3.124 \(\int \frac{\text{csch}^2(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=385 \[ -\frac{2 b (3 a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \text{EllipticF}\left (\tan ^{-1}(\sinh (e+f x)),1-\frac{b}{a}\right )}{3 a^3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{\left (3 a^2-13 a b+8 b^2\right ) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)^2}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)^2}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a^3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{2 b (3 a-2 b) \coth (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{b \coth (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]

[Out]

-(b*Coth[e + f*x])/(3*a*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (2*(3*a - 2*b)*b*Coth[e + f*x])/(3*a^2*(a -
b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) - ((3*a^2 - 13*a*b + 8*b^2)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3
*a^3*(a - b)^2*f) - ((3*a^2 - 13*a*b + 8*b^2)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a +
b*Sinh[e + f*x]^2])/(3*a^3*(a - b)^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (2*(3*a - 2*b)*b*
EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*(a - b)^2*f*Sqrt[(
Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a^2 - 13*a*b + 8*b^2)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e +
f*x])/(3*a^3*(a - b)^2*f)

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Rubi [A]  time = 0.456218, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3188, 472, 579, 583, 531, 418, 492, 411} \[ \frac{\left (3 a^2-13 a b+8 b^2\right ) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)^2}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)^2}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a^3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{2 b (3 a-2 b) \coth (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{2 b (3 a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a^3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{b \coth (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csch[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

-(b*Coth[e + f*x])/(3*a*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (2*(3*a - 2*b)*b*Coth[e + f*x])/(3*a^2*(a -
b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) - ((3*a^2 - 13*a*b + 8*b^2)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3
*a^3*(a - b)^2*f) - ((3*a^2 - 13*a*b + 8*b^2)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a +
b*Sinh[e + f*x]^2])/(3*a^3*(a - b)^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (2*(3*a - 2*b)*b*
EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*(a - b)^2*f*Sqrt[(
Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a^2 - 13*a*b + 8*b^2)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e +
f*x])/(3*a^3*(a - b)^2*f)

Rule 3188

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Sin[e + f*x], x]}, Dist[(ff^(m + 1)*Sqrt[Cos[e + f*x]^2])/(f*Cos[e + f*x]), Subst[Int[(x^m*(a + b*ff^2*
x^2)^p)/Sqrt[1 - ff^2*x^2], x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] &&  !In
tegerQ[p]

Rule 472

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 579

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +
1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(
m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 531

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{\text{csch}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x^2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac{b \coth (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{3 a-4 b-3 b x^2}{x^2 \sqrt{1+x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a (a-b) f}\\ &=-\frac{b \coth (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (3 a-2 b) b \coth (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{3 a^2-13 a b+8 b^2-2 (3 a-2 b) b x^2}{x^2 \sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a^2 (a-b)^2 f}\\ &=-\frac{b \coth (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (3 a-2 b) b \coth (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b)^2 f}-\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{2 a (3 a-2 b) b-b \left (3 a^2-13 a b+8 b^2\right ) x^2}{\sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a^3 (a-b)^2 f}\\ &=-\frac{b \coth (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (3 a-2 b) b \coth (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b)^2 f}-\frac{\left (2 (3 a-2 b) b \sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a^2 (a-b)^2 f}+\frac{\left (b \left (3 a^2-13 a b+8 b^2\right ) \sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a^3 (a-b)^2 f}\\ &=-\frac{b \coth (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (3 a-2 b) b \coth (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b)^2 f}-\frac{2 (3 a-2 b) b F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b)^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{\left (3 a^2-13 a b+8 b^2\right ) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^3 (a-b)^2 f}-\frac{\left (\left (3 a^2-13 a b+8 b^2\right ) \sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{\left (1+x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a^3 (a-b)^2 f}\\ &=-\frac{b \coth (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (3 a-2 b) b \coth (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b)^2 f}-\frac{\left (3 a^2-13 a b+8 b^2\right ) E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b)^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{2 (3 a-2 b) b F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b)^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{\left (3 a^2-13 a b+8 b^2\right ) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^3 (a-b)^2 f}\\ \end{align*}

Mathematica [C]  time = 2.3105, size = 234, normalized size = 0.61 \[ \frac{i \left (4 a^2 \left (\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} \left (\left (3 a^2-7 a b+4 b^2\right ) \text{EllipticF}\left (i (e+f x),\frac{b}{a}\right )+\left (-3 a^2+13 a b-8 b^2\right ) E\left (i (e+f x)\left |\frac{b}{a}\right .\right )\right )+2 i \sqrt{2} \left (-2 a b^2 (a-b) \sinh (2 (e+f x))-b^2 (7 a-5 b) \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)+3 (a-b)^2 \coth (e+f x) (2 a+b \cosh (2 (e+f x))-b)^2\right )\right )}{12 a^3 f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

((I/12)*(4*a^2*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*((-3*a^2 + 13*a*b - 8*b^2)*EllipticE[I*(e + f*x), b/a
] + (3*a^2 - 7*a*b + 4*b^2)*EllipticF[I*(e + f*x), b/a]) + (2*I)*Sqrt[2]*(3*(a - b)^2*(2*a - b + b*Cosh[2*(e +
f*x)])^2*Coth[e + f*x] - 2*a*(a - b)*b^2*Sinh[2*(e + f*x)] - (7*a - 5*b)*b^2*(2*a - b + b*Cosh[2*(e + f*x)])*
Sinh[2*(e + f*x)])))/(a^3*(a - b)^2*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))

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Maple [A]  time = 0.128, size = 747, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(5/2),x)

[Out]

-1/3*((b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*b^2*(9*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b
)^(1/2))*a^2-17*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b+8*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(
a/b)^(1/2))*b^2-3*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^2+13*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2
),(a/b)^(1/2))*a*b-8*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^2)*sinh(f*x+e)*cosh(f*x+e)^2+(b/a*cos
h(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*b*(9*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^3-26*
EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^2*b+25*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a
*b^2-8*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^3-3*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2
))*a^3+16*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^2*b-21*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b
)^(1/2))*a*b^2+8*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^3)*sinh(f*x+e)+(3*(-1/a*b)^(1/2)*a^2*b^2-
13*(-1/a*b)^(1/2)*a*b^3+8*(-1/a*b)^(1/2)*b^4)*cosh(f*x+e)^6+(6*(-1/a*b)^(1/2)*a^3*b-26*(-1/a*b)^(1/2)*a^2*b^2+
38*(-1/a*b)^(1/2)*a*b^3-16*(-1/a*b)^(1/2)*b^4)*cosh(f*x+e)^4+(3*(-1/a*b)^(1/2)*a^4-12*(-1/a*b)^(1/2)*a^3*b+26*
(-1/a*b)^(1/2)*a^2*b^2-25*(-1/a*b)^(1/2)*a*b^3+8*(-1/a*b)^(1/2)*b^4)*cosh(f*x+e)^2)/(a+b*sinh(f*x+e)^2)^(3/2)/
(a-b)^2/(-1/a*b)^(1/2)/sinh(f*x+e)/a^3/cosh(f*x+e)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}\left (f x + e\right )^{2}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(csch(f*x + e)^2/(b*sinh(f*x + e)^2 + a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sinh \left (f x + e\right )^{2} + a} \operatorname{csch}\left (f x + e\right )^{2}}{b^{3} \sinh \left (f x + e\right )^{6} + 3 \, a b^{2} \sinh \left (f x + e\right )^{4} + 3 \, a^{2} b \sinh \left (f x + e\right )^{2} + a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sinh(f*x + e)^2 + a)*csch(f*x + e)^2/(b^3*sinh(f*x + e)^6 + 3*a*b^2*sinh(f*x + e)^4 + 3*a^2*b*
sinh(f*x + e)^2 + a^3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)**2/(a+b*sinh(f*x+e)**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}\left (f x + e\right )^{2}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="giac")

[Out]

integrate(csch(f*x + e)^2/(b*sinh(f*x + e)^2 + a)^(5/2), x)