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(b*x+a)/b-1/3*coth(b*x+a)^3/b+3*tanh(b*x+a)/b-1/3*tanh(b*x+a)^3/b

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Rubi [A]  time = 0.04, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [B]  time = 0.41, size = 330, normalized size = 6.23 \[ -\frac {64 \, {\left (\cosh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{10} + 120 \, b \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{7} + 45 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{8} + 10 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{9} + b \sinh \left (b x + a\right )^{10} - 3 \, b \cosh \left (b x + a\right )^{6} + 3 \, {\left (70 \, b \cosh \left (b x + a\right )^{4} - b\right )} \sinh \left (b x + a\right )^{6} + 18 \, {\left (14 \, b \cosh \left (b x + a\right )^{5} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 15 \, {\left (14 \, b \cosh \left (b x + a\right )^{6} - 3 \, b \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{4} + 60 \, {\left (2 \, b \cosh \left (b x + a\right )^{7} - b \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )^{3} + 2 \, b \cosh \left (b x + a\right )^{2} + {\left (45 \, b \cosh \left (b x + a\right )^{8} - 45 \, b \cosh \left (b x + a\right )^{4} + 2 \, b\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (5 \, b \cosh \left (b x + a\right )^{9} - 9 \, b \cosh \left (b x + a\right )^{5} + 4 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )}} \]

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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giac [A]  time = 0.15, size = 31, normalized size = 0.58 \[ -\frac {32 \, {\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} - 1\right )}}{3 \, b {\left (e^{\left (4 \, b x + 4 \, a\right )} - 1\right )}^{3}} \]

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)

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maple [A]  time = 0.32, size = 62, normalized size = 1.17 \[ \frac {-\frac {1}{3 \sinh \left (b x +a \right )^{3} \cosh \left (b x +a \right )^{3}}+\frac {2}{\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}+8 \left (\frac {2}{3}+\frac {\mathrm {sech}\left (b x +a \right )^{2}}{3}\right ) \tanh \left (b x +a \right )}{b} \]

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 = 0.31, size = 90, normalized size = 1.70 \[ \frac {32 \, e^{\left (-4 \, b x - 4 \, a\right )}}{b {\left (3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )} - 1\right )}} - \frac {32}{3 \, b {\left (3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )} - 1\right )}} \]

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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mupad [B]  time = 0.06, size = 31, normalized size = 0.58 \[ -\frac {32\,\left (3\,{\mathrm {e}}^{4\,a+4\,b\,x}-1\right )}{3\,b\,{\left ({\mathrm {e}}^{4\,a+4\,b\,x}-1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(a + b*x)^4*sinh(a + b*x)^4),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}^{4}{\left (a + b x \right )} \operatorname {sech}^{4}{\left (a + b x \right )}\, dx \]

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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