### 3.70 $$\int \text{csch}^2(c+d x) (a+b \tanh ^3(c+d x))^3 \, dx$$

Optimal. Leaf size=71 $\frac{3 a^2 b \tanh ^2(c+d x)}{2 d}-\frac{a^3 \coth (c+d x)}{d}+\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^8(c+d x)}{8 d}$

[Out]

-((a^3*Coth[c + d*x])/d) + (3*a^2*b*Tanh[c + d*x]^2)/(2*d) + (3*a*b^2*Tanh[c + d*x]^5)/(5*d) + (b^3*Tanh[c + d
*x]^8)/(8*d)

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Rubi [A]  time = 0.0645756, antiderivative size = 71, normalized size of antiderivative = 1., 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, 270} $\frac{3 a^2 b \tanh ^2(c+d x)}{2 d}-\frac{a^3 \coth (c+d x)}{d}+\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^8(c+d x)}{8 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((a^3*Coth[c + d*x])/d) + (3*a^2*b*Tanh[c + d*x]^2)/(2*d) + (3*a*b^2*Tanh[c + d*x]^5)/(5*d) + (b^3*Tanh[c + d
*x]^8)/(8*d)

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]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \text{csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^3\right )^3}{x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^3}{x^2}+3 a^2 b x+3 a b^2 x^4+b^3 x^7\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{a^3 \coth (c+d x)}{d}+\frac{3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^8(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.687407, size = 113, normalized size = 1.59 $\frac{b \left (-4 \text{sech}^2(c+d x) \left (15 a^2+12 a b \tanh (c+d x)+5 b^2\right )+24 a b \tanh (c+d x)+6 b \text{sech}^4(c+d x) (4 a \tanh (c+d x)+5 b)+5 b^2 \text{sech}^8(c+d x)-20 b^2 \text{sech}^6(c+d x)\right )-40 a^3 \coth (c+d x)}{40 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-40*a^3*Coth[c + d*x] + b*(-20*b^2*Sech[c + d*x]^6 + 5*b^2*Sech[c + d*x]^8 + 24*a*b*Tanh[c + d*x] + 6*b*Sech[
c + d*x]^4*(5*b + 4*a*Tanh[c + d*x]) - 4*Sech[c + d*x]^2*(15*a^2 + 5*b^2 + 12*a*b*Tanh[c + d*x])))/(40*d)

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Maple [B]  time = 0.086, size = 223, normalized size = 3.1 \begin{align*}{\frac{1}{d} \left ( -{a}^{3}{\rm coth} \left (dx+c\right )+{\frac{3\,{a}^{2}b \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+3\,a{b}^{2} \left ( -1/2\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}-3/8\,{\frac{\sinh \left ( dx+c \right ) }{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+3/8\, \left ({\frac{8}{15}}+1/5\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}+{\frac{4\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{15}} \right ) \tanh \left ( dx+c \right ) \right ) +{b}^{3} \left ( -{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{8}}}-{\frac{3\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{4\, \left ( \cosh \left ( dx+c \right ) \right ) ^{8}}}-{\frac{3\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{8\, \left ( \cosh \left ( dx+c \right ) \right ) ^{8}}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{8\, \left ( \cosh \left ( dx+c \right ) \right ) ^{6}}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{8\, \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{8\, \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}} \right ) \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-a^3*coth(d*x+c)+3/2*a^2*b*sinh(d*x+c)^2/cosh(d*x+c)^2+3*a*b^2*(-1/2*sinh(d*x+c)^3/cosh(d*x+c)^5-3/8*sinh
(d*x+c)/cosh(d*x+c)^5+3/8*(8/15+1/5*sech(d*x+c)^4+4/15*sech(d*x+c)^2)*tanh(d*x+c))+b^3*(-1/2*sinh(d*x+c)^6/cos
h(d*x+c)^8-3/4*sinh(d*x+c)^4/cosh(d*x+c)^8-3/8*sinh(d*x+c)^2/cosh(d*x+c)^8+1/8*sinh(d*x+c)^2/cosh(d*x+c)^6+1/8
*sinh(d*x+c)^2/cosh(d*x+c)^4+1/8*sinh(d*x+c)^2/cosh(d*x+c)^2))

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Maxima [B]  time = 1.19483, size = 917, normalized size = 12.92 \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)^2*(a+b*tanh(d*x+c)^3)^3,x, algorithm="maxima")

[Out]

-2*b^3*(e^(-2*d*x - 2*c)/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8
*c) + 56*e^(-10*d*x - 10*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)) + 7*e^(-
6*d*x - 6*c)/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(
-10*d*x - 10*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)) + 7*e^(-10*d*x - 10*
c)/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x -
10*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)) + e^(-14*d*x - 14*c)/(d*(8*e^(
-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) + 28*e
^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1))) + 6/5*a*b^2*(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)) + 5
*e^(-8*d*x - 8*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))) + 2*a^3/(d*(e^(-2*d*x - 2*c) - 1)) - 6*a^2*b/(d*(e^(d*x + c) + e^(-d*x - c))^
2)

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Fricas [B]  time = 2.23443, size = 3071, normalized size = 43.25 \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)^2*(a+b*tanh(d*x+c)^3)^3,x, algorithm="fricas")

[Out]

-2/5*((10*a^3 + 15*a^2*b + 12*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 8*(15*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)*s
inh(d*x + c)^7 + (10*a^3 + 15*a^2*b + 12*a*b^2 + 5*b^3)*sinh(d*x + c)^8 + 2*(40*a^3 + 30*a^2*b + 12*a*b^2 - 5*
b^3)*cosh(d*x + c)^6 + 2*(40*a^3 + 30*a^2*b + 12*a*b^2 - 5*b^3 + 14*(10*a^3 + 15*a^2*b + 12*a*b^2 + 5*b^3)*cos
h(d*x + c)^2)*sinh(d*x + c)^6 + 4*(14*(15*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 27*(5*a^2*b + 2*a*b^2)*c
osh(d*x + c))*sinh(d*x + c)^5 + 20*(14*a^3 + 3*a^2*b + 2*b^3)*cosh(d*x + c)^4 + 10*(7*(10*a^3 + 15*a^2*b + 12*
a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 28*a^3 + 6*a^2*b + 4*b^3 + 3*(40*a^3 + 30*a^2*b + 12*a*b^2 - 5*b^3)*cosh(d*x
+ c)^2)*sinh(d*x + c)^4 + 8*(7*(15*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 45*(5*a^2*b + 2*a*b^2)*cosh(d*x
+ c)^3 + 15*(7*a^2*b + 2*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 350*a^3 - 75*a^2*b - 12*a*b^2 + 35*b^3
+ 2*(280*a^3 - 30*a^2*b - 12*a*b^2 - 35*b^3)*cosh(d*x + c)^2 + 2*(14*(10*a^3 + 15*a^2*b + 12*a*b^2 + 5*b^3)*c
osh(d*x + c)^6 + 15*(40*a^3 + 30*a^2*b + 12*a*b^2 - 5*b^3)*cosh(d*x + c)^4 + 280*a^3 - 30*a^2*b - 12*a*b^2 - 3
5*b^3 + 60*(14*a^3 + 3*a^2*b + 2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(2*(15*a^2*b + 18*a*b^2 + 5*b^3)*co
sh(d*x + c)^7 + 27*(5*a^2*b + 2*a*b^2)*cosh(d*x + c)^5 + 30*(7*a^2*b + 2*a*b^2 + b^3)*cosh(d*x + c)^3 + 21*(5*
a^2*b + 2*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^10 + 10*d*cosh(d*x + c)*sinh(d*x + c)^9 + d*si
nh(d*x + c)^10 + 6*d*cosh(d*x + c)^8 + 3*(15*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^8 + 8*(15*d*cosh(d*x + c)^
3 + 8*d*cosh(d*x + c))*sinh(d*x + c)^7 + 13*d*cosh(d*x + c)^6 + (210*d*cosh(d*x + c)^4 + 168*d*cosh(d*x + c)^2
+ 13*d)*sinh(d*x + c)^6 + 2*(126*d*cosh(d*x + c)^5 + 224*d*cosh(d*x + c)^3 + 81*d*cosh(d*x + c))*sinh(d*x + c
)^5 + 8*d*cosh(d*x + c)^4 + (210*d*cosh(d*x + c)^6 + 420*d*cosh(d*x + c)^4 + 195*d*cosh(d*x + c)^2 + 8*d)*sinh
(d*x + c)^4 + 4*(30*d*cosh(d*x + c)^7 + 112*d*cosh(d*x + c)^5 + 135*d*cosh(d*x + c)^3 + 48*d*cosh(d*x + c))*si
nh(d*x + c)^3 - 14*d*cosh(d*x + c)^2 + (45*d*cosh(d*x + c)^8 + 168*d*cosh(d*x + c)^6 + 195*d*cosh(d*x + c)^4 +
48*d*cosh(d*x + c)^2 - 14*d)*sinh(d*x + c)^2 + 2*(5*d*cosh(d*x + c)^9 + 32*d*cosh(d*x + c)^7 + 81*d*cosh(d*x
+ c)^5 + 96*d*cosh(d*x + c)^3 + 42*d*cosh(d*x + c))*sinh(d*x + c) - 14*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 )^{3} \operatorname{csch}^{2}{\left (c + d x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 2.49159, size = 420, normalized size = 5.92 \begin{align*} -\frac{2 \,{\left (\frac{5 \, a^{3}}{e^{\left (2 \, d x + 2 \, c\right )} - 1} + \frac{15 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 15 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 5 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 90 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 45 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 225 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 75 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 35 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 300 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 105 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 225 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 93 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 35 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 39 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}\right )}}{5 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(5*a^3/(e^(2*d*x + 2*c) - 1) + (15*a^2*b*e^(14*d*x + 14*c) + 15*a*b^2*e^(14*d*x + 14*c) + 5*b^3*e^(14*d*x
+ 14*c) + 90*a^2*b*e^(12*d*x + 12*c) + 45*a*b^2*e^(12*d*x + 12*c) + 225*a^2*b*e^(10*d*x + 10*c) + 75*a*b^2*e^
(10*d*x + 10*c) + 35*b^3*e^(10*d*x + 10*c) + 300*a^2*b*e^(8*d*x + 8*c) + 105*a*b^2*e^(8*d*x + 8*c) + 225*a^2*b
*e^(6*d*x + 6*c) + 93*a*b^2*e^(6*d*x + 6*c) + 35*b^3*e^(6*d*x + 6*c) + 90*a^2*b*e^(4*d*x + 4*c) + 39*a*b^2*e^(
4*d*x + 4*c) + 15*a^2*b*e^(2*d*x + 2*c) + 9*a*b^2*e^(2*d*x + 2*c) + 5*b^3*e^(2*d*x + 2*c) + 3*a*b^2)/(e^(2*d*x
+ 2*c) + 1)^8)/d