3.40 \(\int \text{csch}^4(a+b x) \text{sech}^2(a+b x) \, dx\)

Optimal. Leaf size=37 \[ \frac{\tanh (a+b x)}{b}-\frac{\coth ^3(a+b x)}{3 b}+\frac{2 \coth (a+b x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.0367932, antiderivative size = 37, 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, 270} \[ \frac{\tanh (a+b x)}{b}-\frac{\coth ^3(a+b x)}{3 b}+\frac{2 \coth (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]^4*Sech[a + b*x]^2,x]

[Out]

(2*Coth[a + b*x])/b - Coth[a + b*x]^3/(3*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 270

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

Rubi steps

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

Mathematica [A]  time = 0.0391928, size = 45, normalized size = 1.22 \[ \frac{\tanh (a+b x)}{b}+\frac{5 \coth (a+b x)}{3 b}-\frac{\coth (a+b x) \text{csch}^2(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x]^4*Sech[a + b*x]^2,x]

[Out]

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

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Maple [A]  time = 0.02, size = 50, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ( -{\frac{1}{3\,\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}}+{\frac{4}{3\,\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }}+{\frac{8\,\tanh \left ( bx+a \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)^4*sech(b*x+a)^2,x)

[Out]

1/b*(-1/3/sinh(b*x+a)^3/cosh(b*x+a)+4/3/sinh(b*x+a)/cosh(b*x+a)+8/3*tanh(b*x+a))

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Maxima [B]  time = 1.11659, size = 122, normalized size = 3.3 \begin{align*} \frac{32 \, e^{\left (-2 \, b x - 2 \, a\right )}}{3 \, b{\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, e^{\left (-6 \, b x - 6 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )}} - \frac{16}{3 \, b{\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, e^{\left (-6 \, b x - 6 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32/3*e^(-2*b*x - 2*a)/(b*(2*e^(-2*b*x - 2*a) - 2*e^(-6*b*x - 6*a) + e^(-8*b*x - 8*a) - 1)) - 16/3/(b*(2*e^(-2*
b*x - 2*a) - 2*e^(-6*b*x - 6*a) + e^(-8*b*x - 8*a) - 1))

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Fricas [B]  time = 2.16805, size = 622, normalized size = 16.81 \begin{align*} -\frac{16 \,{\left (\cosh \left (b x + a\right ) + 3 \, \sinh \left (b x + a\right )\right )}}{3 \,{\left (b \cosh \left (b x + a\right )^{7} + 7 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{6} + b \sinh \left (b x + a\right )^{7} - 2 \, b \cosh \left (b x + a\right )^{5} +{\left (21 \, b \cosh \left (b x + a\right )^{2} - 2 \, b\right )} \sinh \left (b x + a\right )^{5} + 5 \,{\left (7 \, b \cosh \left (b x + a\right )^{3} - 2 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{4} + 5 \,{\left (7 \, b \cosh \left (b x + a\right )^{4} - 4 \, b \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{3} +{\left (21 \, b \cosh \left (b x + a\right )^{5} - 20 \, b \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )^{2} + b \cosh \left (b x + a\right ) +{\left (7 \, b \cosh \left (b x + a\right )^{6} - 10 \, b \cosh \left (b x + a\right )^{4} + 3 \, b\right )} \sinh \left (b x + a\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/3*(cosh(b*x + a) + 3*sinh(b*x + a))/(b*cosh(b*x + a)^7 + 7*b*cosh(b*x + a)*sinh(b*x + a)^6 + b*sinh(b*x +
a)^7 - 2*b*cosh(b*x + a)^5 + (21*b*cosh(b*x + a)^2 - 2*b)*sinh(b*x + a)^5 + 5*(7*b*cosh(b*x + a)^3 - 2*b*cosh(
b*x + a))*sinh(b*x + a)^4 + 5*(7*b*cosh(b*x + a)^4 - 4*b*cosh(b*x + a)^2)*sinh(b*x + a)^3 + (21*b*cosh(b*x + a
)^5 - 20*b*cosh(b*x + a)^3)*sinh(b*x + a)^2 + b*cosh(b*x + a) + (7*b*cosh(b*x + a)^6 - 10*b*cosh(b*x + a)^4 +
3*b)*sinh(b*x + a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}^{4}{\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)**4*sech(b*x+a)**2,x)

[Out]

Integral(csch(a + b*x)**4*sech(a + b*x)**2, x)

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Giac [A]  time = 1.19365, size = 82, normalized size = 2.22 \begin{align*} -\frac{2}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} + \frac{2 \,{\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} - 12 \, e^{\left (2 \, b x + 2 \, a\right )} + 5\right )}}{3 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^4*sech(b*x+a)^2,x, algorithm="giac")

[Out]

-2/(b*(e^(2*b*x + 2*a) + 1)) + 2/3*(3*e^(4*b*x + 4*a) - 12*e^(2*b*x + 2*a) + 5)/(b*(e^(2*b*x + 2*a) - 1)^3)