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

Optimal. Leaf size=65 \[ \frac{16 \coth (x)}{35 \sqrt{-\text{csch}^2(x)}}+\frac{8 \coth (x)}{35 \left (-\text{csch}^2(x)\right )^{3/2}}+\frac{6 \coth (x)}{35 \left (-\text{csch}^2(x)\right )^{5/2}}+\frac{\coth (x)}{7 \left (-\text{csch}^2(x)\right )^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.0241402, antiderivative size = 65, 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} \[ \frac{16 \coth (x)}{35 \sqrt{-\text{csch}^2(x)}}+\frac{8 \coth (x)}{35 \left (-\text{csch}^2(x)\right )^{3/2}}+\frac{6 \coth (x)}{35 \left (-\text{csch}^2(x)\right )^{5/2}}+\frac{\coth (x)}{7 \left (-\text{csch}^2(x)\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0341773, size = 39, normalized size = 0.6 \[ \frac{(1225 \cosh (x)-245 \cosh (3 x)+49 \cosh (5 x)-5 \cosh (7 x)) \text{csch}(x)}{2240 \sqrt{-\text{csch}^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.044, size = 238, normalized size = 3.7 \begin{align*} -{\frac{{{\rm e}^{8\,x}}}{896\,{{\rm e}^{2\,x}}-896}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{7\,{{\rm e}^{6\,x}}}{640\,{{\rm e}^{2\,x}}-640}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{7\,{{\rm e}^{4\,x}}}{128\,{{\rm e}^{2\,x}}-128}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{35\,{{\rm e}^{2\,x}}}{128\,{{\rm e}^{2\,x}}-128}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{35}{128\,{{\rm e}^{2\,x}}-128}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{7\,{{\rm e}^{-2\,x}}}{128\,{{\rm e}^{2\,x}}-128}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{7\,{{\rm e}^{-4\,x}}}{640\,{{\rm e}^{2\,x}}-640}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{{{\rm e}^{-6\,x}}}{896\,{{\rm e}^{2\,x}}-896}{\frac{1}{\sqrt{-{\frac{{{\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/(-csch(x)^2)^(7/2),x)

[Out]

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

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Maxima [C]  time = 1.70011, size = 63, normalized size = 0.97 \begin{align*} -\frac{1}{896} i \, e^{\left (7 \, x\right )} + \frac{7}{640} i \, e^{\left (5 \, x\right )} - \frac{7}{128} i \, e^{\left (3 \, x\right )} + \frac{35}{128} i \, e^{\left (-x\right )} - \frac{7}{128} i \, e^{\left (-3 \, x\right )} + \frac{7}{640} i \, e^{\left (-5 \, x\right )} - \frac{1}{896} i \, e^{\left (-7 \, x\right )} + \frac{35}{128} i \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 1.70587, size = 184, normalized size = 2.83 \begin{align*} \frac{1}{4480} \,{\left (5 i \, e^{\left (14 \, x\right )} - 49 i \, e^{\left (12 \, x\right )} + 245 i \, e^{\left (10 \, x\right )} - 1225 i \, e^{\left (8 \, x\right )} - 1225 i \, e^{\left (6 \, x\right )} + 245 i \, e^{\left (4 \, x\right )} - 49 i \, e^{\left (2 \, x\right )} + 5 i\right )} e^{\left (-7 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)^2)^(7/2),x, algorithm="fricas")

[Out]

1/4480*(5*I*e^(14*x) - 49*I*e^(12*x) + 245*I*e^(10*x) - 1225*I*e^(8*x) - 1225*I*e^(6*x) + 245*I*e^(4*x) - 49*I
*e^(2*x) + 5*I)*e^(-7*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- \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/(-csch(x)**2)**(7/2),x)

[Out]

Integral((-csch(x)**2)**(-7/2), x)

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Giac [C]  time = 1.19482, size = 103, normalized size = 1.58 \begin{align*} \frac{i \,{\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 )}}{4480 \, \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} - \frac{i \,{\left (5 \, e^{\left (7 \, x\right )} - 49 \, e^{\left (5 \, x\right )} + 245 \, e^{\left (3 \, x\right )} - 1225 \, e^{x}\right )}}{4480 \, \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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