### 3.30 $$\int \text{csch}^2(a+b x) \text{sech}^2(a+b x) \, dx$$

Optimal. Leaf size=23 $-\frac{\tanh (a+b x)}{b}-\frac{\coth (a+b x)}{b}$

[Out]

-(Coth[a + b*x]/b) - Tanh[a + b*x]/b

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Rubi [A]  time = 0.0310056, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {2620, 14} $-\frac{\tanh (a+b x)}{b}-\frac{\coth (a+b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(Coth[a + b*x]/b) - Tanh[a + b*x]/b

Rule 2620

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

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}^2(a+b x) \text{sech}^2(a+b x) \, dx &=\frac{i \operatorname{Subst}\left (\int \frac{1+x^2}{x^2} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac{i \operatorname{Subst}\left (\int \left (1+\frac{1}{x^2}\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=-\frac{\coth (a+b x)}{b}-\frac{\tanh (a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0149412, size = 13, normalized size = 0.57 $-\frac{2 \coth (2 (a+b x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*Coth[2*(a + b*x)])/b

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Maple [A]  time = 0.013, size = 32, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ( -{\frac{1}{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }}-2\,\tanh \left ( bx+a \right ) \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12956, size = 24, normalized size = 1.04 \begin{align*} \frac{4}{b{\left (e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/(b*(e^(-4*b*x - 4*a) - 1))

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Fricas [B]  time = 1.94948, size = 213, normalized size = 9.26 \begin{align*} -\frac{4}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/(b*cosh(b*x + a)^4 + 4*b*cosh(b*x + a)^3*sinh(b*x + a) + 6*b*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*b*cosh(b*x
+ a)*sinh(b*x + a)^3 + b*sinh(b*x + a)^4 - b)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)**2*sech(b*x+a)**2,x)

[Out]

Integral(csch(a + b*x)**2*sech(a + b*x)**2, x)

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Giac [A]  time = 1.17244, size = 24, normalized size = 1.04 \begin{align*} -\frac{4}{b{\left (e^{\left (4 \, b x + 4 \, a\right )} - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/(b*(e^(4*b*x + 4*a) - 1))