3.5.100 \(\int \frac {\coth ^4(e+f x)}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\) [500]

Optimal. Leaf size=341 \[ -\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{a b f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(7 a-8 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 f}+\frac {(3 a-4 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 b f}-\frac {(7 a-8 b) E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a-4 b) F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(7 a-8 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^3 f} \]

[Out]

-(a-b)*coth(f*x+e)*csch(f*x+e)^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/f+1/3*(3*a-4*b)*coth(f*x+e)*csch(f*x+e)^2*(a+b*sinh(f*x+e)^2)^(1/2)/a^2/b/f-1/3*(7*a-8*b)*(1/(1+sin
h(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/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/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/f

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Rubi [A]
time = 0.28, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3275, 479, 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 \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 \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}-\frac {(7 a-8 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 f}+\frac {(3 a-4 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 b f}-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{a b f \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

-(((a - b)*Coth[e + f*x]*Csch[e + f*x]^2)/(a*b*f*Sqrt[a + b*Sinh[e + f*x]^2])) - ((7*a - 8*b)*Coth[e + f*x]*Sq
rt[a + b*Sinh[e + f*x]^2])/(3*a^3*f) + ((3*a - 4*b)*Coth[e + f*x]*Csch[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])
/(3*a^2*b*f) - ((7*a - 8*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2
])/(3*a^3*f*Sqrt[(Sech[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*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*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 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 )^{3/2}} \, 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 )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{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 {3 a-4 b+(2 a-3 b) x^2}{x^4 \sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{a b f}\\ &=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{a b f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(3 a-4 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 b f}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {(7 a-8 b) b+(3 a-4 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 b f}\\ &=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{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 f}+\frac {(3 a-4 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 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) b-(7 a-8 b) b^2 x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a^3 b f}\\ &=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{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 f}+\frac {(3 a-4 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 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 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 f}\\ &=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{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 f}+\frac {(3 a-4 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 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 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 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 f}\\ &=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{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 f}+\frac {(3 a-4 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 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 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 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 f}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.66, size = 214, normalized size = 0.63 \begin {gather*} \frac {-\frac {\left (-8 a^2+37 a b-24 b^2+4 \left (4 a^2-11 a b+8 b^2\right ) \cosh (2 (e+f x))+(7 a-8 b) b \cosh (4 (e+f x))\right ) \coth (e+f x) \text {csch}^2(e+f x)}{2 \sqrt {2}}-2 i a (7 a-8 b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+8 i a (a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )}{6 a^3 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

(-1/2*((-8*a^2 + 37*a*b - 24*b^2 + 4*(4*a^2 - 11*a*b + 8*b^2)*Cosh[2*(e + f*x)] + (7*a - 8*b)*b*Cosh[4*(e + f*
x)])*Coth[e + f*x]*Csch[e + f*x]^2)/Sqrt[2] - (2*I)*a*(7*a - 8*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*Elli
pticE[I*(e + f*x), b/a] + (8*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])
/(6*a^3*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])

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Maple [A]
time = 6.10, size = 522, normalized size = 1.53

method	result	size
default	\(-\frac {7 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{6}\left (f x +e \right )\right )-8 \sqrt {-\frac {b}{a}}\, b^{2} \left (\sinh ^{6}\left (f x +e \right )\right )-3 a^{2} \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) \left (\sinh ^{3}\left (f x +e \right )\right )+11 b \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \left (\sinh ^{3}\left (f x +e \right )\right )-8 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \left (\sinh ^{3}\left (f x +e \right )\right )-7 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b \left (\sinh ^{3}\left (f x +e \right )\right )+8 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \left (\sinh ^{3}\left (f x +e \right )\right )+4 \sqrt {-\frac {b}{a}}\, a^{2} \left (\sinh ^{4}\left (f x +e \right )\right )+3 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{4}\left (f x +e \right )\right )-8 \sqrt {-\frac {b}{a}}\, b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )+5 \sqrt {-\frac {b}{a}}\, a^{2} \left (\sinh ^{2}\left (f x +e \right )\right )-4 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{2}\left (f x +e \right )\right )+\sqrt {-\frac {b}{a}}\, a^{2}}{3 a^{3} \sinh \left (f x +e \right )^{3} \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(522\)
risch	\(\text {Expression too large to display}\)	\(1122878\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(f*x+e)^4/(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(7*(-1/a*b)^(1/2)*a*b*sinh(f*x+e)^6-8*(-1/a*b)^(1/2)*b^2*sinh(f*x+e)^6-3*a^2*((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))*sinh(f*x+e)^3+11*((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*a*sinh(f*x+e)^3-8*((a+b*s
inh(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^2*sinh(f*x+e)
^3-7*((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
*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))*b^2*sinh(f*x+e)^3+4*(-1/a*b)^(1/2)*a^2*sinh(f*x+e)^4+3*(-1/a*b)^(1/2)*a*b*sinh(f*x+e)^4-8*(-1/a*b)^(1
/2)*b^2*sinh(f*x+e)^4+5*(-1/a*b)^(1/2)*a^2*sinh(f*x+e)^2-4*(-1/a*b)^(1/2)*a*b*sinh(f*x+e)^2+(-1/a*b)^(1/2)*a^2
)/a^3/sinh(f*x+e)^3/(-1/a*b)^(1/2)/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/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)^(3/2),x, algorithm="maxima")

[Out]

integrate(coth(f*x + e)^4/(b*sinh(f*x + e)^2 + a)^(3/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6862 vs. \(2 (343) = 686\).
time = 0.20, size = 6862, normalized size = 20.12 \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)^(3/2),x, algorithm="fricas")

[Out]

1/3*(((14*a^2*b^2 - 23*a*b^3 + 8*b^4)*cosh(f*x + e)^10 + 10*(14*a^2*b^2 - 23*a*b^3 + 8*b^4)*cosh(f*x + e)*sinh
(f*x + e)^9 + (14*a^2*b^2 - 23*a*b^3 + 8*b^4)*sinh(f*x + e)^10 + (56*a^3*b - 162*a^2*b^2 + 147*a*b^3 - 40*b^4)
*cosh(f*x + e)^8 + (56*a^3*b - 162*a^2*b^2 + 147*a*b^3 - 40*b^4 + 45*(14*a^2*b^2 - 23*a*b^3 + 8*b^4)*cosh(f*x
+ e)^2)*sinh(f*x + e)^8 + 8*(15*(14*a^2*b^2 - 23*a*b^3 + 8*b^4)*cosh(f*x + e)^3 + (56*a^3*b - 162*a^2*b^2 + 14
7*a*b^3 - 40*b^4)*cosh(f*x + e))*sinh(f*x + e)^7 - 2*(84*a^3*b - 208*a^2*b^2 + 163*a*b^3 - 40*b^4)*cosh(f*x +
e)^6 + 2*(105*(14*a^2*b^2 - 23*a*b^3 + 8*b^4)*cosh(f*x + e)^4 - 84*a^3*b + 208*a^2*b^2 - 163*a*b^3 + 40*b^4 +
14*(56*a^3*b - 162*a^2*b^2 + 147*a*b^3 - 40*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 4*(63*(14*a^2*b^2 - 23*a*b
^3 + 8*b^4)*cosh(f*x + e)^5 + 14*(56*a^3*b - 162*a^2*b^2 + 147*a*b^3 - 40*b^4)*cosh(f*x + e)^3 - 3*(84*a^3*b -
208*a^2*b^2 + 163*a*b^3 - 40*b^4)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(84*a^3*b - 208*a^2*b^2 + 163*a*b^3 - 40
*b^4)*cosh(f*x + e)^4 + 2*(105*(14*a^2*b^2 - 23*a*b^3 + 8*b^4)*cosh(f*x + e)^6 + 35*(56*a^3*b - 162*a^2*b^2 +
147*a*b^3 - 40*b^4)*cosh(f*x + e)^4 + 84*a^3*b - 208*a^2*b^2 + 163*a*b^3 - 40*b^4 - 15*(84*a^3*b - 208*a^2*b^2
+ 163*a*b^3 - 40*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^4 - 14*a^2*b^2 + 23*a*b^3 - 8*b^4 + 8*(15*(14*a^2*b^2 -
23*a*b^3 + 8*b^4)*cosh(f*x + e)^7 + 7*(56*a^3*b - 162*a^2*b^2 + 147*a*b^3 - 40*b^4)*cosh(f*x + e)^5 - 5*(84*a^
3*b - 208*a^2*b^2 + 163*a*b^3 - 40*b^4)*cosh(f*x + e)^3 + (84*a^3*b - 208*a^2*b^2 + 163*a*b^3 - 40*b^4)*cosh(f
*x + e))*sinh(f*x + e)^3 - (56*a^3*b - 162*a^2*b^2 + 147*a*b^3 - 40*b^4)*cosh(f*x + e)^2 + (45*(14*a^2*b^2 - 2
3*a*b^3 + 8*b^4)*cosh(f*x + e)^8 + 28*(56*a^3*b - 162*a^2*b^2 + 147*a*b^3 - 40*b^4)*cosh(f*x + e)^6 - 30*(84*a
^3*b - 208*a^2*b^2 + 163*a*b^3 - 40*b^4)*cosh(f*x + e)^4 - 56*a^3*b + 162*a^2*b^2 - 147*a*b^3 + 40*b^4 + 12*(8
4*a^3*b - 208*a^2*b^2 + 163*a*b^3 - 40*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(5*(14*a^2*b^2 - 23*a*b^3 + 8
*b^4)*cosh(f*x + e)^9 + 4*(56*a^3*b - 162*a^2*b^2 + 147*a*b^3 - 40*b^4)*cosh(f*x + e)^7 - 6*(84*a^3*b - 208*a^
2*b^2 + 163*a*b^3 - 40*b^4)*cosh(f*x + e)^5 + 4*(84*a^3*b - 208*a^2*b^2 + 163*a*b^3 - 40*b^4)*cosh(f*x + e)^3
- (56*a^3*b - 162*a^2*b^2 + 147*a*b^3 - 40*b^4)*cosh(f*x + e))*sinh(f*x + e) - 2*((7*a*b^3 - 8*b^4)*cosh(f*x +
e)^10 + 10*(7*a*b^3 - 8*b^4)*cosh(f*x + e)*sinh(f*x + e)^9 + (7*a*b^3 - 8*b^4)*sinh(f*x + e)^10 + (28*a^2*b^2
- 67*a*b^3 + 40*b^4)*cosh(f*x + e)^8 + (28*a^2*b^2 - 67*a*b^3 + 40*b^4 + 45*(7*a*b^3 - 8*b^4)*cosh(f*x + e)^2
)*sinh(f*x + e)^8 + 8*(15*(7*a*b^3 - 8*b^4)*cosh(f*x + e)^3 + (28*a^2*b^2 - 67*a*b^3 + 40*b^4)*cosh(f*x + e))*
sinh(f*x + e)^7 - 2*(42*a^2*b^2 - 83*a*b^3 + 40*b^4)*cosh(f*x + e)^6 + 2*(105*(7*a*b^3 - 8*b^4)*cosh(f*x + e)^
4 - 42*a^2*b^2 + 83*a*b^3 - 40*b^4 + 14*(28*a^2*b^2 - 67*a*b^3 + 40*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 4*
(63*(7*a*b^3 - 8*b^4)*cosh(f*x + e)^5 + 14*(28*a^2*b^2 - 67*a*b^3 + 40*b^4)*cosh(f*x + e)^3 - 3*(42*a^2*b^2 -
83*a*b^3 + 40*b^4)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(42*a^2*b^2 - 83*a*b^3 + 40*b^4)*cosh(f*x + e)^4 + 2*(10
5*(7*a*b^3 - 8*b^4)*cosh(f*x + e)^6 + 35*(28*a^2*b^2 - 67*a*b^3 + 40*b^4)*cosh(f*x + e)^4 + 42*a^2*b^2 - 83*a*
b^3 + 40*b^4 - 15*(42*a^2*b^2 - 83*a*b^3 + 40*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^4 - 7*a*b^3 + 8*b^4 + 8*(15*
(7*a*b^3 - 8*b^4)*cosh(f*x + e)^7 + 7*(28*a^2*b^2 - 67*a*b^3 + 40*b^4)*cosh(f*x + e)^5 - 5*(42*a^2*b^2 - 83*a*
b^3 + 40*b^4)*cosh(f*x + e)^3 + (42*a^2*b^2 - 83*a*b^3 + 40*b^4)*cosh(f*x + e))*sinh(f*x + e)^3 - (28*a^2*b^2
- 67*a*b^3 + 40*b^4)*cosh(f*x + e)^2 + (45*(7*a*b^3 - 8*b^4)*cosh(f*x + e)^8 + 28*(28*a^2*b^2 - 67*a*b^3 + 40*
b^4)*cosh(f*x + e)^6 - 30*(42*a^2*b^2 - 83*a*b^3 + 40*b^4)*cosh(f*x + e)^4 - 28*a^2*b^2 + 67*a*b^3 - 40*b^4 +
12*(42*a^2*b^2 - 83*a*b^3 + 40*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(5*(7*a*b^3 - 8*b^4)*cosh(f*x + e)^9
+ 4*(28*a^2*b^2 - 67*a*b^3 + 40*b^4)*cosh(f*x + e)^7 - 6*(42*a^2*b^2 - 83*a*b^3 + 40*b^4)*cosh(f*x + e)^5 + 4*
(42*a^2*b^2 - 83*a*b^3 + 40*b^4)*cosh(f*x + e)^3 - (28*a^2*b^2 - 67*a*b^3 + 40*b^4)*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*s
qrt((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)*sq
rt((a^2 - a*b)/b^2))/b^2) - 2*((6*a^3*b - 11*a^2*b^2 + 4*a*b^3)*cosh(f*x + e)^10 + 10*(6*a^3*b - 11*a^2*b^2 +
4*a*b^3)*cosh(f*x + e)*sinh(f*x + e)^9 + (6*a^3*b - 11*a^2*b^2 + 4*a*b^3)*sinh(f*x + e)^10 + (24*a^4 - 74*a^3*
b + 71*a^2*b^2 - 20*a*b^3)*cosh(f*x + e)^8 + (24*a^4 - 74*a^3*b + 71*a^2*b^2 - 20*a*b^3 + 45*(6*a^3*b - 11*a^2
*b^2 + 4*a*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 8*(15*(6*a^3*b - 11*a^2*b^2 + 4*a*b^3)*cosh(f*x + e)^3 + (2
4*a^4 - 74*a^3*b + 71*a^2*b^2 - 20*a*b^3)*cosh(f*x + e))*sinh(f*x + e)^7 - 2*(36*a^4 - 96*a^3*b + 79*a^2*b^2 -
20*a*b^3)*cosh(f*x + e)^6 + 2*(105*(6*a^3*b - 11*a^2*b^2 + 4*a*b^3)*cosh(f*x + e)^4 - 36*a^4 + 96*a^3*b - 79*
a^2*b^2 + 20*a*b^3 + 14*(24*a^4 - 74*a^3*b + 71...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{4}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \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)**(3/2),x)

[Out]

Integral(coth(e + f*x)**4/(a + b*sinh(e + f*x)**2)**(3/2), x)

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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(coth(f*x+e)^4/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Evaluation time:
0.75Error: 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 )}^{3/2}} \,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)^(3/2),x)

[Out]

int(coth(e + f*x)^4/(a + b*sinh(e + f*x)^2)^(3/2), x)

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