*cosh(d*x + c)^14 + 2*(60*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^14 - 2*a*b*co
sh(d*x + c)^12 + 28*(20*a*b*cosh(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c)^13 + 2*(910*a*b*cosh(d*x + c)^4
+ 91*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^12 - 6*a*b*cosh(d*x + c)^10 + 8*(546*a*b*cosh(d*x + c)^5 + 91*a
*b*cosh(d*x + c)^3 - 3*a*b*cosh(d*x + c))*sinh(d*x + c)^11 + 2*(4004*a*b*cosh(d*x + c)^6 + 1001*a*b*cosh(d*x +
c)^4 - 66*a*b*cosh(d*x + c)^2 - 3*a*b)*sinh(d*x + c)^10 + 4*(2860*a*b*cosh(d*x + c)^7 + 1001*a*b*cosh(d*x + c
)^5 - 110*a*b*cosh(d*x + c)^3 - 15*a*b*cosh(d*x + c))*sinh(d*x + c)^9 + 6*(2145*a*b*cosh(d*x + c)^8 + 1001*a*b
*cosh(d*x + c)^6 - 165*a*b*cosh(d*x + c)^4 - 45*a*b*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 6*a*b*cosh(d*x + c)^6 +
16*(715*a*b*cosh(d*x + c)^9 + 429*a*b*cosh(d*x + c)^7 - 99*a*b*cosh(d*x + c)^5 - 45*a*b*cosh(d*x + c)^3)*sinh
(d*x + c)^7 + 2*(4004*a*b*cosh(d*x + c)^10 + 3003*a*b*cosh(d*x + c)^8 - 924*a*b*cosh(d*x + c)^6 - 630*a*b*cosh
(d*x + c)^4 + 3*a*b)*sinh(d*x + c)^6 + 2*a*b*cosh(d*x + c)^4 + 4*(1092*a*b*cosh(d*x + c)^11 + 1001*a*b*cosh(d*
x + c)^9 - 396*a*b*cosh(d*x + c)^7 - 378*a*b*cosh(d*x + c)^5 + 9*a*b*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(910*a
*b*cosh(d*x + c)^12 + 1001*a*b*cosh(d*x + c)^10 - 495*a*b*cosh(d*x + c)^8 - 630*a*b*cosh(d*x + c)^6 + 45*a*b*c
osh(d*x + c)^2 + a*b)*sinh(d*x + c)^4 - 2*a*b*cosh(d*x + c)^2 + 8*(70*a*b*cosh(d*x + c)^13 + 91*a*b*cosh(d*x +
c)^11 - 55*a*b*cosh(d*x + c)^9 - 90*a*b*cosh(d*x + c)^7 + 15*a*b*cosh(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*
x + c)^3 + 2*(60*a*b*cosh(d*x + c)^14 + 91*a*b*cosh(d*x + c)^12 - 66*a*b*cosh(d*x + c)^10 - 135*a*b*cosh(d*x +
c)^8 + 45*a*b*cosh(d*x + c)^4 + 6*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^2 - a*b + 4*(4*a*b*cosh(d*x + c)^1
5 + 7*a*b*cosh(d*x + c)^13 - 6*a*b*cosh(d*x + c)^11 - 15*a*b*cosh(d*x + c)^9 + 9*a*b*cosh(d*x + c)^5 + 2*a*b*c
osh(d*x + c)^3 - a*b*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 15*(
a*b*cosh(d*x + c)^16 + 16*a*b*cosh(d*x + c)*sinh(d*x + c)^15 + a*b*sinh(d*x + c)^16 + 2*a*b*cosh(d*x + c)^14 +
2*(60*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^14 - 2*a*b*cosh(d*x + c)^12 + 28*(20*a*b*cosh(d*x + c)^3 + a*b
*cosh(d*x + c))*sinh(d*x + c)^13 + 2*(910*a*b*cosh(d*x + c)^4 + 91*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^12
- 6*a*b*cosh(d*x + c)^10 + 8*(546*a*b*cosh(d*x + c)^5 + 91*a*b*cosh(d*x + c)^3 - 3*a*b*cosh(d*x + c))*sinh(d*
x + c)^11 + 2*(4004*a*b*cosh(d*x + c)^6 + 1001*a*b*cosh(d*x + c)^4 - 66*a*b*cosh(d*x + c)^2 - 3*a*b)*sinh(d*x
+ c)^10 + 4*(2860*a*b*cosh(d*x + c)^7 + 1001*a*b*cosh(d*x + c)^5 - 110*a*b*cosh(d*x + c)^3 - 15*a*b*cosh(d*x +
c))*sinh(d*x + c)^9 + 6*(2145*a*b*cosh(d*x + c)^8 + 1001*a*b*cosh(d*x + c)^6 - 165*a*b*cosh(d*x + c)^4 - 45*a
*b*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 6*a*b*cosh(d*x + c)^6 + 16*(715*a*b*cosh(d*x + c)^9 + 429*a*b*cosh(d*x +
c)^7 - 99*a*b*cosh(d*x + c)^5 - 45*a*b*cosh(d*x + c)^3)*sinh(d*x + c)^7 + 2*(4004*a*b*cosh(d*x + c)^10 + 3003
*a*b*cosh(d*x + c)^8 - 924*a*b*cosh(d*x + c)^6 - 630*a*b*cosh(d*x + c)^4 + 3*a*b)*sinh(d*x + c)^6 + 2*a*b*cosh
(d*x + c)^4 + 4*(1092*a*b*cosh(d*x + c)^11 + 1001*a*b*cosh(d*x + c)^9 - 396*a*b*cosh(d*x + c)^7 - 378*a*b*cosh
(d*x + c)^5 + 9*a*b*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(910*a*b*cosh(d*x + c)^12 + 1001*a*b*cosh(d*x + c)^10 -
495*a*b*cosh(d*x + c)^8 - 630*a*b*cosh(d*x + c)^6 + 45*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^4 - 2*a*b*cos
h(d*x + c)^2 + 8*(70*a*b*cosh(d*x + c)^13 + 91*a*b*cosh(d*x + c)^11 - 55*a*b*cosh(d*x + c)^9 - 90*a*b*cosh(d*x
+ c)^7 + 15*a*b*cosh(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(60*a*b*cosh(d*x + c)^14 + 91*a*b*co
sh(d*x + c)^12 - 66*a*b*cosh(d*x + c)^10 - 135*a*b*cosh(d*x + c)^8 + 45*a*b*cosh(d*x + c)^4 + 6*a*b*cosh(d*x +
c)^2 - a*b)*sinh(d*x + c)^2 - a*b + 4*(4*a*b*cosh(d*x + c)^15 + 7*a*b*cosh(d*x + c)^13 - 6*a*b*cosh(d*x + c)^
11 - 15*a*b*cosh(d*x + c)^9 + 9*a*b*cosh(d*x + c)^5 + 2*a*b*cosh(d*x + c)^3 - a*b*cosh(d*x + c))*sinh(d*x + c)
)*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 4*(105*a*b*cosh(d*x + c)^13 - 90*(a^2 + b^2)*cosh(d*x
+ c)^11 - 25*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^9 - 20*(25*a^2 + 13*b^2)*cosh(d*x + c)^7 - 3*(100*a^2 -
45*a*b - 44*b^2)*cosh(d*x + c)^5 - 2*(25*a^2 + 17*b^2)*cosh(d*x + c)^3 + (10*a^2 - 15*a*b + 2*b^2)*cosh(d*x +
c))*sinh(d*x + c))/(d*cosh(d*x + c)^16 + 16*d*cosh(d*x + c)*sinh(d*x + c)^15 + d*sinh(d*x + c)^16 + 2*d*cosh(d
*x + c)^14 + 2*(60*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^14 + 28*(20*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(
d*x + c)^13 - 2*d*cosh(d*x + c)^12 + 2*(910*d*cosh(d*x + c)^4 + 91*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^12 + 8
*(546*d*cosh(d*x + c)^5 + 91*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh(d*x + c)^11 - 6*d*cosh(d*x + c)^10 +
2*(4004*d*cosh(d*x + c)^6 + 1001*d*cosh(d*x + c)^4 - 66*d*cosh(d*x + c)^2 - 3*d)*sinh(d*x + c)^10 + 4*(2860*d*
cosh(d*x + c)^7 + 1001*d*cosh(d*x + c)^5 - 110*d*cosh(d*x + c)^3 - 15*d*cosh(d*x + c))*sinh(d*x + c)^9 + 6*(21
45*d*cosh(d*x + c)^8 + 1001*d*cosh(d*x + c)^6 - 165*d*cosh(d*x + c)^4 - 45*d*cosh(d*x + c)^2)*sinh(d*x + c)^8
+ 16*(715*d*cosh(d*x + c)^9 + 429*d*cosh(d*x + c)^7 - 99*d*cosh(d*x + c)^5 - 45*d*cosh(d*x + c)^3)*sinh(d*x +
c)^7 + 6*d*cosh(d*x + c)^6 + 2*(4004*d*cosh(d*x + c)^10 + 3003*d*cosh(d*x + c)^8 - 924*d*cosh(d*x + c)^6 - 630
*d*cosh(d*x + c)^4 + 3*d)*sinh(d*x + c)^6 + 4*(1092*d*cosh(d*x + c)^11 + 1001*d*cosh(d*x + c)^9 - 396*d*cosh(d
*x + c)^7 - 378*d*cosh(d*x + c)^5 + 9*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*d*cosh(d*x + c)^4 + 2*(910*d*cosh(d
*x + c)^12 + 1001*d*cosh(d*x + c)^10 - 495*d*cosh(d*x + c)^8 - 630*d*cosh(d*x + c)^6 + 45*d*cosh(d*x + c)^2 +
d)*sinh(d*x + c)^4 + 8*(70*d*cosh(d*x + c)^13 + 91*d*cosh(d*x + c)^11 - 55*d*cosh(d*x + c)^9 - 90*d*cosh(d*x +
c)^7 + 15*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^3 - 2*d*cosh(d*x + c)^2 + 2*(60*d*cosh(d*x + c)^
14 + 91*d*cosh(d*x + c)^12 - 66*d*cosh(d*x + c)^10 - 135*d*cosh(d*x + c)^8 + 45*d*cosh(d*x + c)^4 + 6*d*cosh(d
*x + c)^2 - d)*sinh(d*x + c)^2 + 4*(4*d*cosh(d*x + c)^15 + 7*d*cosh(d*x + c)^13 - 6*d*cosh(d*x + c)^11 - 15*d*
cosh(d*x + c)^9 + 9*d*cosh(d*x + c)^5 + 2*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c) - d)

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giac [B]  time = 0.33, size = 249, normalized size = 2.57 \[ -\frac {60 \, a b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 60 \, a b \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {10 \, {\left (11 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 33 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 12 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 33 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a^{2} - 11 \, a b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} - \frac {137 \, a b e^{\left (10 \, d x + 10 \, c\right )} + 805 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 1730 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 120 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1730 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 40 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 805 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 40 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 137 \, a b - 8 \, b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{30 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^2,x, algorithm="giac")

[Out]

-1/30*(60*a*b*log(e^(2*d*x + 2*c) + 1) - 60*a*b*log(abs(e^(2*d*x + 2*c) - 1)) + 10*(11*a*b*e^(6*d*x + 6*c) - 3
3*a*b*e^(4*d*x + 4*c) + 12*a^2*e^(2*d*x + 2*c) + 33*a*b*e^(2*d*x + 2*c) - 4*a^2 - 11*a*b)/(e^(2*d*x + 2*c) - 1
)^3 - (137*a*b*e^(10*d*x + 10*c) + 805*a*b*e^(8*d*x + 8*c) + 1730*a*b*e^(6*d*x + 6*c) - 120*b^2*e^(6*d*x + 6*c
) + 1730*a*b*e^(4*d*x + 4*c) + 40*b^2*e^(4*d*x + 4*c) + 805*a*b*e^(2*d*x + 2*c) - 40*b^2*e^(2*d*x + 2*c) + 137
*a*b - 8*b^2)/(e^(2*d*x + 2*c) + 1)^5)/d

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maple [A]  time = 0.49, size = 146, normalized size = 1.51 \[ \frac {2 a^{2} \coth \left (d x +c \right )}{3 d}-\frac {a^{2} \coth \left (d x +c \right ) \mathrm {csch}\left (d x +c \right )^{2}}{3 d}+\frac {a b}{d \cosh \left (d x +c \right )^{2}}+\frac {2 a b \ln \left (\tanh \left (d x +c \right )\right )}{d}-\frac {b^{2} \sinh \left (d x +c \right )}{4 d \cosh \left (d x +c \right )^{5}}+\frac {2 b^{2} \tanh \left (d x +c \right )}{15 d}+\frac {b^{2} \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{4}}{20 d}+\frac {b^{2} \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{2}}{15 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^2,x)

[Out]

2/3*a^2*coth(d*x+c)/d-1/3/d*a^2*coth(d*x+c)*csch(d*x+c)^2+1/d*a*b/cosh(d*x+c)^2+2*a*b*ln(tanh(d*x+c))/d-1/4/d*
b^2*sinh(d*x+c)/cosh(d*x+c)^5+2/15*b^2*tanh(d*x+c)/d+1/20/d*b^2*tanh(d*x+c)*sech(d*x+c)^4+1/15/d*b^2*tanh(d*x+
c)*sech(d*x+c)^2

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maxima [B]  time = 0.41, size = 468, normalized size = 4.82 \[ 2 \, a b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {4}{15} \, b^{2} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} - \frac {5 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {15 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + \frac {4}{3} \, a^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

2*a*b*(log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d - log(e^(-2*d*x - 2*c) + 1)/d + 2*e^(-2*d*x - 2*c)/(d
*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1))) + 4/15*b^2*(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)) - 5*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)) +
15*e^(-6*d*x - 6*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))) + 4/3*a^2*(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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mupad [B]  time = 0.26, size = 344, normalized size = 3.55 \[ \frac {40\,b^2}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {4\,a^2}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,a^2}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {4\,\left (b^2+a\,b\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {16\,b^2}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {32\,b^2}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {4\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-d^2}}{d\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {-d^2}}+\frac {4\,a\,b}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tanh(c + d*x)^3)^2/sinh(c + d*x)^4,x)

[Out]

(40*b^2)/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) - (4*a^2)/(d*(exp(4*c + 4*d*x)
- 2*exp(2*c + 2*d*x) + 1)) - (8*a^2)/(3*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) -
(4*(a*b + b^2))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (16*b^2)/(d*(4*exp(2*c + 2*d*x) + 6*exp(4*c
+ 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) + (32*b^2)/(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)) - (4*atan((a*b*exp(2*c)*exp(2*d*x)*(
-d^2)^(1/2))/(d*(a^2*b^2)^(1/2)))*(a^2*b^2)^(1/2))/(-d^2)^(1/2) + (4*a*b)/(d*(exp(2*c + 2*d*x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{2} \operatorname {csch}^{4}{\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**4*(a+b*tanh(d*x+c)**3)**2,x)

[Out]

Integral((a + b*tanh(c + d*x)**3)**2*csch(c + d*x)**4, x)

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