### 3.16 $$\int \text{csch}^4(c+d x) (a+b \tanh ^2(c+d x))^2 \, dx$$

Optimal. Leaf size=72 $-\frac{a^2 \coth ^3(c+d x)}{3 d}-\frac{b (2 a-b) \tanh (c+d x)}{d}+\frac{a (a-2 b) \coth (c+d x)}{d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}$

[Out]

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

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Rubi [A]  time = 0.0789086, antiderivative size = 72, 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, 448} $-\frac{a^2 \coth ^3(c+d x)}{3 d}-\frac{b (2 a-b) \tanh (c+d x)}{d}+\frac{a (a-2 b) \coth (c+d x)}{d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a*(a - 2*b)*Coth[c + d*x])/d - (a^2*Coth[c + d*x]^3)/(3*d) - ((2*a - b)*b*Tanh[c + d*x])/d - (b^2*Tanh[c + d*
x]^3)/(3*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 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

Mathematica [A]  time = 0.48016, size = 59, normalized size = 0.82 $\frac{b \tanh (c+d x) \left (-6 a+b \text{sech}^2(c+d x)+2 b\right )-a \coth (c+d x) \left (a \text{csch}^2(c+d x)-2 a+6 b\right )}{3 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-(a*Coth[c + d*x]*(-2*a + 6*b + a*Csch[c + d*x]^2)) + b*(-6*a + 2*b + b*Sech[c + d*x]^2)*Tanh[c + d*x])/(3*d)

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Maple [A]  time = 0.059, size = 81, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ({\frac{2}{3}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{3}} \right ){\rm coth} \left (dx+c\right )+2\,ab \left ( -{\frac{1}{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }}-2\,\tanh \left ( dx+c \right ) \right ) +{b}^{2} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{3}} \right ) \tanh \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*tanh(d*x+c)^2)^2,x)

[Out]

1/d*(a^2*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)+2*a*b*(-1/sinh(d*x+c)/cosh(d*x+c)-2*tanh(d*x+c))+b^2*(2/3+1/3*sec
h(d*x+c)^2)*tanh(d*x+c))

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Maxima [B]  time = 1.05125, size = 284, normalized size = 3.94 \begin{align*} \frac{4}{3} \, b^{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 )} + \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 )} + \frac{8 \, a b}{d{\left (e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*b^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))) + 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))) + 8*a*b/(d*(e^(-4*d*x - 4*c) - 1))

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Fricas [B]  time = 1.93208, size = 1011, normalized size = 14.04 \begin{align*} -\frac{8 \,{\left ({\left (a^{2} + 6 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 8 \,{\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (a^{2} + 6 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \,{\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \,{\left (a^{2} + 6 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} - 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, a^{2} - 6 \, a b + 3 \, b^{2} + 8 \,{\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{3} +{\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}{3 \,{\left (d \cosh \left (d x + c\right )^{8} + 56 \, d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{5} + 28 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{6} + 8 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + d \sinh \left (d x + c\right )^{8} - 4 \, d \cosh \left (d x + c\right )^{4} + 2 \,{\left (35 \, d \cosh \left (d x + c\right )^{4} - 2 \, d\right )} \sinh \left (d x + c\right )^{4} + 8 \,{\left (7 \, d \cosh \left (d x + c\right )^{5} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \,{\left (7 \, d \cosh \left (d x + c\right )^{6} - 6 \, d \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \,{\left (d \cosh \left (d x + c\right )^{7} - d \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right ) + 3 \, d\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/3*((a^2 + 6*a*b + b^2)*cosh(d*x + c)^4 + 8*(a^2 + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 6*a*b + b^2)*
sinh(d*x + c)^4 + 4*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 6*a*b + b^2)*cosh(d*x + c)^2 + 2*a^2 - 2*b^2)*si
nh(d*x + c)^2 + 3*a^2 - 6*a*b + 3*b^2 + 8*((a^2 + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*x +
c))/(d*cosh(d*x + c)^8 + 56*d*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*d*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*d*co
sh(d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 - 4*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4 - 2*d)*sinh(d*
x + c)^4 + 8*(7*d*cosh(d*x + c)^5 - d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*d*cosh(d*x + c)^6 - 6*d*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + 8*(d*cosh(d*x + c)^7 - d*cosh(d*x + c)^3)*sinh(d*x + c) + 3*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2} \operatorname{csch}^{4}{\left (c + d x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.41702, size = 193, normalized size = 2.68 \begin{align*} -\frac{4 \,{\left (3 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 6 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 3 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 8 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} - 8 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 6 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 12 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - a^{2} + 6 \, a b - b^{2}\right )}}{3 \, d{\left (e^{\left (4 \, d x + 4 \, 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*tanh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

-4/3*(3*a^2*e^(8*d*x + 8*c) + 6*a*b*e^(8*d*x + 8*c) + 3*b^2*e^(8*d*x + 8*c) + 8*a^2*e^(6*d*x + 6*c) - 8*b^2*e^
(6*d*x + 6*c) + 6*a^2*e^(4*d*x + 4*c) - 12*a*b*e^(4*d*x + 4*c) + 6*b^2*e^(4*d*x + 4*c) - a^2 + 6*a*b - b^2)/(d
*(e^(4*d*x + 4*c) - 1)^3)