3.2.24 \(\int \frac {\text {csch}^2(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [124]

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

[Out]

-1/3*b*coth(f*x+e)/a/(a-b)/f/(a+b*sinh(f*x+e)^2)^(3/2)-2/3*(3*a-2*b)*b*coth(f*x+e)/a^2/(a-b)^2/f/(a+b*sinh(f*x
+e)^2)^(1/2)-1/3*(3*a^2-13*a*b+8*b^2)*coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a^3/(a-b)^2/f-1/3*(3*a^2-13*a*b+8*
b^2)*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2),(1-b/a)
^(1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a^3/(a-b)^2/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)-2/3*(3
*a-2*b)*b*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2),(1
-b/a)^(1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a^3/(a-b)^2/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1
/3*(3*a^2-13*a*b+8*b^2)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/a^3/(a-b)^2/f

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Rubi [A]
time = 0.33, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3267, 483, 593, 597, 545, 429, 506, 422} \begin {gather*} -\frac {2 b (3 a-2 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\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 {\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 (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\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 {b \coth (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(b*Coth[e + f*x])/(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 422

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

Rule 429

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

Rule 483

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 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[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 545

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 593

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 597

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 3267

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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.61, size = 234, normalized size = 0.61 \begin {gather*} \frac {i \left (4 a^2 \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} \left (\left (-3 a^2+13 a b-8 b^2\right ) E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+\left (3 a^2-7 a b+4 b^2\right ) F\left (i (e+f x)\left |\frac {b}{a}\right .\right )\right )+2 i \sqrt {2} \left (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))\right )\right )}{12 a^3 (a-b)^2 f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \end {gather*}

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 = 1.91, size = 747, normalized size = 1.94

method	result	size
default	\(-\frac {\left (3 \sqrt {-\frac {b}{a}}\, a^{2} b^{2}-13 \sqrt {-\frac {b}{a}}\, a \,b^{3}+8 \sqrt {-\frac {b}{a}}\, b^{4}\right ) \left (\cosh ^{6}\left (f x +e \right )\right )+\left (6 \sqrt {-\frac {b}{a}}\, a^{3} b -26 \sqrt {-\frac {b}{a}}\, a^{2} b^{2}+38 \sqrt {-\frac {b}{a}}\, a \,b^{3}-16 \sqrt {-\frac {b}{a}}\, b^{4}\right ) \left (\cosh ^{4}\left (f x +e \right )\right )+\sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, b^{2} \left (9 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2}-17 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b +8 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}-3 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2}+13 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b -8 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}\right ) \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\left (3 \sqrt {-\frac {b}{a}}\, a^{4}-12 \sqrt {-\frac {b}{a}}\, a^{3} b +26 \sqrt {-\frac {b}{a}}\, a^{2} b^{2}-25 \sqrt {-\frac {b}{a}}\, a \,b^{3}+8 \sqrt {-\frac {b}{a}}\, b^{4}\right ) \left (\cosh ^{2}\left (f x +e \right )\right )+\sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, b \left (9 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{3}-26 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b +25 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}-8 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}-3 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{3}+16 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b -21 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}+8 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}\right ) \sinh \left (f x +e \right )}{3 \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}\, \left (a -b \right )^{2} a^{3} \cosh \left (f x +e \right ) f}\)	\(747\)
risch	\(\text {Expression too large to display}\)	\(83949\)

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,method=_RETURNVERBOSE)

[Out]

-1/3*((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+(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*Ellipti
cF(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*Ellip
ticE(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*El
lipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^2)*cosh(f*x+e)^2*sinh(f*x+e)+(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+(b/a*cosh
(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*E
llipticF(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))/(a+b*sinh(f*x+e)^2)^(3/2)/
sinh(f*x+e)/(-1/a*b)^(1/2)/(a-b)^2/a^3/cosh(f*x+e)/f

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 8769 vs. \(2 (385) = 770\).
time = 0.31, size = 8769, normalized size = 22.78 \begin {gather*} \text {too large to display} \end {gather*}

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]

1/3*(((6*a^3*b^2 - 29*a^2*b^3 + 29*a*b^4 - 8*b^5)*cosh(f*x + e)^10 + 10*(6*a^3*b^2 - 29*a^2*b^3 + 29*a*b^4 - 8
*b^5)*cosh(f*x + e)*sinh(f*x + e)^9 + (6*a^3*b^2 - 29*a^2*b^3 + 29*a*b^4 - 8*b^5)*sinh(f*x + e)^10 + (48*a^4*b
- 262*a^3*b^2 + 377*a^2*b^3 - 209*a*b^4 + 40*b^5)*cosh(f*x + e)^8 + (48*a^4*b - 262*a^3*b^2 + 377*a^2*b^3 - 2
09*a*b^4 + 40*b^5 + 45*(6*a^3*b^2 - 29*a^2*b^3 + 29*a*b^4 - 8*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 8*(15*(6
*a^3*b^2 - 29*a^2*b^3 + 29*a*b^4 - 8*b^5)*cosh(f*x + e)^3 + (48*a^4*b - 262*a^3*b^2 + 377*a^2*b^3 - 209*a*b^4
+ 40*b^5)*cosh(f*x + e))*sinh(f*x + e)^7 + 2*(48*a^5 - 304*a^4*b + 610*a^3*b^2 - 557*a^2*b^3 + 241*a*b^4 - 40*
b^5)*cosh(f*x + e)^6 + 2*(48*a^5 - 304*a^4*b + 610*a^3*b^2 - 557*a^2*b^3 + 241*a*b^4 - 40*b^5 + 105*(6*a^3*b^2
- 29*a^2*b^3 + 29*a*b^4 - 8*b^5)*cosh(f*x + e)^4 + 14*(48*a^4*b - 262*a^3*b^2 + 377*a^2*b^3 - 209*a*b^4 + 40*
b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 4*(63*(6*a^3*b^2 - 29*a^2*b^3 + 29*a*b^4 - 8*b^5)*cosh(f*x + e)^5 + 14
*(48*a^4*b - 262*a^3*b^2 + 377*a^2*b^3 - 209*a*b^4 + 40*b^5)*cosh(f*x + e)^3 + 3*(48*a^5 - 304*a^4*b + 610*a^3
*b^2 - 557*a^2*b^3 + 241*a*b^4 - 40*b^5)*cosh(f*x + e))*sinh(f*x + e)^5 - 6*a^3*b^2 + 29*a^2*b^3 - 29*a*b^4 +
8*b^5 - 2*(48*a^5 - 304*a^4*b + 610*a^3*b^2 - 557*a^2*b^3 + 241*a*b^4 - 40*b^5)*cosh(f*x + e)^4 + 2*(105*(6*a^
3*b^2 - 29*a^2*b^3 + 29*a*b^4 - 8*b^5)*cosh(f*x + e)^6 - 48*a^5 + 304*a^4*b - 610*a^3*b^2 + 557*a^2*b^3 - 241*
a*b^4 + 40*b^5 + 35*(48*a^4*b - 262*a^3*b^2 + 377*a^2*b^3 - 209*a*b^4 + 40*b^5)*cosh(f*x + e)^4 + 15*(48*a^5 -
304*a^4*b + 610*a^3*b^2 - 557*a^2*b^3 + 241*a*b^4 - 40*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(15*(6*a^3*b
^2 - 29*a^2*b^3 + 29*a*b^4 - 8*b^5)*cosh(f*x + e)^7 + 7*(48*a^4*b - 262*a^3*b^2 + 377*a^2*b^3 - 209*a*b^4 + 40
*b^5)*cosh(f*x + e)^5 + 5*(48*a^5 - 304*a^4*b + 610*a^3*b^2 - 557*a^2*b^3 + 241*a*b^4 - 40*b^5)*cosh(f*x + e)^
3 - (48*a^5 - 304*a^4*b + 610*a^3*b^2 - 557*a^2*b^3 + 241*a*b^4 - 40*b^5)*cosh(f*x + e))*sinh(f*x + e)^3 - (48
*a^4*b - 262*a^3*b^2 + 377*a^2*b^3 - 209*a*b^4 + 40*b^5)*cosh(f*x + e)^2 + (45*(6*a^3*b^2 - 29*a^2*b^3 + 29*a*
b^4 - 8*b^5)*cosh(f*x + e)^8 + 28*(48*a^4*b - 262*a^3*b^2 + 377*a^2*b^3 - 209*a*b^4 + 40*b^5)*cosh(f*x + e)^6
- 48*a^4*b + 262*a^3*b^2 - 377*a^2*b^3 + 209*a*b^4 - 40*b^5 + 30*(48*a^5 - 304*a^4*b + 610*a^3*b^2 - 557*a^2*b
^3 + 241*a*b^4 - 40*b^5)*cosh(f*x + e)^4 - 12*(48*a^5 - 304*a^4*b + 610*a^3*b^2 - 557*a^2*b^3 + 241*a*b^4 - 40
*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(5*(6*a^3*b^2 - 29*a^2*b^3 + 29*a*b^4 - 8*b^5)*cosh(f*x + e)^9 + 4*
(48*a^4*b - 262*a^3*b^2 + 377*a^2*b^3 - 209*a*b^4 + 40*b^5)*cosh(f*x + e)^7 + 6*(48*a^5 - 304*a^4*b + 610*a^3*
b^2 - 557*a^2*b^3 + 241*a*b^4 - 40*b^5)*cosh(f*x + e)^5 - 4*(48*a^5 - 304*a^4*b + 610*a^3*b^2 - 557*a^2*b^3 +
241*a*b^4 - 40*b^5)*cosh(f*x + e)^3 - (48*a^4*b - 262*a^3*b^2 + 377*a^2*b^3 - 209*a*b^4 + 40*b^5)*cosh(f*x + e
))*sinh(f*x + e) - 2*((3*a^2*b^3 - 13*a*b^4 + 8*b^5)*cosh(f*x + e)^10 + 10*(3*a^2*b^3 - 13*a*b^4 + 8*b^5)*cosh
(f*x + e)*sinh(f*x + e)^9 + (3*a^2*b^3 - 13*a*b^4 + 8*b^5)*sinh(f*x + e)^10 + (24*a^3*b^2 - 119*a^2*b^3 + 129*
a*b^4 - 40*b^5)*cosh(f*x + e)^8 + (24*a^3*b^2 - 119*a^2*b^3 + 129*a*b^4 - 40*b^5 + 45*(3*a^2*b^3 - 13*a*b^4 +
8*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 8*(15*(3*a^2*b^3 - 13*a*b^4 + 8*b^5)*cosh(f*x + e)^3 + (24*a^3*b^2 -
119*a^2*b^3 + 129*a*b^4 - 40*b^5)*cosh(f*x + e))*sinh(f*x + e)^7 + 2*(24*a^4*b - 140*a^3*b^2 + 235*a^2*b^3 -
161*a*b^4 + 40*b^5)*cosh(f*x + e)^6 + 2*(24*a^4*b - 140*a^3*b^2 + 235*a^2*b^3 - 161*a*b^4 + 40*b^5 + 105*(3*a^
2*b^3 - 13*a*b^4 + 8*b^5)*cosh(f*x + e)^4 + 14*(24*a^3*b^2 - 119*a^2*b^3 + 129*a*b^4 - 40*b^5)*cosh(f*x + e)^2
)*sinh(f*x + e)^6 + 4*(63*(3*a^2*b^3 - 13*a*b^4 + 8*b^5)*cosh(f*x + e)^5 + 14*(24*a^3*b^2 - 119*a^2*b^3 + 129*
a*b^4 - 40*b^5)*cosh(f*x + e)^3 + 3*(24*a^4*b - 140*a^3*b^2 + 235*a^2*b^3 - 161*a*b^4 + 40*b^5)*cosh(f*x + e))
*sinh(f*x + e)^5 - 3*a^2*b^3 + 13*a*b^4 - 8*b^5 - 2*(24*a^4*b - 140*a^3*b^2 + 235*a^2*b^3 - 161*a*b^4 + 40*b^5
)*cosh(f*x + e)^4 + 2*(105*(3*a^2*b^3 - 13*a*b^4 + 8*b^5)*cosh(f*x + e)^6 - 24*a^4*b + 140*a^3*b^2 - 235*a^2*b
^3 + 161*a*b^4 - 40*b^5 + 35*(24*a^3*b^2 - 119*a^2*b^3 + 129*a*b^4 - 40*b^5)*cosh(f*x + e)^4 + 15*(24*a^4*b -
140*a^3*b^2 + 235*a^2*b^3 - 161*a*b^4 + 40*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(15*(3*a^2*b^3 - 13*a*b^4
+ 8*b^5)*cosh(f*x + e)^7 + 7*(24*a^3*b^2 - 119*a^2*b^3 + 129*a*b^4 - 40*b^5)*cosh(f*x + e)^5 + 5*(24*a^4*b -
140*a^3*b^2 + 235*a^2*b^3 - 161*a*b^4 + 40*b^5)*cosh(f*x + e)^3 - (24*a^4*b - 140*a^3*b^2 + 235*a^2*b^3 - 161*
a*b^4 + 40*b^5)*cosh(f*x + e))*sinh(f*x + e)^3 - (24*a^3*b^2 - 119*a^2*b^3 + 129*a*b^4 - 40*b^5)*cosh(f*x + e)
^2 + (45*(3*a^2*b^3 - 13*a*b^4 + 8*b^5)*cosh(f*x + e)^8 + 28*(24*a^3*b^2 - 119*a^2*b^3 + 129*a*b^4 - 40*b^5)*c
osh(f*x + e)^6 - 24*a^3*b^2 + 119*a^2*b^3 - 129*a*b^4 + 40*b^5 + 30*(24*a^4*b - 140*a^3*b^2 + 235*a^2*b^3 - 16
1*a*b^4 + 40*b^5)*cosh(f*x + e)^4 - 12*(24*a^4*b - 140*a^3*b^2 + 235*a^2*b^3 - 161*a*b^4 + 40*b^5)*cosh(f*x +
e)^2)*sinh(f*x + e)^2 + 2*(5*(3*a^2*b^3 - 13*a*...

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 5007 deep

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Evaluation time:
0.71Error: Bad Argument Type

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\mathrm {sinh}\left (e+f\,x\right )}^2\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(e + f*x)^2*(a + b*sinh(e + f*x)^2)^(5/2)),x)

[Out]

int(1/(sinh(e + f*x)^2*(a + b*sinh(e + f*x)^2)^(5/2)), x)

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