3.104 \(\int \cosh (a+b x) \coth ^4(a+b x) \, dx\)

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

[Out]

-2*csch(b*x+a)/b-1/3*csch(b*x+a)^3/b+sinh(b*x+a)/b

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Rubi [A]  time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2590, 270} \[ \frac {\sinh (a+b x)}{b}-\frac {\text {csch}^3(a+b x)}{3 b}-\frac {2 \text {csch}(a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]*Coth[a + b*x]^4,x]

[Out]

(-2*Csch[a + b*x])/b - Csch[a + b*x]^3/(3*b) + Sinh[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 2590

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 37, normalized size = 1.00 \[ \frac {\sinh (a+b x)}{b}-\frac {\text {csch}^3(a+b x)}{3 b}-\frac {2 \text {csch}(a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]*Coth[a + b*x]^4,x]

[Out]

(-2*Csch[a + b*x])/b - Csch[a + b*x]^3/(3*b) + Sinh[a + b*x]/b

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fricas [B]  time = 0.46, size = 89, normalized size = 2.41 \[ \frac {3 \, \cosh \left (b x + a\right )^{4} + 3 \, \sinh \left (b x + a\right )^{4} + 18 \, {\left (\cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right )^{2} - 36 \, \cosh \left (b x + a\right )^{2} + 25}{6 \, {\left (b \sinh \left (b x + a\right )^{3} + 3 \, {\left (b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*coth(b*x+a)^4,x, algorithm="fricas")

[Out]

1/6*(3*cosh(b*x + a)^4 + 3*sinh(b*x + a)^4 + 18*(cosh(b*x + a)^2 - 2)*sinh(b*x + a)^2 - 36*cosh(b*x + a)^2 + 2
5)/(b*sinh(b*x + a)^3 + 3*(b*cosh(b*x + a)^2 - b)*sinh(b*x + a))

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giac [B]  time = 0.19, size = 71, normalized size = 1.92 \[ -\frac {\frac {8 \, {\left (3 \, e^{\left (5 \, b x + 5 \, a\right )} - 4 \, e^{\left (3 \, b x + 3 \, a\right )} + 3 \, e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} - 3 \, e^{\left (b x + a\right )} + 3 \, e^{\left (-b x - a\right )}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*coth(b*x+a)^4,x, algorithm="giac")

[Out]

-1/6*(8*(3*e^(5*b*x + 5*a) - 4*e^(3*b*x + 3*a) + 3*e^(b*x + a))/(e^(2*b*x + 2*a) - 1)^3 - 3*e^(b*x + a) + 3*e^
(-b*x - a))/b

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maple [A]  time = 0.15, size = 51, normalized size = 1.38 \[ \frac {\frac {\cosh ^{4}\left (b x +a \right )}{\sinh \left (b x +a \right )^{3}}-\frac {4 \left (\cosh ^{2}\left (b x +a \right )\right )}{\sinh \left (b x +a \right )^{3}}+\frac {8}{3 \sinh \left (b x +a \right )^{3}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)*coth(b*x+a)^4,x)

[Out]

1/b*(cosh(b*x+a)^4/sinh(b*x+a)^3-4/sinh(b*x+a)^3*cosh(b*x+a)^2+8/3/sinh(b*x+a)^3)

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maxima [B]  time = 0.32, size = 100, normalized size = 2.70 \[ -\frac {e^{\left (-b x - a\right )}}{2 \, b} - \frac {33 \, e^{\left (-2 \, b x - 2 \, a\right )} - 41 \, e^{\left (-4 \, b x - 4 \, a\right )} + 27 \, e^{\left (-6 \, b x - 6 \, a\right )} - 3}{6 \, b {\left (e^{\left (-b x - a\right )} - 3 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )} - e^{\left (-7 \, b x - 7 \, a\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*coth(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/2*e^(-b*x - a)/b - 1/6*(33*e^(-2*b*x - 2*a) - 41*e^(-4*b*x - 4*a) + 27*e^(-6*b*x - 6*a) - 3)/(b*(e^(-b*x -
a) - 3*e^(-3*b*x - 3*a) + 3*e^(-5*b*x - 5*a) - e^(-7*b*x - 7*a)))

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mupad [B]  time = 1.45, size = 131, normalized size = 3.54 \[ \frac {{\mathrm {e}}^{a+b\,x}}{2\,b}-\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}-\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {4\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)*coth(a + b*x)^4,x)

[Out]

exp(a + b*x)/(2*b) - exp(- a - b*x)/(2*b) - (8*exp(a + b*x))/(3*b*(exp(4*a + 4*b*x) - 2*exp(2*a + 2*b*x) + 1))
- (8*exp(a + 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)) - (4*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 \[ \int \cosh {\left (a + b x \right )} \coth ^{4}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*coth(b*x+a)**4,x)

[Out]

Integral(cosh(a + b*x)*coth(a + b*x)**4, x)

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