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

Optimal. Leaf size=24 \[ -\frac {\text {csch}(a+b x)}{b}-\frac {\tan ^{-1}(\sinh (a+b x))}{b} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2621, 321, 207} \[ -\frac {\text {csch}(a+b x)}{b}-\frac {\tan ^{-1}(\sinh (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]^2*Sech[a + b*x],x]

[Out]

-(ArcTan[Sinh[a + b*x]]/b) - Csch[a + b*x]/b

Rule 207

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2621

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rubi steps

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

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Mathematica [C]  time = 0.02, size = 29, normalized size = 1.21 \[ -\frac {\text {csch}(a+b x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\sinh ^2(a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Csch[a + b*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Sinh[a + b*x]^2])/b)

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fricas [B]  time = 0.42, size = 103, normalized size = 4.29 \[ -\frac {2 \, {\left ({\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.17, size = 54, normalized size = 2.25 \[ -\frac {\pi + \frac {4}{e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}} + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 27, normalized size = 1.12 \[ -\frac {1}{b \sinh \left (b x +a \right )}-\frac {2 \arctan \left ({\mathrm e}^{b x +a}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b/sinh(b*x+a)-2*arctan(exp(b*x+a))/b

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maxima [A]  time = 0.51, size = 43, normalized size = 1.79 \[ \frac {2 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} + \frac {2 \, e^{\left (-b x - a\right )}}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.47, size = 48, normalized size = 2.00 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(a + b*x)*sinh(a + b*x)^2),x)

[Out]

- (2*atan((exp(b*x)*exp(a)*(b^2)^(1/2))/b))/(b^2)^(1/2) - (2*exp(a + b*x))/(b*(exp(2*a + 2*b*x) - 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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