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

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Rubi [A]  time = 0.04, antiderivative size = 37, 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} \[ \frac {\tanh (a+b x)}{b}-\frac {\coth ^3(a+b x)}{3 b}+\frac {2 \coth (a+b x)}{b} \]

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 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}^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*}

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Mathematica [A]  time = 0.04, 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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fricas [B]  time = 0.39, size = 229, normalized size = 6.19 \[ -\frac {16 \, {\left (\cosh \left (b x + a\right ) + 3 \, \sinh \left (b x + a\right )\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{7} + 7 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{6} + b \sinh \left (b x + a\right )^{7} - 2 \, b \cosh \left (b x + a\right )^{5} + {\left (21 \, b \cosh \left (b x + a\right )^{2} - 2 \, b\right )} \sinh \left (b x + a\right )^{5} + 5 \, {\left (7 \, b \cosh \left (b x + a\right )^{3} - 2 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{4} + 5 \, {\left (7 \, b \cosh \left (b x + a\right )^{4} - 4 \, b \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{3} + {\left (21 \, b \cosh \left (b x + a\right )^{5} - 20 \, b \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )^{2} + b \cosh \left (b x + a\right ) + {\left (7 \, b \cosh \left (b x + a\right )^{6} - 10 \, b \cosh \left (b x + a\right )^{4} + 3 \, b\right )} \sinh \left (b x + a\right )\right )}} \]

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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giac [A]  time = 0.14, size = 60, normalized size = 1.62 \[ -\frac {2 \, {\left (\frac {3}{e^{\left (2 \, b x + 2 \, a\right )} + 1} - \frac {3 \, e^{\left (4 \, b x + 4 \, a\right )} - 12 \, e^{\left (2 \, b x + 2 \, a\right )} + 5}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}}\right )}}{3 \, b} \]

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]

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

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maple [A]  time = 0.32, size = 50, normalized size = 1.35 \[ \frac {-\frac {1}{3 \sinh \left (b x +a \right )^{3} \cosh \left (b x +a \right )}+\frac {4}{3 \sinh \left (b x +a \right ) \cosh \left (b x +a \right )}+\frac {8 \tanh \left (b x +a \right )}{3}}{b} \]

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 = 0.31, size = 90, normalized size = 2.43 \[ \frac {32 \, e^{\left (-2 \, b x - 2 \, a\right )}}{3 \, b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, e^{\left (-6 \, b x - 6 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )}} - \frac {16}{3 \, b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, e^{\left (-6 \, b x - 6 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )}} \]

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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mupad [B]  time = 1.48, size = 153, normalized size = 4.14 \[ \frac {\frac {2}{3\,b}-\frac {4\,{\mathrm {e}}^{2\,a+2\,b\,x}}{b}+\frac {2\,{\mathrm {e}}^{4\,a+4\,b\,x}}{3\,b}}{3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1}-\frac {\frac {2}{b}-\frac {2\,{\mathrm {e}}^{2\,a+2\,b\,x}}{3\,b}}{{\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1}+\frac {2}{3\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(a + b*x)^2*sinh(a + b*x)^4),x)

[Out]

(2/(3*b) - (4*exp(2*a + 2*b*x))/b + (2*exp(4*a + 4*b*x))/(3*b))/(3*exp(2*a + 2*b*x) - 3*exp(4*a + 4*b*x) + exp
(6*a + 6*b*x) - 1) - (2/b - (2*exp(2*a + 2*b*x))/(3*b))/(exp(4*a + 4*b*x) - 2*exp(2*a + 2*b*x) + 1) + 2/(3*b*(
exp(2*a + 2*b*x) - 1)) - 2/(b*(exp(2*a + 2*b*x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}^{4}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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