### 3.63 $$\int \text{csch}^3(c+d x) (a+b \tanh ^3(c+d x))^2 \, dx$$

Optimal. Leaf size=107 $\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{a b \tan ^{-1}(\sinh (c+d x))}{d}+\frac{a b \tanh (c+d x) \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^5(c+d x)}{5 d}-\frac{b^2 \text{sech}^3(c+d x)}{3 d}$

[Out]

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

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Rubi [A]  time = 0.159902, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.217, Rules used = {3666, 3768, 3770, 2606, 14} $\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{a b \tan ^{-1}(\sinh (c+d x))}{d}+\frac{a b \tanh (c+d x) \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^5(c+d x)}{5 d}-\frac{b^2 \text{sech}^3(c+d x)}{3 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 3666

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n},
x] && IGtQ[p, 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]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.150823, size = 138, normalized size = 1.29 $-\frac{a^2 \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a^2 \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a^2 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+\frac{2 a b \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{d}+\frac{a b \tanh (c+d x) \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^5(c+d x)}{5 d}-\frac{b^2 \text{sech}^3(c+d x)}{3 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*a*b*ArcTan[Tanh[(c + d*x)/2]])/d - (a^2*Csch[(c + d*x)/2]^2)/(8*d) - (a^2*Log[Tanh[(c + d*x)/2]])/(2*d) - (
a^2*Sech[(c + d*x)/2]^2)/(8*d) - (b^2*Sech[c + d*x]^3)/(3*d) + (b^2*Sech[c + d*x]^5)/(5*d) + (a*b*Sech[c + d*x
]*Tanh[c + d*x])/d

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Maple [A]  time = 0.085, size = 154, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2}{\rm coth} \left (dx+c\right ){\rm csch} \left (dx+c\right )}{2\,d}}+{\frac{{a}^{2}{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{d}}+{\frac{ab{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{d}}+2\,{\frac{ab\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}-{\frac{{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{5\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+{\frac{2\,{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+{\frac{2\,{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15\,d\cosh \left ( dx+c \right ) }}-{\frac{2\,{b}^{2}\cosh \left ( dx+c \right ) }{15\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^2*coth(d*x+c)*csch(d*x+c)/d+1/d*a^2*arctanh(exp(d*x+c))+1/d*a*b*sech(d*x+c)*tanh(d*x+c)+2/d*a*b*arctan(
exp(d*x+c))-1/5/d*b^2*sinh(d*x+c)^2/cosh(d*x+c)^5+2/15/d*b^2*sinh(d*x+c)^2/cosh(d*x+c)^3+2/15/d*b^2*sinh(d*x+c
)^2/cosh(d*x+c)-2/15*b^2*cosh(d*x+c)/d

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Maxima [B]  time = 1.5607, size = 510, normalized size = 4.77 \begin{align*} -2 \, a b{\left (\frac{\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, 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{1}{2} \, a^{2}{\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{8}{15} \, b^{2}{\left (\frac{5 \, e^{\left (-3 \, d x - 3 \, 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{2 \, e^{\left (-5 \, d x - 5 \, 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 (-7 \, d x - 7 \, 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 )}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*b*(arctan(e^(-d*x - c))/d - (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))) + 1/2*a^2*(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))) - 8/15*b^2*(5*e^(-3*d*x - 3*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)) - 2*e^(-5*d*x - 5*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^(-7*d*x - 7*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)))

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Fricas [B]  time = 2.99178, size = 12523, normalized size = 117.04 \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)^3*(a+b*tanh(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

-1/30*(30*(a^2 - 2*a*b)*cosh(d*x + c)^13 + 390*(a^2 - 2*a*b)*cosh(d*x + c)*sinh(d*x + c)^12 + 30*(a^2 - 2*a*b)
*sinh(d*x + c)^13 + 20*(9*a^2 + 4*b^2)*cosh(d*x + c)^11 + 20*(117*(a^2 - 2*a*b)*cosh(d*x + c)^2 + 9*a^2 + 4*b^
2)*sinh(d*x + c)^11 + 220*(39*(a^2 - 2*a*b)*cosh(d*x + c)^3 + (9*a^2 + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^10
+ 6*(75*a^2 + 30*a*b - 32*b^2)*cosh(d*x + c)^9 + 2*(10725*(a^2 - 2*a*b)*cosh(d*x + c)^4 + 550*(9*a^2 + 4*b^2)*
cosh(d*x + c)^2 + 225*a^2 + 90*a*b - 96*b^2)*sinh(d*x + c)^9 + 6*(6435*(a^2 - 2*a*b)*cosh(d*x + c)^5 + 550*(9*
a^2 + 4*b^2)*cosh(d*x + c)^3 + 9*(75*a^2 + 30*a*b - 32*b^2)*cosh(d*x + c))*sinh(d*x + c)^8 + 8*(75*a^2 + 28*b^
2)*cosh(d*x + c)^7 + 8*(6435*(a^2 - 2*a*b)*cosh(d*x + c)^6 + 825*(9*a^2 + 4*b^2)*cosh(d*x + c)^4 + 27*(75*a^2
+ 30*a*b - 32*b^2)*cosh(d*x + c)^2 + 75*a^2 + 28*b^2)*sinh(d*x + c)^7 + 8*(6435*(a^2 - 2*a*b)*cosh(d*x + c)^7
+ 1155*(9*a^2 + 4*b^2)*cosh(d*x + c)^5 + 63*(75*a^2 + 30*a*b - 32*b^2)*cosh(d*x + c)^3 + 7*(75*a^2 + 28*b^2)*c
osh(d*x + c))*sinh(d*x + c)^6 + 6*(75*a^2 - 30*a*b - 32*b^2)*cosh(d*x + c)^5 + 6*(6435*(a^2 - 2*a*b)*cosh(d*x
+ c)^8 + 1540*(9*a^2 + 4*b^2)*cosh(d*x + c)^6 + 126*(75*a^2 + 30*a*b - 32*b^2)*cosh(d*x + c)^4 + 28*(75*a^2 +
28*b^2)*cosh(d*x + c)^2 + 75*a^2 - 30*a*b - 32*b^2)*sinh(d*x + c)^5 + 2*(10725*(a^2 - 2*a*b)*cosh(d*x + c)^9 +
3300*(9*a^2 + 4*b^2)*cosh(d*x + c)^7 + 378*(75*a^2 + 30*a*b - 32*b^2)*cosh(d*x + c)^5 + 140*(75*a^2 + 28*b^2)
*cosh(d*x + c)^3 + 15*(75*a^2 - 30*a*b - 32*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 20*(9*a^2 + 4*b^2)*cosh(d*x
+ c)^3 + 4*(2145*(a^2 - 2*a*b)*cosh(d*x + c)^10 + 825*(9*a^2 + 4*b^2)*cosh(d*x + c)^8 + 126*(75*a^2 + 30*a*b -
32*b^2)*cosh(d*x + c)^6 + 70*(75*a^2 + 28*b^2)*cosh(d*x + c)^4 + 15*(75*a^2 - 30*a*b - 32*b^2)*cosh(d*x + c)^
2 + 45*a^2 + 20*b^2)*sinh(d*x + c)^3 + 4*(585*(a^2 - 2*a*b)*cosh(d*x + c)^11 + 275*(9*a^2 + 4*b^2)*cosh(d*x +
c)^9 + 54*(75*a^2 + 30*a*b - 32*b^2)*cosh(d*x + c)^7 + 42*(75*a^2 + 28*b^2)*cosh(d*x + c)^5 + 15*(75*a^2 - 30*
a*b - 32*b^2)*cosh(d*x + c)^3 + 15*(9*a^2 + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 60*(a*b*cosh(d*x + c)^14 +
14*a*b*cosh(d*x + c)*sinh(d*x + c)^13 + a*b*sinh(d*x + c)^14 + 3*a*b*cosh(d*x + c)^12 + (91*a*b*cosh(d*x + c)
^2 + 3*a*b)*sinh(d*x + c)^12 + a*b*cosh(d*x + c)^10 + 4*(91*a*b*cosh(d*x + c)^3 + 9*a*b*cosh(d*x + c))*sinh(d*
x + c)^11 + (1001*a*b*cosh(d*x + c)^4 + 198*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^10 - 5*a*b*cosh(d*x + c)^
8 + 2*(1001*a*b*cosh(d*x + c)^5 + 330*a*b*cosh(d*x + c)^3 + 5*a*b*cosh(d*x + c))*sinh(d*x + c)^9 + (3003*a*b*c
osh(d*x + c)^6 + 1485*a*b*cosh(d*x + c)^4 + 45*a*b*cosh(d*x + c)^2 - 5*a*b)*sinh(d*x + c)^8 - 5*a*b*cosh(d*x +
c)^6 + 8*(429*a*b*cosh(d*x + c)^7 + 297*a*b*cosh(d*x + c)^5 + 15*a*b*cosh(d*x + c)^3 - 5*a*b*cosh(d*x + c))*s
inh(d*x + c)^7 + (3003*a*b*cosh(d*x + c)^8 + 2772*a*b*cosh(d*x + c)^6 + 210*a*b*cosh(d*x + c)^4 - 140*a*b*cosh
(d*x + c)^2 - 5*a*b)*sinh(d*x + c)^6 + a*b*cosh(d*x + c)^4 + 2*(1001*a*b*cosh(d*x + c)^9 + 1188*a*b*cosh(d*x +
c)^7 + 126*a*b*cosh(d*x + c)^5 - 140*a*b*cosh(d*x + c)^3 - 15*a*b*cosh(d*x + c))*sinh(d*x + c)^5 + (1001*a*b*
cosh(d*x + c)^10 + 1485*a*b*cosh(d*x + c)^8 + 210*a*b*cosh(d*x + c)^6 - 350*a*b*cosh(d*x + c)^4 - 75*a*b*cosh(
d*x + c)^2 + a*b)*sinh(d*x + c)^4 + 3*a*b*cosh(d*x + c)^2 + 4*(91*a*b*cosh(d*x + c)^11 + 165*a*b*cosh(d*x + c)
^9 + 30*a*b*cosh(d*x + c)^7 - 70*a*b*cosh(d*x + c)^5 - 25*a*b*cosh(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x +
c)^3 + (91*a*b*cosh(d*x + c)^12 + 198*a*b*cosh(d*x + c)^10 + 45*a*b*cosh(d*x + c)^8 - 140*a*b*cosh(d*x + c)^6
- 75*a*b*cosh(d*x + c)^4 + 6*a*b*cosh(d*x + c)^2 + 3*a*b)*sinh(d*x + c)^2 + a*b + 2*(7*a*b*cosh(d*x + c)^13 +
18*a*b*cosh(d*x + c)^11 + 5*a*b*cosh(d*x + c)^9 - 20*a*b*cosh(d*x + c)^7 - 15*a*b*cosh(d*x + c)^5 + 2*a*b*cosh
(d*x + c)^3 + 3*a*b*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) + 30*(a^2 + 2*a*b)*cos
h(d*x + c) - 15*(a^2*cosh(d*x + c)^14 + 14*a^2*cosh(d*x + c)*sinh(d*x + c)^13 + a^2*sinh(d*x + c)^14 + 3*a^2*c
osh(d*x + c)^12 + (91*a^2*cosh(d*x + c)^2 + 3*a^2)*sinh(d*x + c)^12 + a^2*cosh(d*x + c)^10 + 4*(91*a^2*cosh(d*
x + c)^3 + 9*a^2*cosh(d*x + c))*sinh(d*x + c)^11 + (1001*a^2*cosh(d*x + c)^4 + 198*a^2*cosh(d*x + c)^2 + a^2)*
sinh(d*x + c)^10 - 5*a^2*cosh(d*x + c)^8 + 2*(1001*a^2*cosh(d*x + c)^5 + 330*a^2*cosh(d*x + c)^3 + 5*a^2*cosh(
d*x + c))*sinh(d*x + c)^9 + (3003*a^2*cosh(d*x + c)^6 + 1485*a^2*cosh(d*x + c)^4 + 45*a^2*cosh(d*x + c)^2 - 5*
a^2)*sinh(d*x + c)^8 - 5*a^2*cosh(d*x + c)^6 + 8*(429*a^2*cosh(d*x + c)^7 + 297*a^2*cosh(d*x + c)^5 + 15*a^2*c
osh(d*x + c)^3 - 5*a^2*cosh(d*x + c))*sinh(d*x + c)^7 + (3003*a^2*cosh(d*x + c)^8 + 2772*a^2*cosh(d*x + c)^6 +
210*a^2*cosh(d*x + c)^4 - 140*a^2*cosh(d*x + c)^2 - 5*a^2)*sinh(d*x + c)^6 + a^2*cosh(d*x + c)^4 + 2*(1001*a^
2*cosh(d*x + c)^9 + 1188*a^2*cosh(d*x + c)^7 + 126*a^2*cosh(d*x + c)^5 - 140*a^2*cosh(d*x + c)^3 - 15*a^2*cosh
(d*x + c))*sinh(d*x + c)^5 + (1001*a^2*cosh(d*x + c)^10 + 1485*a^2*cosh(d*x + c)^8 + 210*a^2*cosh(d*x + c)^6 -
350*a^2*cosh(d*x + c)^4 - 75*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^4 + 3*a^2*cosh(d*x + c)^2 + 4*(91*a^2*c
osh(d*x + c)^11 + 165*a^2*cosh(d*x + c)^9 + 30*a^2*cosh(d*x + c)^7 - 70*a^2*cosh(d*x + c)^5 - 25*a^2*cosh(d*x
+ c)^3 + a^2*cosh(d*x + c))*sinh(d*x + c)^3 + (91*a^2*cosh(d*x + c)^12 + 198*a^2*cosh(d*x + c)^10 + 45*a^2*cos
h(d*x + c)^8 - 140*a^2*cosh(d*x + c)^6 - 75*a^2*cosh(d*x + c)^4 + 6*a^2*cosh(d*x + c)^2 + 3*a^2)*sinh(d*x + c)
^2 + a^2 + 2*(7*a^2*cosh(d*x + c)^13 + 18*a^2*cosh(d*x + c)^11 + 5*a^2*cosh(d*x + c)^9 - 20*a^2*cosh(d*x + c)^
7 - 15*a^2*cosh(d*x + c)^5 + 2*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + s
inh(d*x + c) + 1) + 15*(a^2*cosh(d*x + c)^14 + 14*a^2*cosh(d*x + c)*sinh(d*x + c)^13 + a^2*sinh(d*x + c)^14 +
3*a^2*cosh(d*x + c)^12 + (91*a^2*cosh(d*x + c)^2 + 3*a^2)*sinh(d*x + c)^12 + a^2*cosh(d*x + c)^10 + 4*(91*a^2*
cosh(d*x + c)^3 + 9*a^2*cosh(d*x + c))*sinh(d*x + c)^11 + (1001*a^2*cosh(d*x + c)^4 + 198*a^2*cosh(d*x + c)^2
+ a^2)*sinh(d*x + c)^10 - 5*a^2*cosh(d*x + c)^8 + 2*(1001*a^2*cosh(d*x + c)^5 + 330*a^2*cosh(d*x + c)^3 + 5*a^
2*cosh(d*x + c))*sinh(d*x + c)^9 + (3003*a^2*cosh(d*x + c)^6 + 1485*a^2*cosh(d*x + c)^4 + 45*a^2*cosh(d*x + c)
^2 - 5*a^2)*sinh(d*x + c)^8 - 5*a^2*cosh(d*x + c)^6 + 8*(429*a^2*cosh(d*x + c)^7 + 297*a^2*cosh(d*x + c)^5 + 1
5*a^2*cosh(d*x + c)^3 - 5*a^2*cosh(d*x + c))*sinh(d*x + c)^7 + (3003*a^2*cosh(d*x + c)^8 + 2772*a^2*cosh(d*x +
c)^6 + 210*a^2*cosh(d*x + c)^4 - 140*a^2*cosh(d*x + c)^2 - 5*a^2)*sinh(d*x + c)^6 + a^2*cosh(d*x + c)^4 + 2*(
1001*a^2*cosh(d*x + c)^9 + 1188*a^2*cosh(d*x + c)^7 + 126*a^2*cosh(d*x + c)^5 - 140*a^2*cosh(d*x + c)^3 - 15*a
^2*cosh(d*x + c))*sinh(d*x + c)^5 + (1001*a^2*cosh(d*x + c)^10 + 1485*a^2*cosh(d*x + c)^8 + 210*a^2*cosh(d*x +
c)^6 - 350*a^2*cosh(d*x + c)^4 - 75*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^4 + 3*a^2*cosh(d*x + c)^2 + 4*(9
1*a^2*cosh(d*x + c)^11 + 165*a^2*cosh(d*x + c)^9 + 30*a^2*cosh(d*x + c)^7 - 70*a^2*cosh(d*x + c)^5 - 25*a^2*co
sh(d*x + c)^3 + a^2*cosh(d*x + c))*sinh(d*x + c)^3 + (91*a^2*cosh(d*x + c)^12 + 198*a^2*cosh(d*x + c)^10 + 45*
a^2*cosh(d*x + c)^8 - 140*a^2*cosh(d*x + c)^6 - 75*a^2*cosh(d*x + c)^4 + 6*a^2*cosh(d*x + c)^2 + 3*a^2)*sinh(d
*x + c)^2 + a^2 + 2*(7*a^2*cosh(d*x + c)^13 + 18*a^2*cosh(d*x + c)^11 + 5*a^2*cosh(d*x + c)^9 - 20*a^2*cosh(d*
x + c)^7 - 15*a^2*cosh(d*x + c)^5 + 2*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x +
c) + sinh(d*x + c) - 1) + 2*(195*(a^2 - 2*a*b)*cosh(d*x + c)^12 + 110*(9*a^2 + 4*b^2)*cosh(d*x + c)^10 + 27*(
75*a^2 + 30*a*b - 32*b^2)*cosh(d*x + c)^8 + 28*(75*a^2 + 28*b^2)*cosh(d*x + c)^6 + 15*(75*a^2 - 30*a*b - 32*b^
2)*cosh(d*x + c)^4 + 30*(9*a^2 + 4*b^2)*cosh(d*x + c)^2 + 15*a^2 + 30*a*b)*sinh(d*x + c))/(d*cosh(d*x + c)^14
+ 14*d*cosh(d*x + c)*sinh(d*x + c)^13 + d*sinh(d*x + c)^14 + 3*d*cosh(d*x + c)^12 + (91*d*cosh(d*x + c)^2 + 3*
d)*sinh(d*x + c)^12 + 4*(91*d*cosh(d*x + c)^3 + 9*d*cosh(d*x + c))*sinh(d*x + c)^11 + d*cosh(d*x + c)^10 + (10
01*d*cosh(d*x + c)^4 + 198*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^10 + 2*(1001*d*cosh(d*x + c)^5 + 330*d*cosh(d*
x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)^9 - 5*d*cosh(d*x + c)^8 + (3003*d*cosh(d*x + c)^6 + 1485*d*cosh(d*
x + c)^4 + 45*d*cosh(d*x + c)^2 - 5*d)*sinh(d*x + c)^8 + 8*(429*d*cosh(d*x + c)^7 + 297*d*cosh(d*x + c)^5 + 15
*d*cosh(d*x + c)^3 - 5*d*cosh(d*x + c))*sinh(d*x + c)^7 - 5*d*cosh(d*x + c)^6 + (3003*d*cosh(d*x + c)^8 + 2772
*d*cosh(d*x + c)^6 + 210*d*cosh(d*x + c)^4 - 140*d*cosh(d*x + c)^2 - 5*d)*sinh(d*x + c)^6 + 2*(1001*d*cosh(d*x
+ c)^9 + 1188*d*cosh(d*x + c)^7 + 126*d*cosh(d*x + c)^5 - 140*d*cosh(d*x + c)^3 - 15*d*cosh(d*x + c))*sinh(d*
x + c)^5 + d*cosh(d*x + c)^4 + (1001*d*cosh(d*x + c)^10 + 1485*d*cosh(d*x + c)^8 + 210*d*cosh(d*x + c)^6 - 350
*d*cosh(d*x + c)^4 - 75*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^4 + 4*(91*d*cosh(d*x + c)^11 + 165*d*cosh(d*x + c
)^9 + 30*d*cosh(d*x + c)^7 - 70*d*cosh(d*x + c)^5 - 25*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^3 +
3*d*cosh(d*x + c)^2 + (91*d*cosh(d*x + c)^12 + 198*d*cosh(d*x + c)^10 + 45*d*cosh(d*x + c)^8 - 140*d*cosh(d*x
+ c)^6 - 75*d*cosh(d*x + c)^4 + 6*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^2 + 2*(7*d*cosh(d*x + c)^13 + 18*d*co
sh(d*x + c)^11 + 5*d*cosh(d*x + c)^9 - 20*d*cosh(d*x + c)^7 - 15*d*cosh(d*x + c)^5 + 2*d*cosh(d*x + c)^3 + 3*d
*cosh(d*x + c))*sinh(d*x + c) + d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{2} \operatorname{csch}^{3}{\left (c + d x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.5929, size = 259, normalized size = 2.42 \begin{align*} \frac{60 \, a b \arctan \left (e^{\left (d x + c\right )}\right ) + 15 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) - 15 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac{30 \,{\left (a^{2} e^{\left (3 \, d x + 3 \, c\right )} + a^{2} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} + \frac{4 \,{\left (15 \, a b e^{\left (9 \, d x + 9 \, c\right )} + 30 \, a b e^{\left (7 \, d x + 7 \, c\right )} - 20 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 8 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 30 \, a b e^{\left (3 \, d x + 3 \, c\right )} - 20 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 15 \, a b e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{30 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(60*a*b*arctan(e^(d*x + c)) + 15*a^2*log(e^(d*x + c) + 1) - 15*a^2*log(abs(e^(d*x + c) - 1)) - 30*(a^2*e^
(3*d*x + 3*c) + a^2*e^(d*x + c))/(e^(2*d*x + 2*c) - 1)^2 + 4*(15*a*b*e^(9*d*x + 9*c) + 30*a*b*e^(7*d*x + 7*c)
- 20*b^2*e^(7*d*x + 7*c) + 8*b^2*e^(5*d*x + 5*c) - 30*a*b*e^(3*d*x + 3*c) - 20*b^2*e^(3*d*x + 3*c) - 15*a*b*e^
(d*x + c))/(e^(2*d*x + 2*c) + 1)^5)/d