3.6.11 \(\int \frac {\coth ^4(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [511]

Optimal. Leaf size=385 \[ -\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (a-3 b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a^2 b f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {8 (a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac {(3 a-8 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 b f}-\frac {8 (a-2 b) E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^4 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a-8 b) F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^4 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {8 (a-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^4 f} \]

[Out]

-1/3*(a-b)*coth(f*x+e)*csch(f*x+e)^2/a/b/f/(a+b*sinh(f*x+e)^2)^(3/2)-2/3*(a-3*b)*coth(f*x+e)*csch(f*x+e)^2/a^2
/b/f/(a+b*sinh(f*x+e)^2)^(1/2)-8/3*(a-2*b)*coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a^4/f+1/3*(3*a-8*b)*coth(f*x+
e)*csch(f*x+e)^2*(a+b*sinh(f*x+e)^2)^(1/2)/a^3/b/f-8/3*(a-2*b)*(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^4/f/
(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/3*(3*a-8*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^4/f/(sech
(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+8/3*(a-2*b)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/a^4/f

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Rubi [A]
time = 0.38, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3275, 479, 593, 597, 545, 429, 506, 422} \begin {gather*} \frac {(3 a-8 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^4 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {8 (a-2 b) \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^4 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {8 (a-2 b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^4 f}-\frac {8 (a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac {(3 a-8 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 b f}-\frac {2 (a-3 b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a^2 b f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

-1/3*((a - b)*Coth[e + f*x]*Csch[e + f*x]^2)/(a*b*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (2*(a - 3*b)*Coth[e + f*x
]*Csch[e + f*x]^2)/(3*a^2*b*f*Sqrt[a + b*Sinh[e + f*x]^2]) - (8*(a - 2*b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*
x]^2])/(3*a^4*f) + ((3*a - 8*b)*Coth[e + f*x]*Csch[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*b*f) - (8*(a
- 2*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^4*f*Sqrt[(Se
ch[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a - 8*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f
*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^4*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (8*(a - 2*b)*Sq
rt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a^4*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 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(c*b -
a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Dist[1/(a*b*n*(p + 1)),
Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(
p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 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 3275

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_), 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/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2]
&&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\coth ^4(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 {\left (1+x^2\right )^{3/2}}{x^4 \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{3 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-2 b)+(2 a-5 b) x^2}{x^4 \sqrt {1+x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a b f}\\ &=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (a-3 b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a^2 b 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 (3 a-8 b) (a-b)+6 (a-3 b) (a-b) x^2}{x^4 \sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a^2 (a-b) b f}\\ &=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (a-3 b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a^2 b f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(3 a-8 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 b f}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {24 (a-2 b) (a-b) b+3 (3 a-8 b) (a-b) b x^2}{x^2 \sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{9 a^3 (a-b) b f}\\ &=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (a-3 b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a^2 b f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {8 (a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac {(3 a-8 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 b f}-\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {-3 a (3 a-8 b) (a-b) b-24 (a-2 b) (a-b) b^2 x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{9 a^4 (a-b) b f}\\ &=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (a-3 b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a^2 b f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {8 (a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac {(3 a-8 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 b f}+\frac {\left ((3 a-8 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^3 f}+\frac {\left (8 (a-2 b) b \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^4 f}\\ &=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (a-3 b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a^2 b f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {8 (a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac {(3 a-8 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 b f}+\frac {(3 a-8 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^4 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {8 (a-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^4 f}-\frac {\left (8 (a-2 b) \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^4 f}\\ &=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (a-3 b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a^2 b f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {8 (a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac {(3 a-8 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 b f}-\frac {8 (a-2 b) 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^4 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a-8 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^4 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {8 (a-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^4 f}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.39, size = 247, normalized size = 0.64 \begin {gather*} -\frac {i \left (\frac {i b \left (8 a^3-63 a^2 b+92 a b^2-40 b^3-2 \left (8 a^3-38 a^2 b+63 a b^2-30 b^3\right ) \cosh (2 (e+f x))-b \left (13 a^2-36 a b+24 b^2\right ) \cosh (4 (e+f x))-2 a b^2 \cosh (6 (e+f x))+4 b^3 \cosh (6 (e+f x))\right ) \coth (e+f x) \text {csch}^2(e+f x)}{\sqrt {2}}+2 a^2 b \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} \left (8 (a-2 b) E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+(-5 a+8 b) F\left (i (e+f x)\left |\frac {b}{a}\right .\right )\right )\right )}{6 a^4 b f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

((-1/6*I)*((I*b*(8*a^3 - 63*a^2*b + 92*a*b^2 - 40*b^3 - 2*(8*a^3 - 38*a^2*b + 63*a*b^2 - 30*b^3)*Cosh[2*(e + f
*x)] - b*(13*a^2 - 36*a*b + 24*b^2)*Cosh[4*(e + f*x)] - 2*a*b^2*Cosh[6*(e + f*x)] + 4*b^3*Cosh[6*(e + f*x)])*C
oth[e + f*x]*Csch[e + f*x]^2)/Sqrt[2] + 2*a^2*b*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*(8*(a - 2*b)*Ellipti
cE[I*(e + f*x), b/a] + (-5*a + 8*b)*EllipticF[I*(e + f*x), b/a])))/(a^4*b*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3
/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(922\) vs. \(2(433)=866\).
time = 2.94, size = 923, normalized size = 2.40

method	result	size
default	\(\text {Expression too large to display}\)	\(923\)
risch	\(\text {Expression too large to display}\)	\(1124572\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(f*x+e)^4/(a+b*sinh(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(8*(-1/a*b)^(1/2)*a*b^2*sinh(f*x+e)^8-16*(-1/a*b)^(1/2)*b^3*sinh(f*x+e)^8-3*((a+b*sinh(f*x+e)^2)/a)^(1/2)
*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^2*b*sinh(f*x+e)^5+16*((a+b*sinh(f*x
+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b^2*sinh(f*x+e)^5-16
*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^3*sin
h(f*x+e)^5-8*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1
/2))*a*b^2*sinh(f*x+e)^5+16*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)
^(1/2),(a/b)^(1/2))*b^3*sinh(f*x+e)^5+13*(-1/a*b)^(1/2)*a^2*b*sinh(f*x+e)^6-16*(-1/a*b)^(1/2)*a*b^2*sinh(f*x+e
)^6-16*(-1/a*b)^(1/2)*b^3*sinh(f*x+e)^6-3*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f
*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^3*sinh(f*x+e)^3+16*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*Ell
ipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^2*b*sinh(f*x+e)^3-16*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+
e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b^2*sinh(f*x+e)^3-8*((a+b*sinh(f*x+e)^2)/a)^(1
/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^2*b*sinh(f*x+e)^3+16*((a+b*sinh(
f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b^2*sinh(f*x+e)^3
+4*(-1/a*b)^(1/2)*a^3*sinh(f*x+e)^4+7*(-1/a*b)^(1/2)*a^2*b*sinh(f*x+e)^4-24*(-1/a*b)^(1/2)*a*b^2*sinh(f*x+e)^4
+5*(-1/a*b)^(1/2)*a^3*sinh(f*x+e)^2-6*(-1/a*b)^(1/2)*a^2*b*sinh(f*x+e)^2+(-1/a*b)^(1/2)*a^3)/a^4/(-1/a*b)^(1/2
)/(a+b*sinh(f*x+e)^2)^(3/2)/sinh(f*x+e)^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(coth(f*x+e)^4/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(coth(f*x + e)^4/(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. 13823 vs. \(2 (381) = 762\).
time = 0.41, size = 13823, normalized size = 35.90 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^4/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

2/3*(4*((2*a^2*b^3 - 5*a*b^4 + 2*b^5)*cosh(f*x + e)^14 + 14*(2*a^2*b^3 - 5*a*b^4 + 2*b^5)*cosh(f*x + e)*sinh(f
*x + e)^13 + (2*a^2*b^3 - 5*a*b^4 + 2*b^5)*sinh(f*x + e)^14 + (16*a^3*b^2 - 54*a^2*b^3 + 51*a*b^4 - 14*b^5)*co
sh(f*x + e)^12 + (16*a^3*b^2 - 54*a^2*b^3 + 51*a*b^4 - 14*b^5 + 91*(2*a^2*b^3 - 5*a*b^4 + 2*b^5)*cosh(f*x + e)
^2)*sinh(f*x + e)^12 + 4*(91*(2*a^2*b^3 - 5*a*b^4 + 2*b^5)*cosh(f*x + e)^3 + 3*(16*a^3*b^2 - 54*a^2*b^3 + 51*a
*b^4 - 14*b^5)*cosh(f*x + e))*sinh(f*x + e)^11 + (32*a^4*b - 160*a^3*b^2 + 274*a^2*b^3 - 185*a*b^4 + 42*b^5)*c
osh(f*x + e)^10 + (32*a^4*b - 160*a^3*b^2 + 274*a^2*b^3 - 185*a*b^4 + 42*b^5 + 1001*(2*a^2*b^3 - 5*a*b^4 + 2*b
^5)*cosh(f*x + e)^4 + 66*(16*a^3*b^2 - 54*a^2*b^3 + 51*a*b^4 - 14*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^10 + 2*(
1001*(2*a^2*b^3 - 5*a*b^4 + 2*b^5)*cosh(f*x + e)^5 + 110*(16*a^3*b^2 - 54*a^2*b^3 + 51*a*b^4 - 14*b^5)*cosh(f*
x + e)^3 + 5*(32*a^4*b - 160*a^3*b^2 + 274*a^2*b^3 - 185*a*b^4 + 42*b^5)*cosh(f*x + e))*sinh(f*x + e)^9 - (96*
a^4*b - 40 ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)**4/(a+b*sinh(f*x+e)**2)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^4/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 1.24Error: Bad Argument Type

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {coth}\left (e+f\,x\right )}^4}{{\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(coth(e + f*x)^4/(a + b*sinh(e + f*x)^2)^(5/2),x)

[Out]

int(coth(e + f*x)^4/(a + b*sinh(e + f*x)^2)^(5/2), x)

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