3.35 \(\int \frac{1}{(a \text{csch}^2(x))^{7/2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{16 \coth (x)}{35 a^3 \sqrt{a \text{csch}^2(x)}}+\frac{8 \coth (x)}{35 a^2 \left (a \text{csch}^2(x)\right )^{3/2}}-\frac{6 \coth (x)}{35 a \left (a \text{csch}^2(x)\right )^{5/2}}+\frac{\coth (x)}{7 \left (a \text{csch}^2(x)\right )^{7/2}} \]

[Out]

Coth[x]/(7*(a*Csch[x]^2)^(7/2)) - (6*Coth[x])/(35*a*(a*Csch[x]^2)^(5/2)) + (8*Coth[x])/(35*a^2*(a*Csch[x]^2)^(
3/2)) - (16*Coth[x])/(35*a^3*Sqrt[a*Csch[x]^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0401579, antiderivative size = 74, 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 = {4122, 192, 191} \[ -\frac{16 \coth (x)}{35 a^3 \sqrt{a \text{csch}^2(x)}}+\frac{8 \coth (x)}{35 a^2 \left (a \text{csch}^2(x)\right )^{3/2}}-\frac{6 \coth (x)}{35 a \left (a \text{csch}^2(x)\right )^{5/2}}+\frac{\coth (x)}{7 \left (a \text{csch}^2(x)\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csch[x]^2)^(-7/2),x]

[Out]

Coth[x]/(7*(a*Csch[x]^2)^(7/2)) - (6*Coth[x])/(35*a*(a*Csch[x]^2)^(5/2)) + (8*Coth[x])/(35*a^2*(a*Csch[x]^2)^(
3/2)) - (16*Coth[x])/(35*a^3*Sqrt[a*Csch[x]^2])

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (a \text{csch}^2(x)\right )^{7/2}} \, dx &=-\left (a \operatorname{Subst}\left (\int \frac{1}{\left (-a+a x^2\right )^{9/2}} \, dx,x,\coth (x)\right )\right )\\ &=\frac{\coth (x)}{7 \left (a \text{csch}^2(x)\right )^{7/2}}+\frac{6}{7} \operatorname{Subst}\left (\int \frac{1}{\left (-a+a x^2\right )^{7/2}} \, dx,x,\coth (x)\right )\\ &=\frac{\coth (x)}{7 \left (a \text{csch}^2(x)\right )^{7/2}}-\frac{6 \coth (x)}{35 a \left (a \text{csch}^2(x)\right )^{5/2}}-\frac{24 \operatorname{Subst}\left (\int \frac{1}{\left (-a+a x^2\right )^{5/2}} \, dx,x,\coth (x)\right )}{35 a}\\ &=\frac{\coth (x)}{7 \left (a \text{csch}^2(x)\right )^{7/2}}-\frac{6 \coth (x)}{35 a \left (a \text{csch}^2(x)\right )^{5/2}}+\frac{8 \coth (x)}{35 a^2 \left (a \text{csch}^2(x)\right )^{3/2}}+\frac{16 \operatorname{Subst}\left (\int \frac{1}{\left (-a+a x^2\right )^{3/2}} \, dx,x,\coth (x)\right )}{35 a^2}\\ &=\frac{\coth (x)}{7 \left (a \text{csch}^2(x)\right )^{7/2}}-\frac{6 \coth (x)}{35 a \left (a \text{csch}^2(x)\right )^{5/2}}+\frac{8 \coth (x)}{35 a^2 \left (a \text{csch}^2(x)\right )^{3/2}}-\frac{16 \coth (x)}{35 a^3 \sqrt{a \text{csch}^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.052438, size = 42, normalized size = 0.57 \[ \frac{\sinh (x) (-1225 \cosh (x)+245 \cosh (3 x)-49 \cosh (5 x)+5 \cosh (7 x)) \sqrt{a \text{csch}^2(x)}}{2240 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csch[x]^2)^(-7/2),x]

[Out]

((-1225*Cosh[x] + 245*Cosh[3*x] - 49*Cosh[5*x] + 5*Cosh[7*x])*Sqrt[a*Csch[x]^2]*Sinh[x])/(2240*a^4)

________________________________________________________________________________________

Maple [B]  time = 0.052, size = 262, normalized size = 3.5 \begin{align*}{\frac{{{\rm e}^{8\,x}}}{896\,{a}^{3} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{7\,{{\rm e}^{6\,x}}}{640\,{a}^{3} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{7\,{{\rm e}^{4\,x}}}{128\,{a}^{3} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{35\,{{\rm e}^{2\,x}}}{128\,{a}^{3} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{35}{128\,{a}^{3} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{7\,{{\rm e}^{-2\,x}}}{128\,{a}^{3} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{7\,{{\rm e}^{-4\,x}}}{640\,{a}^{3} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{{{\rm e}^{-6\,x}}}{896\,{a}^{3} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*csch(x)^2)^(7/2),x)

[Out]

1/896/a^3*exp(8*x)/(exp(2*x)-1)/(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)-7/640/a^3*exp(6*x)/(exp(2*x)-1)/(a*exp(2*x)/
(exp(2*x)-1)^2)^(1/2)+7/128/a^3*exp(4*x)/(exp(2*x)-1)/(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)-35/128/a^3*exp(2*x)/(e
xp(2*x)-1)/(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)-35/128/a^3/(exp(2*x)-1)/(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)+7/128/a
^3*exp(-2*x)/(exp(2*x)-1)/(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)-7/640/a^3*exp(-4*x)/(exp(2*x)-1)/(a*exp(2*x)/(exp(
2*x)-1)^2)^(1/2)+1/896/a^3*exp(-6*x)/(exp(2*x)-1)/(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.71474, size = 96, normalized size = 1.3 \begin{align*} -\frac{e^{\left (7 \, x\right )}}{896 \, a^{\frac{7}{2}}} + \frac{7 \, e^{\left (5 \, x\right )}}{640 \, a^{\frac{7}{2}}} - \frac{7 \, e^{\left (3 \, x\right )}}{128 \, a^{\frac{7}{2}}} + \frac{35 \, e^{\left (-x\right )}}{128 \, a^{\frac{7}{2}}} - \frac{7 \, e^{\left (-3 \, x\right )}}{128 \, a^{\frac{7}{2}}} + \frac{7 \, e^{\left (-5 \, x\right )}}{640 \, a^{\frac{7}{2}}} - \frac{e^{\left (-7 \, x\right )}}{896 \, a^{\frac{7}{2}}} + \frac{35 \, e^{x}}{128 \, a^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)^2)^(7/2),x, algorithm="maxima")

[Out]

-1/896*e^(7*x)/a^(7/2) + 7/640*e^(5*x)/a^(7/2) - 7/128*e^(3*x)/a^(7/2) + 35/128*e^(-x)/a^(7/2) - 7/128*e^(-3*x
)/a^(7/2) + 7/640*e^(-5*x)/a^(7/2) - 1/896*e^(-7*x)/a^(7/2) + 35/128*e^x/a^(7/2)

________________________________________________________________________________________

Fricas [B]  time = 2.00259, size = 3340, normalized size = 45.14 \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)^2)^(7/2),x, algorithm="fricas")

[Out]

1/4480*(5*(e^(2*x) - 1)*sinh(x)^14 - 5*cosh(x)^14 + 70*(cosh(x)*e^(2*x) - cosh(x))*sinh(x)^13 - 7*(65*cosh(x)^
2 - (65*cosh(x)^2 - 7)*e^(2*x) - 7)*sinh(x)^12 + 49*cosh(x)^12 - 28*(65*cosh(x)^3 - (65*cosh(x)^3 - 21*cosh(x)
)*e^(2*x) - 21*cosh(x))*sinh(x)^11 - 7*(715*cosh(x)^4 - 462*cosh(x)^2 - (715*cosh(x)^4 - 462*cosh(x)^2 + 35)*e
^(2*x) + 35)*sinh(x)^10 - 245*cosh(x)^10 - 70*(143*cosh(x)^5 - 154*cosh(x)^3 - (143*cosh(x)^5 - 154*cosh(x)^3
+ 35*cosh(x))*e^(2*x) + 35*cosh(x))*sinh(x)^9 - 35*(429*cosh(x)^6 - 693*cosh(x)^4 + 315*cosh(x)^2 - (429*cosh(
x)^6 - 693*cosh(x)^4 + 315*cosh(x)^2 - 35)*e^(2*x) - 35)*sinh(x)^8 + 1225*cosh(x)^8 - 8*(2145*cosh(x)^7 - 4851
*cosh(x)^5 + 3675*cosh(x)^3 - (2145*cosh(x)^7 - 4851*cosh(x)^5 + 3675*cosh(x)^3 - 1225*cosh(x))*e^(2*x) - 1225
*cosh(x))*sinh(x)^7 - 7*(2145*cosh(x)^8 - 6468*cosh(x)^6 + 7350*cosh(x)^4 - 4900*cosh(x)^2 - (2145*cosh(x)^8 -
6468*cosh(x)^6 + 7350*cosh(x)^4 - 4900*cosh(x)^2 - 175)*e^(2*x) - 175)*sinh(x)^6 + 1225*cosh(x)^6 - 14*(715*c
osh(x)^9 - 2772*cosh(x)^7 + 4410*cosh(x)^5 - 4900*cosh(x)^3 - (715*cosh(x)^9 - 2772*cosh(x)^7 + 4410*cosh(x)^5
- 4900*cosh(x)^3 - 525*cosh(x))*e^(2*x) - 525*cosh(x))*sinh(x)^5 - 35*(143*cosh(x)^10 - 693*cosh(x)^8 + 1470*
cosh(x)^6 - 2450*cosh(x)^4 - 525*cosh(x)^2 - (143*cosh(x)^10 - 693*cosh(x)^8 + 1470*cosh(x)^6 - 2450*cosh(x)^4
- 525*cosh(x)^2 + 7)*e^(2*x) + 7)*sinh(x)^4 - 245*cosh(x)^4 - 140*(13*cosh(x)^11 - 77*cosh(x)^9 + 210*cosh(x)
^7 - 490*cosh(x)^5 - 175*cosh(x)^3 - (13*cosh(x)^11 - 77*cosh(x)^9 + 210*cosh(x)^7 - 490*cosh(x)^5 - 175*cosh(
x)^3 + 7*cosh(x))*e^(2*x) + 7*cosh(x))*sinh(x)^3 - 7*(65*cosh(x)^12 - 462*cosh(x)^10 + 1575*cosh(x)^8 - 4900*c
osh(x)^6 - 2625*cosh(x)^4 + 210*cosh(x)^2 - (65*cosh(x)^12 - 462*cosh(x)^10 + 1575*cosh(x)^8 - 4900*cosh(x)^6
- 2625*cosh(x)^4 + 210*cosh(x)^2 - 7)*e^(2*x) - 7)*sinh(x)^2 + 49*cosh(x)^2 + (5*cosh(x)^14 - 49*cosh(x)^12 +
245*cosh(x)^10 - 1225*cosh(x)^8 - 1225*cosh(x)^6 + 245*cosh(x)^4 - 49*cosh(x)^2 + 5)*e^(2*x) - 14*(5*cosh(x)^1
3 - 42*cosh(x)^11 + 175*cosh(x)^9 - 700*cosh(x)^7 - 525*cosh(x)^5 + 70*cosh(x)^3 - (5*cosh(x)^13 - 42*cosh(x)^
11 + 175*cosh(x)^9 - 700*cosh(x)^7 - 525*cosh(x)^5 + 70*cosh(x)^3 - 7*cosh(x))*e^(2*x) - 7*cosh(x))*sinh(x) -
5)*sqrt(a/(e^(4*x) - 2*e^(2*x) + 1))*e^x/(a^4*cosh(x)^7*e^x + 7*a^4*cosh(x)^6*e^x*sinh(x) + 21*a^4*cosh(x)^5*e
^x*sinh(x)^2 + 35*a^4*cosh(x)^4*e^x*sinh(x)^3 + 35*a^4*cosh(x)^3*e^x*sinh(x)^4 + 21*a^4*cosh(x)^2*e^x*sinh(x)^
5 + 7*a^4*cosh(x)*e^x*sinh(x)^6 + a^4*e^x*sinh(x)^7)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{csch}^{2}{\left (x \right )}\right )^{\frac{7}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)**2)**(7/2),x)

[Out]

Integral((a*csch(x)**2)**(-7/2), x)

________________________________________________________________________________________

Giac [A]  time = 1.1931, size = 108, normalized size = 1.46 \begin{align*} -\frac{\frac{{\left (1225 \, e^{\left (6 \, x\right )} - 245 \, e^{\left (4 \, x\right )} + 49 \, e^{\left (2 \, x\right )} - 5\right )} e^{\left (-7 \, x\right )}}{\mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} - \frac{5 \, e^{\left (7 \, x\right )} - 49 \, e^{\left (5 \, x\right )} + 245 \, e^{\left (3 \, x\right )} - 1225 \, e^{x}}{\mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )}}{4480 \, a^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)^2)^(7/2),x, algorithm="giac")

[Out]

-1/4480*((1225*e^(6*x) - 245*e^(4*x) + 49*e^(2*x) - 5)*e^(-7*x)/sgn(e^(3*x) - e^x) - (5*e^(7*x) - 49*e^(5*x) +
245*e^(3*x) - 1225*e^x)/sgn(e^(3*x) - e^x))/a^(7/2)