### 3.1057 $$\int \frac{-\text{csch}^2(a+b x)+\text{sech}^2(a+b x)}{\text{csch}^2(a+b x)+\text{sech}^2(a+b x)} \, dx$$

Optimal. Leaf size=12 $-\frac{\tan ^{-1}(\tanh (a+b x))}{b}$

[Out]

-(ArcTan[Tanh[a + b*x]]/b)

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Rubi [A]  time = 0.263468, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 39, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.026, Rules used = {204} $-\frac{\tan ^{-1}(\tanh (a+b x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[(-Csch[a + b*x]^2 + Sech[a + b*x]^2)/(Csch[a + b*x]^2 + Sech[a + b*x]^2),x]

[Out]

-(ArcTan[Tanh[a + b*x]]/b)

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0076216, size = 17, normalized size = 1.42 $-\frac{\tan ^{-1}(\sinh (2 a+2 b x))}{2 b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-Csch[a + b*x]^2 + Sech[a + b*x]^2)/(Csch[a + b*x]^2 + Sech[a + b*x]^2),x]

[Out]

-ArcTan[Sinh[2*a + 2*b*x]]/(2*b)

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Maple [B]  time = 0.101, size = 148, normalized size = 12.3 \begin{align*} 2\,{\frac{\sqrt{2}}{b \left ( 2+2\,\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\tanh \left ( 1/2\,bx+a/2 \right ) }{2+2\,\sqrt{2}}} \right ) }+2\,{\frac{1}{b \left ( 2+2\,\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\tanh \left ( 1/2\,bx+a/2 \right ) }{2+2\,\sqrt{2}}} \right ) }-2\,{\frac{\sqrt{2}}{b \left ( -2+2\,\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\tanh \left ( 1/2\,bx+a/2 \right ) }{-2+2\,\sqrt{2}}} \right ) }+2\,{\frac{1}{b \left ( -2+2\,\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\tanh \left ( 1/2\,bx+a/2 \right ) }{-2+2\,\sqrt{2}}} \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)/(csch(b*x+a)^2+sech(b*x+a)^2),x)

[Out]

2/b*2^(1/2)/(2+2*2^(1/2))*arctan(2*tanh(1/2*b*x+1/2*a)/(2+2*2^(1/2)))+2/b/(2+2*2^(1/2))*arctan(2*tanh(1/2*b*x+
1/2*a)/(2+2*2^(1/2)))-2/b*2^(1/2)/(-2+2*2^(1/2))*arctan(2*tanh(1/2*b*x+1/2*a)/(-2+2*2^(1/2)))+2/b/(-2+2*2^(1/2
))*arctan(2*tanh(1/2*b*x+1/2*a)/(-2+2*2^(1/2)))

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Maxima [B]  time = 1.58104, size = 68, normalized size = 5.67 \begin{align*} -\frac{\arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{\left (-b x - a\right )}\right )}\right ) - \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{\left (-b x - a\right )}\right )}\right )}{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)/(csch(b*x+a)^2+sech(b*x+a)^2),x, algorithm="maxima")

[Out]

-(arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^(-b*x - a))) - arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^(-b*x - a))))/b

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Fricas [B]  time = 1.94676, size = 103, normalized size = 8.58 \begin{align*} \frac{\arctan \left (-\frac{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{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)/(csch(b*x+a)^2+sech(b*x+a)^2),x, algorithm="fricas")

[Out]

arctan(-(cosh(b*x + a) + sinh(b*x + a))/(cosh(b*x + a) - sinh(b*x + a)))/b

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

[Out]

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

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Giac [A]  time = 1.16565, size = 20, normalized size = 1.67 \begin{align*} -\frac{\arctan \left (e^{\left (2 \, b x + 2 \, a\right )}\right )}{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)/(csch(b*x+a)^2+sech(b*x+a)^2),x, algorithm="giac")

[Out]

-arctan(e^(2*b*x + 2*a))/b