3.44 \(\int (a \text {csch}^4(x))^{3/2} \, dx\)

Optimal. Leaf size=62 \[ -\frac {1}{5} a \cosh ^2(x) \coth ^3(x) \sqrt {a \text {csch}^4(x)}+\frac {2}{3} a \cosh ^2(x) \coth (x) \sqrt {a \text {csch}^4(x)}-a \sinh (x) \cosh (x) \sqrt {a \text {csch}^4(x)} \]

[Out]

2/3*a*cosh(x)^2*coth(x)*(a*csch(x)^4)^(1/2)-1/5*a*cosh(x)^2*coth(x)^3*(a*csch(x)^4)^(1/2)-a*cosh(x)*sinh(x)*(a
*csch(x)^4)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4123, 3767} \[ -\frac {1}{5} a \cosh ^2(x) \coth ^3(x) \sqrt {a \text {csch}^4(x)}+\frac {2}{3} a \cosh ^2(x) \coth (x) \sqrt {a \text {csch}^4(x)}-a \sinh (x) \cosh (x) \sqrt {a \text {csch}^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csch[x]^4)^(3/2),x]

[Out]

(2*a*Cosh[x]^2*Coth[x]*Sqrt[a*Csch[x]^4])/3 - (a*Cosh[x]^2*Coth[x]^3*Sqrt[a*Csch[x]^4])/5 - a*Cosh[x]*Sqrt[a*C
sch[x]^4]*Sinh[x]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4123

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^
FracPart[p])/(c*Sec[e + f*x])^(n*FracPart[p]), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \left (a \text {csch}^4(x)\right )^{3/2} \, dx &=\left (a \sqrt {a \text {csch}^4(x)} \sinh ^2(x)\right ) \int \text {csch}^6(x) \, dx\\ &=-\left (\left (i a \sqrt {a \text {csch}^4(x)} \sinh ^2(x)\right ) \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \coth (x)\right )\right )\\ &=\frac {2}{3} a \cosh ^2(x) \coth (x) \sqrt {a \text {csch}^4(x)}-\frac {1}{5} a \cosh ^2(x) \coth ^3(x) \sqrt {a \text {csch}^4(x)}-a \cosh (x) \sqrt {a \text {csch}^4(x)} \sinh (x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.53 \[ -\frac {1}{15} a \sinh (x) \cosh (x) \left (3 \text {csch}^4(x)-4 \text {csch}^2(x)+8\right ) \sqrt {a \text {csch}^4(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csch[x]^4)^(3/2),x]

[Out]

-1/15*(a*Cosh[x]*Sqrt[a*Csch[x]^4]*(8 - 4*Csch[x]^2 + 3*Csch[x]^4)*Sinh[x])

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fricas [B]  time = 1.57, size = 529, normalized size = 8.53 \[ -\frac {16 \, {\left (10 \, a \cosh \relax (x)^{4} + 10 \, {\left (a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a\right )} \sinh \relax (x)^{4} + 40 \, {\left (a \cosh \relax (x) e^{\left (4 \, x\right )} - 2 \, a \cosh \relax (x) e^{\left (2 \, x\right )} + a \cosh \relax (x)\right )} \sinh \relax (x)^{3} - 5 \, a \cosh \relax (x)^{2} + 5 \, {\left (12 \, a \cosh \relax (x)^{2} + {\left (12 \, a \cosh \relax (x)^{2} - a\right )} e^{\left (4 \, x\right )} - 2 \, {\left (12 \, a \cosh \relax (x)^{2} - a\right )} e^{\left (2 \, x\right )} - a\right )} \sinh \relax (x)^{2} + {\left (10 \, a \cosh \relax (x)^{4} - 5 \, a \cosh \relax (x)^{2} + a\right )} e^{\left (4 \, x\right )} - 2 \, {\left (10 \, a \cosh \relax (x)^{4} - 5 \, a \cosh \relax (x)^{2} + a\right )} e^{\left (2 \, x\right )} + 10 \, {\left (4 \, a \cosh \relax (x)^{3} - a \cosh \relax (x) + {\left (4 \, a \cosh \relax (x)^{3} - a \cosh \relax (x)\right )} e^{\left (4 \, x\right )} - 2 \, {\left (4 \, a \cosh \relax (x)^{3} - a \cosh \relax (x)\right )} e^{\left (2 \, x\right )}\right )} \sinh \relax (x) + a\right )} \sqrt {\frac {a}{e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1}} e^{\left (2 \, x\right )}}{15 \, {\left (10 \, \cosh \relax (x) e^{\left (2 \, x\right )} \sinh \relax (x)^{9} + e^{\left (2 \, x\right )} \sinh \relax (x)^{10} + 5 \, {\left (9 \, \cosh \relax (x)^{2} - 1\right )} e^{\left (2 \, x\right )} \sinh \relax (x)^{8} + 40 \, {\left (3 \, \cosh \relax (x)^{3} - \cosh \relax (x)\right )} e^{\left (2 \, x\right )} \sinh \relax (x)^{7} + 10 \, {\left (21 \, \cosh \relax (x)^{4} - 14 \, \cosh \relax (x)^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \relax (x)^{6} + 4 \, {\left (63 \, \cosh \relax (x)^{5} - 70 \, \cosh \relax (x)^{3} + 15 \, \cosh \relax (x)\right )} e^{\left (2 \, x\right )} \sinh \relax (x)^{5} + 10 \, {\left (21 \, \cosh \relax (x)^{6} - 35 \, \cosh \relax (x)^{4} + 15 \, \cosh \relax (x)^{2} - 1\right )} e^{\left (2 \, x\right )} \sinh \relax (x)^{4} + 40 \, {\left (3 \, \cosh \relax (x)^{7} - 7 \, \cosh \relax (x)^{5} + 5 \, \cosh \relax (x)^{3} - \cosh \relax (x)\right )} e^{\left (2 \, x\right )} \sinh \relax (x)^{3} + 5 \, {\left (9 \, \cosh \relax (x)^{8} - 28 \, \cosh \relax (x)^{6} + 30 \, \cosh \relax (x)^{4} - 12 \, \cosh \relax (x)^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \relax (x)^{2} + 10 \, {\left (\cosh \relax (x)^{9} - 4 \, \cosh \relax (x)^{7} + 6 \, \cosh \relax (x)^{5} - 4 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} e^{\left (2 \, x\right )} \sinh \relax (x) + {\left (\cosh \relax (x)^{10} - 5 \, \cosh \relax (x)^{8} + 10 \, \cosh \relax (x)^{6} - 10 \, \cosh \relax (x)^{4} + 5 \, \cosh \relax (x)^{2} - 1\right )} e^{\left (2 \, x\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csch(x)^4)^(3/2),x, algorithm="fricas")

[Out]

-16/15*(10*a*cosh(x)^4 + 10*(a*e^(4*x) - 2*a*e^(2*x) + a)*sinh(x)^4 + 40*(a*cosh(x)*e^(4*x) - 2*a*cosh(x)*e^(2
*x) + a*cosh(x))*sinh(x)^3 - 5*a*cosh(x)^2 + 5*(12*a*cosh(x)^2 + (12*a*cosh(x)^2 - a)*e^(4*x) - 2*(12*a*cosh(x
)^2 - a)*e^(2*x) - a)*sinh(x)^2 + (10*a*cosh(x)^4 - 5*a*cosh(x)^2 + a)*e^(4*x) - 2*(10*a*cosh(x)^4 - 5*a*cosh(
x)^2 + a)*e^(2*x) + 10*(4*a*cosh(x)^3 - a*cosh(x) + (4*a*cosh(x)^3 - a*cosh(x))*e^(4*x) - 2*(4*a*cosh(x)^3 - a
*cosh(x))*e^(2*x))*sinh(x) + a)*sqrt(a/(e^(8*x) - 4*e^(6*x) + 6*e^(4*x) - 4*e^(2*x) + 1))*e^(2*x)/(10*cosh(x)*
e^(2*x)*sinh(x)^9 + e^(2*x)*sinh(x)^10 + 5*(9*cosh(x)^2 - 1)*e^(2*x)*sinh(x)^8 + 40*(3*cosh(x)^3 - cosh(x))*e^
(2*x)*sinh(x)^7 + 10*(21*cosh(x)^4 - 14*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^6 + 4*(63*cosh(x)^5 - 70*cosh(x)^3 + 15
*cosh(x))*e^(2*x)*sinh(x)^5 + 10*(21*cosh(x)^6 - 35*cosh(x)^4 + 15*cosh(x)^2 - 1)*e^(2*x)*sinh(x)^4 + 40*(3*co
sh(x)^7 - 7*cosh(x)^5 + 5*cosh(x)^3 - cosh(x))*e^(2*x)*sinh(x)^3 + 5*(9*cosh(x)^8 - 28*cosh(x)^6 + 30*cosh(x)^
4 - 12*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^2 + 10*(cosh(x)^9 - 4*cosh(x)^7 + 6*cosh(x)^5 - 4*cosh(x)^3 + cosh(x))*e
^(2*x)*sinh(x) + (cosh(x)^10 - 5*cosh(x)^8 + 10*cosh(x)^6 - 10*cosh(x)^4 + 5*cosh(x)^2 - 1)*e^(2*x))

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giac [A]  time = 0.13, size = 27, normalized size = 0.44 \[ -\frac {16 \, a^{\frac {3}{2}} {\left (10 \, e^{\left (4 \, x\right )} - 5 \, e^{\left (2 \, x\right )} + 1\right )}}{15 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csch(x)^4)^(3/2),x, algorithm="giac")

[Out]

-16/15*a^(3/2)*(10*e^(4*x) - 5*e^(2*x) + 1)/(e^(2*x) - 1)^5

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maple [A]  time = 0.20, size = 46, normalized size = 0.74 \[ -\frac {16 a \,{\mathrm e}^{-2 x} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}\, \left (10 \,{\mathrm e}^{4 x}-5 \,{\mathrm e}^{2 x}+1\right )}{15 \left ({\mathrm e}^{2 x}-1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*csch(x)^4)^(3/2),x)

[Out]

-16/15*a*exp(-2*x)/(exp(2*x)-1)^3*(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)*(10*exp(4*x)-5*exp(2*x)+1)

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maxima [B]  time = 0.51, size = 120, normalized size = 1.94 \[ -\frac {16 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} + \frac {32 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} + \frac {16 \, a^{\frac {3}{2}}}{15 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csch(x)^4)^(3/2),x, algorithm="maxima")

[Out]

-16/3*a^(3/2)*e^(-2*x)/(5*e^(-2*x) - 10*e^(-4*x) + 10*e^(-6*x) - 5*e^(-8*x) + e^(-10*x) - 1) + 32/3*a^(3/2)*e^
(-4*x)/(5*e^(-2*x) - 10*e^(-4*x) + 10*e^(-6*x) - 5*e^(-8*x) + e^(-10*x) - 1) + 16/15*a^(3/2)/(5*e^(-2*x) - 10*
e^(-4*x) + 10*e^(-6*x) - 5*e^(-8*x) + e^(-10*x) - 1)

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mupad [B]  time = 1.46, size = 46, normalized size = 0.74 \[ -\frac {4\,a\,{\mathrm {e}}^{-2\,x}\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (10\,{\mathrm {e}}^{4\,x}-5\,{\mathrm {e}}^{2\,x}+1\right )}{15\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/sinh(x)^4)^(3/2),x)

[Out]

-(4*a*exp(-2*x)*(a/(exp(-x)/2 - exp(x)/2)^4)^(1/2)*(10*exp(4*x) - 5*exp(2*x) + 1))/(15*(exp(2*x) - 1)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \operatorname {csch}^{4}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csch(x)**4)**(3/2),x)

[Out]

Integral((a*csch(x)**4)**(3/2), x)

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