3.6.10 \(\int \frac {\coth ^2(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [510]

Optimal. Leaf size=351 \[ \frac {\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(3 a-4 b) \coth (e+f x)}{3 a^2 (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(7 a-8 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f}-\frac {(7 a-8 b) E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a-4 b) F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(7 a-8 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^3 (a-b) f} \]

[Out]

1/3*coth(f*x+e)/a/f/(a+b*sinh(f*x+e)^2)^(3/2)+1/3*(3*a-4*b)*coth(f*x+e)/a^2/(a-b)/f/(a+b*sinh(f*x+e)^2)^(1/2)-
1/3*(7*a-8*b)*coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a^3/(a-b)/f-1/3*(7*a-8*b)*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+s
inh(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)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/3*(3*a-4*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)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/3*(7*a-8*b)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh
(f*x+e)/a^3/(a-b)/f

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Rubi [A]
time = 0.28, antiderivative size = 351, 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 = {3275, 480, 593, 597, 545, 429, 506, 422} \begin {gather*} \frac {(3 a-4 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) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(7 a-8 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^3 f (a-b) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(7 a-8 b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)}-\frac {(7 a-8 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)}+\frac {(3 a-4 b) \coth (e+f x)}{3 a^2 f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Coth[e + f*x]/(3*a*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + ((3*a - 4*b)*Coth[e + f*x])/(3*a^2*(a - b)*f*Sqrt[a + b*
Sinh[e + f*x]^2]) - ((7*a - 8*b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*(a - b)*f) - ((7*a - 8*b)*E
llipticE[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)*f*Sqrt[(Sec
h[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a - 4*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)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((7*a -
8*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a^3*(a - b)*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 480

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(e*x)^
(m + 1))*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*e*n*(p + 1))), x] + Dist[1/(a*n*(p + 1)), Int[(e*x)^m*(a + b*x^
n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + 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] && LtQ[0, 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 ^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 {\sqrt {1+x^2}}{x^2 \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac {\coth (e+f x)}{3 a 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 {-4-3 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 f}\\ &=\frac {\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(3 a-4 b) \coth (e+f x)}{3 a^2 (a-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 {-7 a+8 b+(-3 a+4 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) f}\\ &=\frac {\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(3 a-4 b) \coth (e+f x)}{3 a^2 (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(7 a-8 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {a (3 a-4 b)+(7 a-8 b) b x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a^3 (a-b) f}\\ &=\frac {\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(3 a-4 b) \coth (e+f x)}{3 a^2 (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(7 a-8 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f}+\frac {\left ((3 a-4 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) f}+\frac {\left ((7 a-8 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^3 (a-b) f}\\ &=\frac {\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(3 a-4 b) \coth (e+f x)}{3 a^2 (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(7 a-8 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f}+\frac {(3 a-4 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) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(7 a-8 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^3 (a-b) f}-\frac {\left ((7 a-8 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^3 (a-b) f}\\ &=\frac {\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(3 a-4 b) \coth (e+f x)}{3 a^2 (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(7 a-8 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f}-\frac {(7 a-8 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^3 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a-4 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) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(7 a-8 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^3 (a-b) f}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.28, size = 226, normalized size = 0.64 \begin {gather*} \frac {-\frac {\left (24 a^3-68 a^2 b+69 a b^2-24 b^3+4 b \left (11 a^2-19 a b+8 b^2\right ) \cosh (2 (e+f x))+(7 a-8 b) b^2 \cosh (4 (e+f x))\right ) \coth (e+f x)}{\sqrt {2}}-2 i a^2 (7 a-8 b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+8 i a^2 (a-b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )}{6 a^3 (a-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]^2/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

(-(((24*a^3 - 68*a^2*b + 69*a*b^2 - 24*b^3 + 4*b*(11*a^2 - 19*a*b + 8*b^2)*Cosh[2*(e + f*x)] + (7*a - 8*b)*b^2
*Cosh[4*(e + f*x)])*Coth[e + f*x])/Sqrt[2]) - (2*I)*a^2*(7*a - 8*b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*
EllipticE[I*(e + f*x), b/a] + (8*I)*a^2*(a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x
), b/a])/(6*a^3*(a - b)*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))

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Maple [A]
time = 2.15, size = 642, normalized size = 1.83

method	result	size
default	\(-\frac {\left (7 \sqrt {-\frac {b}{a}}\, a \,b^{2}-8 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \left (\cosh ^{6}\left (f x +e \right )\right )+\left (11 \sqrt {-\frac {b}{a}}\, a^{2} b -26 \sqrt {-\frac {b}{a}}\, a \,b^{2}+16 \sqrt {-\frac {b}{a}}\, b^{3}\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 \left (3 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2}-11 \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}+7 \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^{3}-14 \sqrt {-\frac {b}{a}}\, a^{2} b +19 \sqrt {-\frac {b}{a}}\, a \,b^{2}-8 \sqrt {-\frac {b}{a}}\, b^{3}\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}}\, \left (3 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{3}-14 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b +19 \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}+7 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b -15 \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 \sqrt {-\frac {b}{a}}\, \left (a -b \right ) a^{3} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \sinh \left (f x +e \right ) \cosh \left (f x +e \right ) f}\)	\(642\)
risch	\(\text {Expression too large to display}\)	\(76488\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*((7*(-1/a*b)^(1/2)*a*b^2-8*(-1/a*b)^(1/2)*b^3)*cosh(f*x+e)^6+(11*(-1/a*b)^(1/2)*a^2*b-26*(-1/a*b)^(1/2)*a
*b^2+16*(-1/a*b)^(1/2)*b^3)*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*(3*Ellipti
cF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^2-11*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b+8*Elli
pticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^2+7*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^3-14*(-1/a*
b)^(1/2)*a^2*b+19*(-1/a*b)^(1/2)*a*b^2-8*(-1/a*b)^(1/2)*b^3)*cosh(f*x+e)^2-(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(
cosh(f*x+e)^2)^(1/2)*(3*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^3-14*EllipticF(sinh(f*x+e)*(-1/a*b
)^(1/2),(a/b)^(1/2))*a^2*b+19*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+7*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^2*b-15*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))/(
-1/a*b)^(1/2)/(a-b)/a^3/(a+b*sinh(f*x+e)^2)^(3/2)/sinh(f*x+e)/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)^2/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(coth(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. 7847 vs. \(2 (351) = 702\).
time = 0.26, size = 7847, normalized size = 22.36 \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)^2/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

1/3*(((14*a^2*b^3 - 23*a*b^4 + 8*b^5)*cosh(f*x + e)^10 + 10*(14*a^2*b^3 - 23*a*b^4 + 8*b^5)*cosh(f*x + e)*sinh
(f*x + e)^9 + (14*a^2*b^3 - 23*a*b^4 + 8*b^5)*sinh(f*x + e)^10 + (112*a^3*b^2 - 254*a^2*b^3 + 179*a*b^4 - 40*b
^5)*cosh(f*x + e)^8 + (112*a^3*b^2 - 254*a^2*b^3 + 179*a*b^4 - 40*b^5 + 45*(14*a^2*b^3 - 23*a*b^4 + 8*b^5)*cos
h(f*x + e)^2)*sinh(f*x + e)^8 + 8*(15*(14*a^2*b^3 - 23*a*b^4 + 8*b^5)*cosh(f*x + e)^3 + (112*a^3*b^2 - 254*a^2
*b^3 + 179*a*b^4 - 40*b^5)*cosh(f*x + e))*sinh(f*x + e)^7 + 2*(112*a^4*b - 352*a^3*b^2 + 410*a^2*b^3 - 211*a*b
^4 + 40*b^5)*cosh(f*x + e)^6 + 2*(112*a^4*b - 352*a^3*b^2 + 410*a^2*b^3 - 211*a*b^4 + 40*b^5 + 105*(14*a^2*b^3
- 23*a*b^4 + 8*b^5)*cosh(f*x + e)^4 + 14*(112*a^3*b^2 - 254*a^2*b^3 + 179*a*b^4 - 40*b^5)*cosh(f*x + e)^2)*si
nh(f*x + e)^6 + 4*(63*(14*a^2*b^3 - 23*a*b^4 + 8*b^5)*cosh(f*x + e)^5 + 14*(112*a^3*b^2 - 254*a^2*b^3 + 179*a*
b^4 - 40*b^5)*cosh(f*x + e)^3 + 3*(112*a^4*b - 352*a^3*b^2 + 410*a^2*b^3 - 211*a*b^4 + 40*b^5)*cosh(f*x + e))*
sinh(f*x + e)^5 - 14*a^2*b^3 + 23*a*b^4 - 8*b^5 - 2*(112*a^4*b - 352*a^3*b^2 + 410*a^2*b^3 - 211*a*b^4 + 40*b^
5)*cosh(f*x + e)^4 + 2*(105*(14*a^2*b^3 - 23*a*b^4 + 8*b^5)*cosh(f*x + e)^6 - 112*a^4*b + 352*a^3*b^2 - 410*a^
2*b^3 + 211*a*b^4 - 40*b^5 + 35*(112*a^3*b^2 - 254*a^2*b^3 + 179*a*b^4 - 40*b^5)*cosh(f*x + e)^4 + 15*(112*a^4
*b - 352*a^3*b^2 + 410*a^2*b^3 - 211*a*b^4 + 40*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(15*(14*a^2*b^3 - 23
*a*b^4 + 8*b^5)*cosh(f*x + e)^7 + 7*(112*a^3*b^2 - 254*a^2*b^3 + 179*a*b^4 - 40*b^5)*cosh(f*x + e)^5 + 5*(112*
a^4*b - 352*a^3*b^2 + 410*a^2*b^3 - 211*a*b^4 + 40*b^5)*cosh(f*x + e)^3 - (112*a^4*b - 352*a^3*b^2 + 410*a^2*b
^3 - 211*a*b^4 + 40*b^5)*cosh(f*x + e))*sinh(f*x + e)^3 - (112*a^3*b^2 - 254*a^2*b^3 + 179*a*b^4 - 40*b^5)*cos
h(f*x + e)^2 + (45*(14*a^2*b^3 - 23*a*b^4 + 8*b^5)*cosh(f*x + e)^8 + 28*(112*a^3*b^2 - 254*a^2*b^3 + 179*a*b^4
- 40*b^5)*cosh(f*x + e)^6 - 112*a^3*b^2 + 254*a^2*b^3 - 179*a*b^4 + 40*b^5 + 30*(112*a^4*b - 352*a^3*b^2 + 41
0*a^2*b^3 - 211*a*b^4 + 40*b^5)*cosh(f*x + e)^4 - 12*(112*a^4*b - 352*a^3*b^2 + 410*a^2*b^3 - 211*a*b^4 + 40*b
^5)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(5*(14*a^2*b^3 - 23*a*b^4 + 8*b^5)*cosh(f*x + e)^9 + 4*(112*a^3*b^2 -
254*a^2*b^3 + 179*a*b^4 - 40*b^5)*cosh(f*x + e)^7 + 6*(112*a^4*b - 352*a^3*b^2 + 410*a^2*b^3 - 211*a*b^4 + 40
*b^5)*cosh(f*x + e)^5 - 4*(112*a^4*b - 352*a^3*b^2 + 410*a^2*b^3 - 211*a*b^4 + 40*b^5)*cosh(f*x + e)^3 - (112*
a^3*b^2 - 254*a^2*b^3 + 179*a*b^4 - 40*b^5)*cosh(f*x + e))*sinh(f*x + e) - 2*((7*a*b^4 - 8*b^5)*cosh(f*x + e)^
10 + 10*(7*a*b^4 - 8*b^5)*cosh(f*x + e)*sinh(f*x + e)^9 + (7*a*b^4 - 8*b^5)*sinh(f*x + e)^10 + (56*a^2*b^3 - 9
9*a*b^4 + 40*b^5)*cosh(f*x + e)^8 + (56*a^2*b^3 - 99*a*b^4 + 40*b^5 + 45*(7*a*b^4 - 8*b^5)*cosh(f*x + e)^2)*si
nh(f*x + e)^8 + 8*(15*(7*a*b^4 - 8*b^5)*cosh(f*x + e)^3 + (56*a^2*b^3 - 99*a*b^4 + 40*b^5)*cosh(f*x + e))*sinh
(f*x + e)^7 + 2*(56*a^3*b^2 - 148*a^2*b^3 + 131*a*b^4 - 40*b^5)*cosh(f*x + e)^6 + 2*(56*a^3*b^2 - 148*a^2*b^3
+ 131*a*b^4 - 40*b^5 + 105*(7*a*b^4 - 8*b^5)*cosh(f*x + e)^4 + 14*(56*a^2*b^3 - 99*a*b^4 + 40*b^5)*cosh(f*x +
e)^2)*sinh(f*x + e)^6 + 4*(63*(7*a*b^4 - 8*b^5)*cosh(f*x + e)^5 + 14*(56*a^2*b^3 - 99*a*b^4 + 40*b^5)*cosh(f*x
+ e)^3 + 3*(56*a^3*b^2 - 148*a^2*b^3 + 131*a*b^4 - 40*b^5)*cosh(f*x + e))*sinh(f*x + e)^5 - 7*a*b^4 + 8*b^5 -
2*(56*a^3*b^2 - 148*a^2*b^3 + 131*a*b^4 - 40*b^5)*cosh(f*x + e)^4 + 2*(105*(7*a*b^4 - 8*b^5)*cosh(f*x + e)^6
- 56*a^3*b^2 + 148*a^2*b^3 - 131*a*b^4 + 40*b^5 + 35*(56*a^2*b^3 - 99*a*b^4 + 40*b^5)*cosh(f*x + e)^4 + 15*(56
*a^3*b^2 - 148*a^2*b^3 + 131*a*b^4 - 40*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(15*(7*a*b^4 - 8*b^5)*cosh(f
*x + e)^7 + 7*(56*a^2*b^3 - 99*a*b^4 + 40*b^5)*cosh(f*x + e)^5 + 5*(56*a^3*b^2 - 148*a^2*b^3 + 131*a*b^4 - 40*
b^5)*cosh(f*x + e)^3 - (56*a^3*b^2 - 148*a^2*b^3 + 131*a*b^4 - 40*b^5)*cosh(f*x + e))*sinh(f*x + e)^3 - (56*a^
2*b^3 - 99*a*b^4 + 40*b^5)*cosh(f*x + e)^2 + (45*(7*a*b^4 - 8*b^5)*cosh(f*x + e)^8 + 28*(56*a^2*b^3 - 99*a*b^4
+ 40*b^5)*cosh(f*x + e)^6 - 56*a^2*b^3 + 99*a*b^4 - 40*b^5 + 30*(56*a^3*b^2 - 148*a^2*b^3 + 131*a*b^4 - 40*b^
5)*cosh(f*x + e)^4 - 12*(56*a^3*b^2 - 148*a^2*b^3 + 131*a*b^4 - 40*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(
5*(7*a*b^4 - 8*b^5)*cosh(f*x + e)^9 + 4*(56*a^2*b^3 - 99*a*b^4 + 40*b^5)*cosh(f*x + e)^7 + 6*(56*a^3*b^2 - 148
*a^2*b^3 + 131*a*b^4 - 40*b^5)*cosh(f*x + e)^5 - 4*(56*a^3*b^2 - 148*a^2*b^3 + 131*a*b^4 - 40*b^5)*cosh(f*x +
e)^3 - (56*a^2*b^3 - 99*a*b^4 + 40*b^5)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2))*sqrt(b)*sqrt((2*b
*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*(cosh(f*x
+ e) + sinh(f*x + e))), (8*a^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b)/b^2))/b^2) - 2*((6*a^3*b^2 -
11*a^2*b^3 + 4*a*b^4)*cosh(f*x + e)^10 + 10*(6*a^3*b^2 - 11*a^2*b^3 + 4*a*b^4)*cosh(f*x + e)*sinh(f*x + e)^9 +
(6*a^3*b^2 - 11*a^2*b^3 + 4*a*b^4)*sinh(f*x + e)^10 + (48*a^4*b - 118*a^3*b^2 + 87*a^2*b^3 - 20*a*b^4)*cosh(f
*x + e)^8 + (48*a^4*b - 118*a^3*b^2 + 87*a^2*b^...

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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)**2/(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)^2/(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: 0.87Error: 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 )}^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(coth(e + f*x)^2/(a + b*sinh(e + f*x)^2)^(5/2),x)

[Out]

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

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