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

Optimal. Leaf size=53 \[ -\frac{\tanh ^3(a+b x)}{3 b}+\frac{3 \tanh (a+b x)}{b}-\frac{\coth ^3(a+b x)}{3 b}+\frac{3 \coth (a+b x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.0395316, antiderivative size = 53, 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} \[ -\frac{\tanh ^3(a+b x)}{3 b}+\frac{3 \tanh (a+b x)}{b}-\frac{\coth ^3(a+b x)}{3 b}+\frac{3 \coth (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Coth[a + b*x])/b - Coth[a + b*x]^3/(3*b) + (3*Tanh[a + b*x])/b - Tanh[a + b*x]^3/(3*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}^4(a+b x) \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x^4} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (3+\frac{1}{x^4}+\frac{3}{x^2}+x^2\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac{3 \coth (a+b x)}{b}-\frac{\coth ^3(a+b x)}{3 b}+\frac{3 \tanh (a+b x)}{b}-\frac{\tanh ^3(a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.0192102, size = 43, normalized size = 0.81 \[ 16 \left (\frac{\coth (2 (a+b x))}{3 b}-\frac{\coth (2 (a+b x)) \text{csch}^2(2 (a+b x))}{6 b}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

16*(Coth[2*(a + b*x)]/(3*b) - (Coth[2*(a + b*x)]*Csch[2*(a + b*x)]^2)/(6*b))

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Maple [A]  time = 0.025, size = 62, normalized size = 1.2 \begin{align*}{\frac{1}{b} \left ( -{\frac{1}{3\, \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}}+2\,{\frac{1}{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }}+8\, \left ( 2/3+1/3\, \left ({\rm sech} \left (bx+a\right ) \right ) ^{2} \right ) \tanh \left ( bx+a \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(-1/3/sinh(b*x+a)^3/cosh(b*x+a)^3+2/sinh(b*x+a)/cosh(b*x+a)^3+8*(2/3+1/3*sech(b*x+a)^2)*tanh(b*x+a))

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Maxima [A]  time = 1.17381, size = 122, normalized size = 2.3 \begin{align*} \frac{32 \, e^{\left (-4 \, b x - 4 \, a\right )}}{b{\left (3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )} - 1\right )}} - \frac{32}{3 \, b{\left (3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-12 \, b x - 12 \, 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)^4,x, algorithm="maxima")

[Out]

32*e^(-4*b*x - 4*a)/(b*(3*e^(-4*b*x - 4*a) - 3*e^(-8*b*x - 8*a) + e^(-12*b*x - 12*a) - 1)) - 32/3/(b*(3*e^(-4*
b*x - 4*a) - 3*e^(-8*b*x - 8*a) + e^(-12*b*x - 12*a) - 1))

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Fricas [B]  time = 2.26766, size = 900, normalized size = 16.98 \begin{align*} -\frac{64 \,{\left (\cosh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )}}{3 \,{\left (b \cosh \left (b x + a\right )^{10} + 120 \, b \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{7} + 45 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{8} + 10 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{9} + b \sinh \left (b x + a\right )^{10} - 3 \, b \cosh \left (b x + a\right )^{6} + 3 \,{\left (70 \, b \cosh \left (b x + a\right )^{4} - b\right )} \sinh \left (b x + a\right )^{6} + 18 \,{\left (14 \, b \cosh \left (b x + a\right )^{5} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 15 \,{\left (14 \, b \cosh \left (b x + a\right )^{6} - 3 \, b \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{4} + 60 \,{\left (2 \, b \cosh \left (b x + a\right )^{7} - b \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )^{3} + 2 \, b \cosh \left (b x + a\right )^{2} +{\left (45 \, b \cosh \left (b x + a\right )^{8} - 45 \, b \cosh \left (b x + a\right )^{4} + 2 \, b\right )} \sinh \left (b x + a\right )^{2} + 2 \,{\left (5 \, b \cosh \left (b x + a\right )^{9} - 9 \, b \cosh \left (b x + a\right )^{5} + 4 \, b \cosh \left (b x + a\right )\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)^4,x, algorithm="fricas")

[Out]

-64/3*(cosh(b*x + a)^2 + 4*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2)/(b*cosh(b*x + a)^10 + 120*b*cosh(b*x
+ a)^3*sinh(b*x + a)^7 + 45*b*cosh(b*x + a)^2*sinh(b*x + a)^8 + 10*b*cosh(b*x + a)*sinh(b*x + a)^9 + b*sinh(b
*x + a)^10 - 3*b*cosh(b*x + a)^6 + 3*(70*b*cosh(b*x + a)^4 - b)*sinh(b*x + a)^6 + 18*(14*b*cosh(b*x + a)^5 - b
*cosh(b*x + a))*sinh(b*x + a)^5 + 15*(14*b*cosh(b*x + a)^6 - 3*b*cosh(b*x + a)^2)*sinh(b*x + a)^4 + 60*(2*b*co
sh(b*x + a)^7 - b*cosh(b*x + a)^3)*sinh(b*x + a)^3 + 2*b*cosh(b*x + a)^2 + (45*b*cosh(b*x + a)^8 - 45*b*cosh(b
*x + a)^4 + 2*b)*sinh(b*x + a)^2 + 2*(5*b*cosh(b*x + a)^9 - 9*b*cosh(b*x + a)^5 + 4*b*cosh(b*x + a))*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}^{4}{\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)**4,x)

[Out]

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

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Giac [A]  time = 1.1614, size = 42, normalized size = 0.79 \begin{align*} -\frac{32 \,{\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} - 1\right )}}{3 \, b{\left (e^{\left (4 \, b x + 4 \, 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)^4,x, algorithm="giac")

[Out]

-32/3*(3*e^(4*b*x + 4*a) - 1)/(b*(e^(4*b*x + 4*a) - 1)^3)