### 3.116 $$\int \coth ^3(a+b x) \text{csch}^2(a+b x) \, dx$$

Optimal. Leaf size=15 $-\frac{\coth ^4(a+b x)}{4 b}$

[Out]

-Coth[a + b*x]^4/(4*b)

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Rubi [A]  time = 0.0290491, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {2607, 30} $-\frac{\coth ^4(a+b x)}{4 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[a + b*x]^3*Csch[a + b*x]^2,x]

[Out]

-Coth[a + b*x]^4/(4*b)

Rule 2607

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.004639, size = 15, normalized size = 1. $-\frac{\coth ^4(a+b x)}{4 b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Coth[a + b*x]^3*Csch[a + b*x]^2,x]

[Out]

-Coth[a + b*x]^4/(4*b)

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Maple [B]  time = 0.022, size = 42, normalized size = 2.8 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{4\, \left ( \sinh \left ( bx+a \right ) \right ) ^{4}}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{4\, \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(coth(b*x+a)^3*csch(b*x+a)^2,x)

[Out]

1/b*(-1/4/sinh(b*x+a)^4*cosh(b*x+a)^2-1/4*cosh(b*x+a)^2/sinh(b*x+a)^2)

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Maxima [A]  time = 1.04203, size = 18, normalized size = 1.2 \begin{align*} -\frac{\coth \left (b x + a\right )^{4}}{4 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*coth(b*x + a)^4/b

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Fricas [B]  time = 1.85037, size = 568, normalized size = 37.87 \begin{align*} -\frac{2 \,{\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} +{\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{5} + 5 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + b \sinh \left (b x + a\right )^{5} - 3 \, b \cosh \left (b x + a\right )^{3} + 5 \,{\left (2 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{3} +{\left (10 \, b \cosh \left (b x + a\right )^{3} - 9 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) + 5 \,{\left (b \cosh \left (b x + a\right )^{4} - 3 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)^3*csch(b*x+a)^2,x, algorithm="fricas")

[Out]

-2*(cosh(b*x + a)^3 + 3*cosh(b*x + a)*sinh(b*x + a)^2 + sinh(b*x + a)^3 + (3*cosh(b*x + a)^2 - 1)*sinh(b*x + a
) + cosh(b*x + a))/(b*cosh(b*x + a)^5 + 5*b*cosh(b*x + a)*sinh(b*x + a)^4 + b*sinh(b*x + a)^5 - 3*b*cosh(b*x +
a)^3 + 5*(2*b*cosh(b*x + a)^2 - b)*sinh(b*x + a)^3 + (10*b*cosh(b*x + a)^3 - 9*b*cosh(b*x + a))*sinh(b*x + a)
^2 + 2*b*cosh(b*x + a) + 5*(b*cosh(b*x + a)^4 - 3*b*cosh(b*x + a)^2 + 2*b)*sinh(b*x + a))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)**3*csch(b*x+a)**2,x)

[Out]

Integral(coth(a + b*x)**3*csch(a + b*x)**2, x)

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Giac [B]  time = 1.25696, size = 50, normalized size = 3.33 \begin{align*} -\frac{2 \,{\left (e^{\left (6 \, b x + 6 \, a\right )} + e^{\left (2 \, b x + 2 \, a\right )}\right )}}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(e^(6*b*x + 6*a) + e^(2*b*x + 2*a))/(b*(e^(2*b*x + 2*a) - 1)^4)