3.47 \(\int \frac{1}{(a \text{csch}^4(x))^{3/2}} \, dx\)

Optimal. Leaf size=86 \[ -\frac{5 x \text{csch}^2(x)}{16 a \sqrt{a \text{csch}^4(x)}}+\frac{5 \coth (x)}{16 a \sqrt{a \text{csch}^4(x)}}+\frac{\sinh ^3(x) \cosh (x)}{6 a \sqrt{a \text{csch}^4(x)}}-\frac{5 \sinh (x) \cosh (x)}{24 a \sqrt{a \text{csch}^4(x)}} \]

[Out]

(5*Coth[x])/(16*a*Sqrt[a*Csch[x]^4]) - (5*x*Csch[x]^2)/(16*a*Sqrt[a*Csch[x]^4]) - (5*Cosh[x]*Sinh[x])/(24*a*Sq
rt[a*Csch[x]^4]) + (Cosh[x]*Sinh[x]^3)/(6*a*Sqrt[a*Csch[x]^4])

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Rubi [A]  time = 0.0327562, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4123, 2635, 8} \[ -\frac{5 x \text{csch}^2(x)}{16 a \sqrt{a \text{csch}^4(x)}}+\frac{5 \coth (x)}{16 a \sqrt{a \text{csch}^4(x)}}+\frac{\sinh ^3(x) \cosh (x)}{6 a \sqrt{a \text{csch}^4(x)}}-\frac{5 \sinh (x) \cosh (x)}{24 a \sqrt{a \text{csch}^4(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Coth[x])/(16*a*Sqrt[a*Csch[x]^4]) - (5*x*Csch[x]^2)/(16*a*Sqrt[a*Csch[x]^4]) - (5*Cosh[x]*Sinh[x])/(24*a*Sq
rt[a*Csch[x]^4]) + (Cosh[x]*Sinh[x]^3)/(6*a*Sqrt[a*Csch[x]^4])

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]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (a \text{csch}^4(x)\right )^{3/2}} \, dx &=\frac{\text{csch}^2(x) \int \sinh ^6(x) \, dx}{a \sqrt{a \text{csch}^4(x)}}\\ &=\frac{\cosh (x) \sinh ^3(x)}{6 a \sqrt{a \text{csch}^4(x)}}-\frac{\left (5 \text{csch}^2(x)\right ) \int \sinh ^4(x) \, dx}{6 a \sqrt{a \text{csch}^4(x)}}\\ &=-\frac{5 \cosh (x) \sinh (x)}{24 a \sqrt{a \text{csch}^4(x)}}+\frac{\cosh (x) \sinh ^3(x)}{6 a \sqrt{a \text{csch}^4(x)}}+\frac{\left (5 \text{csch}^2(x)\right ) \int \sinh ^2(x) \, dx}{8 a \sqrt{a \text{csch}^4(x)}}\\ &=\frac{5 \coth (x)}{16 a \sqrt{a \text{csch}^4(x)}}-\frac{5 \cosh (x) \sinh (x)}{24 a \sqrt{a \text{csch}^4(x)}}+\frac{\cosh (x) \sinh ^3(x)}{6 a \sqrt{a \text{csch}^4(x)}}-\frac{\left (5 \text{csch}^2(x)\right ) \int 1 \, dx}{16 a \sqrt{a \text{csch}^4(x)}}\\ &=\frac{5 \coth (x)}{16 a \sqrt{a \text{csch}^4(x)}}-\frac{5 x \text{csch}^2(x)}{16 a \sqrt{a \text{csch}^4(x)}}-\frac{5 \cosh (x) \sinh (x)}{24 a \sqrt{a \text{csch}^4(x)}}+\frac{\cosh (x) \sinh ^3(x)}{6 a \sqrt{a \text{csch}^4(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0384692, size = 38, normalized size = 0.44 \[ \frac{(-60 x+45 \sinh (2 x)-9 \sinh (4 x)+\sinh (6 x)) \text{csch}^6(x)}{192 \left (a \text{csch}^4(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csch[x]^6*(-60*x + 45*Sinh[2*x] - 9*Sinh[4*x] + Sinh[6*x]))/(192*(a*Csch[x]^4)^(3/2))

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Maple [B]  time = 0.053, size = 230, normalized size = 2.7 \begin{align*} -{\frac{5\,{{\rm e}^{2\,x}}x}{16\,a \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{4}}}}}}}+{\frac{{{\rm e}^{8\,x}}}{384\,a \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{4}}}}}}}-{\frac{3\,{{\rm e}^{6\,x}}}{128\,a \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{4}}}}}}}+{\frac{15\,{{\rm e}^{4\,x}}}{128\,a \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{4}}}}}}}-{\frac{15}{128\,a \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{4}}}}}}}+{\frac{3\,{{\rm e}^{-2\,x}}}{128\,a \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{4}}}}}}}-{\frac{{{\rm e}^{-4\,x}}}{384\,a \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{4}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*csch(x)^4)^(3/2),x)

[Out]

-5/16/a*exp(2*x)/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)*x+1/384/a*exp(8*x)/(exp(2*x)-1)^2/(a*exp(4*x
)/(exp(2*x)-1)^4)^(1/2)-3/128/a*exp(6*x)/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)+15/128/a*exp(4*x)/(e
xp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)-15/128/a/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)+3/128
/a*exp(-2*x)/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)-1/384/a*exp(-4*x)/(exp(2*x)-1)^2/(a*exp(4*x)/(ex
p(2*x)-1)^4)^(1/2)

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Maxima [A]  time = 1.57918, size = 62, normalized size = 0.72 \begin{align*} -\frac{{\left (9 \, e^{\left (-2 \, x\right )} - 45 \, e^{\left (-4 \, x\right )} + 45 \, e^{\left (-8 \, x\right )} - 9 \, e^{\left (-10 \, x\right )} + e^{\left (-12 \, x\right )} - 1\right )} e^{\left (6 \, x\right )}}{384 \, a^{\frac{3}{2}}} - \frac{5 \, x}{16 \, a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)^4)^(3/2),x, algorithm="maxima")

[Out]

-1/384*(9*e^(-2*x) - 45*e^(-4*x) + 45*e^(-8*x) - 9*e^(-10*x) + e^(-12*x) - 1)*e^(6*x)/a^(3/2) - 5/16*x/a^(3/2)

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Fricas [B]  time = 1.77925, size = 3687, normalized size = 42.87 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)^4)^(3/2),x, algorithm="fricas")

[Out]

1/384*((e^(4*x) - 2*e^(2*x) + 1)*sinh(x)^12 + cosh(x)^12 + 12*(cosh(x)*e^(4*x) - 2*cosh(x)*e^(2*x) + cosh(x))*
sinh(x)^11 + 3*(22*cosh(x)^2 + (22*cosh(x)^2 - 3)*e^(4*x) - 2*(22*cosh(x)^2 - 3)*e^(2*x) - 3)*sinh(x)^10 - 9*c
osh(x)^10 + 10*(22*cosh(x)^3 + (22*cosh(x)^3 - 9*cosh(x))*e^(4*x) - 2*(22*cosh(x)^3 - 9*cosh(x))*e^(2*x) - 9*c
osh(x))*sinh(x)^9 + 45*(11*cosh(x)^4 - 9*cosh(x)^2 + (11*cosh(x)^4 - 9*cosh(x)^2 + 1)*e^(4*x) - 2*(11*cosh(x)^
4 - 9*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^8 + 45*cosh(x)^8 + 72*(11*cosh(x)^5 - 15*cosh(x)^3 + (11*cosh(x)^5 -
15*cosh(x)^3 + 5*cosh(x))*e^(4*x) - 2*(11*cosh(x)^5 - 15*cosh(x)^3 + 5*cosh(x))*e^(2*x) + 5*cosh(x))*sinh(x)^
7 - 120*x*cosh(x)^6 + 6*(154*cosh(x)^6 - 315*cosh(x)^4 + 210*cosh(x)^2 + (154*cosh(x)^6 - 315*cosh(x)^4 + 210*
cosh(x)^2 - 20*x)*e^(4*x) - 2*(154*cosh(x)^6 - 315*cosh(x)^4 + 210*cosh(x)^2 - 20*x)*e^(2*x) - 20*x)*sinh(x)^6
+ 36*(22*cosh(x)^7 - 63*cosh(x)^5 + 70*cosh(x)^3 - 20*x*cosh(x) + (22*cosh(x)^7 - 63*cosh(x)^5 + 70*cosh(x)^3
- 20*x*cosh(x))*e^(4*x) - 2*(22*cosh(x)^7 - 63*cosh(x)^5 + 70*cosh(x)^3 - 20*x*cosh(x))*e^(2*x))*sinh(x)^5 +
45*(11*cosh(x)^8 - 42*cosh(x)^6 + 70*cosh(x)^4 - 40*x*cosh(x)^2 + (11*cosh(x)^8 - 42*cosh(x)^6 + 70*cosh(x)^4
- 40*x*cosh(x)^2 - 1)*e^(4*x) - 2*(11*cosh(x)^8 - 42*cosh(x)^6 + 70*cosh(x)^4 - 40*x*cosh(x)^2 - 1)*e^(2*x) -
1)*sinh(x)^4 - 45*cosh(x)^4 + 20*(11*cosh(x)^9 - 54*cosh(x)^7 + 126*cosh(x)^5 - 120*x*cosh(x)^3 + (11*cosh(x)^
9 - 54*cosh(x)^7 + 126*cosh(x)^5 - 120*x*cosh(x)^3 - 9*cosh(x))*e^(4*x) - 2*(11*cosh(x)^9 - 54*cosh(x)^7 + 126
*cosh(x)^5 - 120*x*cosh(x)^3 - 9*cosh(x))*e^(2*x) - 9*cosh(x))*sinh(x)^3 + 3*(22*cosh(x)^10 - 135*cosh(x)^8 +
420*cosh(x)^6 - 600*x*cosh(x)^4 - 90*cosh(x)^2 + (22*cosh(x)^10 - 135*cosh(x)^8 + 420*cosh(x)^6 - 600*x*cosh(x
)^4 - 90*cosh(x)^2 + 3)*e^(4*x) - 2*(22*cosh(x)^10 - 135*cosh(x)^8 + 420*cosh(x)^6 - 600*x*cosh(x)^4 - 90*cosh
(x)^2 + 3)*e^(2*x) + 3)*sinh(x)^2 + 9*cosh(x)^2 + (cosh(x)^12 - 9*cosh(x)^10 + 45*cosh(x)^8 - 120*x*cosh(x)^6
- 45*cosh(x)^4 + 9*cosh(x)^2 - 1)*e^(4*x) - 2*(cosh(x)^12 - 9*cosh(x)^10 + 45*cosh(x)^8 - 120*x*cosh(x)^6 - 45
*cosh(x)^4 + 9*cosh(x)^2 - 1)*e^(2*x) + 6*(2*cosh(x)^11 - 15*cosh(x)^9 + 60*cosh(x)^7 - 120*x*cosh(x)^5 - 30*c
osh(x)^3 + (2*cosh(x)^11 - 15*cosh(x)^9 + 60*cosh(x)^7 - 120*x*cosh(x)^5 - 30*cosh(x)^3 + 3*cosh(x))*e^(4*x) -
2*(2*cosh(x)^11 - 15*cosh(x)^9 + 60*cosh(x)^7 - 120*x*cosh(x)^5 - 30*cosh(x)^3 + 3*cosh(x))*e^(2*x) + 3*cosh(
x))*sinh(x) - 1)*sqrt(a/(e^(8*x) - 4*e^(6*x) + 6*e^(4*x) - 4*e^(2*x) + 1))*e^(2*x)/(a^2*cosh(x)^6*e^(2*x) + 6*
a^2*cosh(x)^5*e^(2*x)*sinh(x) + 15*a^2*cosh(x)^4*e^(2*x)*sinh(x)^2 + 20*a^2*cosh(x)^3*e^(2*x)*sinh(x)^3 + 15*a
^2*cosh(x)^2*e^(2*x)*sinh(x)^4 + 6*a^2*cosh(x)*e^(2*x)*sinh(x)^5 + a^2*e^(2*x)*sinh(x)^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{csch}^{4}{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)**4)**(3/2),x)

[Out]

Integral((a*csch(x)**4)**(-3/2), x)

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Giac [A]  time = 1.2494, size = 68, normalized size = 0.79 \begin{align*} \frac{{\left (110 \, e^{\left (6 \, x\right )} - 45 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-6 \, x\right )} - 120 \, x + e^{\left (6 \, x\right )} - 9 \, e^{\left (4 \, x\right )} + 45 \, e^{\left (2 \, x\right )}}{384 \, a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)^4)^(3/2),x, algorithm="giac")

[Out]

1/384*((110*e^(6*x) - 45*e^(4*x) + 9*e^(2*x) - 1)*e^(-6*x) - 120*x + e^(6*x) - 9*e^(4*x) + 45*e^(2*x))/a^(3/2)