3.3.10 \(\int \coth (x) \sqrt {a+b \coth ^2(x)+c \coth ^4(x)} \, dx\) [210]
Optimal. Leaf size=132 \[ -\frac {(b+2 c) \tanh ^{-1}\left (\frac {b+2 c \coth ^2(x)}{2 \sqrt {c} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{4 \sqrt {c}}+\frac {1}{2} \sqrt {a+b+c} \tanh ^{-1}\left (\frac {2 a+b+(b+2 c) \coth ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)} \]
[Out]
-1/4*(b+2*c)*arctanh(1/2*(b+2*c*coth(x)^2)/c^(1/2)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2))/c^(1/2)+1/2*arctanh(1/2*
(2*a+b+(b+2*c)*coth(x)^2)/(a+b+c)^(1/2)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2))*(a+b+c)^(1/2)-1/2*(a+b*coth(x)^2+c*
coth(x)^4)^(1/2)
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Rubi [A]
time = 0.16, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3782, 1261,
748, 857, 635, 212, 738} \begin {gather*} -\frac {1}{2} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}-\frac {(b+2 c) \tanh ^{-1}\left (\frac {b+2 c \coth ^2(x)}{2 \sqrt {c} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{4 \sqrt {c}}+\frac {1}{2} \sqrt {a+b+c} \tanh ^{-1}\left (\frac {2 a+(b+2 c) \coth ^2(x)+b}{2 \sqrt {a+b+c} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Coth[x]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4],x]
[Out]
-1/4*((b + 2*c)*ArcTanh[(b + 2*c*Coth[x]^2)/(2*Sqrt[c]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])])/Sqrt[c] + (Sqrt[
a + b + c]*ArcTanh[(2*a + b + (b + 2*c)*Coth[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])])/2
- Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4]/2
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 635
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 738
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]
Rule 748
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) && !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Rule 857
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& !IGtQ[m, 0]
Rule 1261
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Rule 3782
Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n_.) + (c_.)*(cot[(d_.) + (e
_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] :> Dist[-f/e, Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2
)), x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Rubi steps
\begin {align*} \int \coth (x) \sqrt {a+b \coth ^2(x)+c \coth ^4(x)} \, dx &=-\text {Subst}\left (\int \frac {x \sqrt {a-b x^2+c x^4}}{1+x^2} \, dx,x,-i \coth (x)\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a-b x+c x^2}}{1+x} \, dx,x,-\coth ^2(x)\right )\right )\\ &=-\frac {1}{2} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}+\frac {1}{4} \text {Subst}\left (\int \frac {-2 a-b+(b+2 c) x}{(1+x) \sqrt {a-b x+c x^2}} \, dx,x,-\coth ^2(x)\right )\\ &=-\frac {1}{2} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}+\frac {1}{2} (-a-b-c) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-b x+c x^2}} \, dx,x,-\coth ^2(x)\right )+\frac {1}{4} (b+2 c) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x+c x^2}} \, dx,x,-\coth ^2(x)\right )\\ &=-\frac {1}{2} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}+(a+b+c) \text {Subst}\left (\int \frac {1}{4 a+4 b+4 c-x^2} \, dx,x,\frac {2 a+b+(b+2 c) \coth ^2(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )+\frac {1}{2} (b+2 c) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {-b-2 c \coth ^2(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )\\ &=-\frac {(b+2 c) \tanh ^{-1}\left (\frac {b+2 c \coth ^2(x)}{2 \sqrt {c} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{4 \sqrt {c}}+\frac {1}{2} \sqrt {a+b+c} \tanh ^{-1}\left (\frac {2 a+b+(b+2 c) \coth ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(304\) vs. \(2(132)=264\).
time = 5.84, size = 304, normalized size = 2.30 \begin {gather*} \frac {\text {csch}^2(x) \left (4 \sqrt {c} (a+b+c) \tanh ^{-1}\left (\frac {-a+c+(a+b+c) \cosh (2 x)}{2 \sqrt {a+b+c} \sqrt {c+(b+2 c) \sinh ^2(x)+(a+b+c) \sinh ^4(x)}}\right ) \sqrt {3 a-b+3 c-4 (a-c) \cosh (2 x)+(a+b+c) \cosh (4 x)}+\sqrt {a+b+c} \left (-2 (b+2 c) \tanh ^{-1}\left (\frac {2 c+(b+2 c) \sinh ^2(x)}{2 \sqrt {c} \sqrt {c+(b+2 c) \sinh ^2(x)+(a+b+c) \sinh ^4(x)}}\right ) \sqrt {3 a-b+3 c-4 (a-c) \cosh (2 x)+(a+b+c) \cosh (4 x)}-\sqrt {2} \sqrt {c} (3 a-b+3 c-4 (a-c) \cosh (2 x)+(a+b+c) \cosh (4 x)) \text {csch}^2(x)\right )\right )}{8 \sqrt {c} \sqrt {a+b+c} \sqrt {(3 a-b+3 c-4 (a-c) \cosh (2 x)+(a+b+c) \cosh (4 x)) \text {csch}^4(x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Coth[x]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4],x]
[Out]
(Csch[x]^2*(4*Sqrt[c]*(a + b + c)*ArcTanh[(-a + c + (a + b + c)*Cosh[2*x])/(2*Sqrt[a + b + c]*Sqrt[c + (b + 2*
c)*Sinh[x]^2 + (a + b + c)*Sinh[x]^4])]*Sqrt[3*a - b + 3*c - 4*(a - c)*Cosh[2*x] + (a + b + c)*Cosh[4*x]] + Sq
rt[a + b + c]*(-2*(b + 2*c)*ArcTanh[(2*c + (b + 2*c)*Sinh[x]^2)/(2*Sqrt[c]*Sqrt[c + (b + 2*c)*Sinh[x]^2 + (a +
b + c)*Sinh[x]^4])]*Sqrt[3*a - b + 3*c - 4*(a - c)*Cosh[2*x] + (a + b + c)*Cosh[4*x]] - Sqrt[2]*Sqrt[c]*(3*a
- b + 3*c - 4*(a - c)*Cosh[2*x] + (a + b + c)*Cosh[4*x])*Csch[x]^2)))/(8*Sqrt[c]*Sqrt[a + b + c]*Sqrt[(3*a - b
+ 3*c - 4*(a - c)*Cosh[2*x] + (a + b + c)*Cosh[4*x])*Csch[x]^4])
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.01, size = 850, normalized size = 6.44 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)*(a+b*coth(x)^2+c*coth(x)^4)^(1/2),x,method=_RETURNVERBOSE)
[Out]
-1/2*(a+b*coth(x)^2+c*coth(x)^4)^(1/2)-1/8*(b+c)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^
2)^(1/2))/a*coth(x)^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*coth(x)^2)^(1/2)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2)*
EllipticF(1/2*coth(x)*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))
-1/2*(c+1/2*b)*ln((b+2*c*coth(x)^2)/c^(1/2)+2*(a+b*coth(x)^2+c*coth(x)^4)^(1/2))/c^(1/2)-1/2*(a+b+c)*(-1/2/(a+
b+c)^(1/2)*arctanh(1/2*(b*coth(x)^2+2*c*coth(x)^2+2*a+b)/(a+b+c)^(1/2)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2))-2^(1
/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(1-1/2*(-b+(-4*a*c+b^2)^(1/2))/a*coth(x)^2)^(1/2)*(1+1/2*(b+(-4*a*c+b^2)
^(1/2))/a*coth(x)^2)^(1/2)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2)*EllipticPi(1/2*coth(x)*2^(1/2)*((-b+(-4*a*c+b^2)^
(1/2))/a)^(1/2),2/(-b+(-4*a*c+b^2)^(1/2))*a,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1
/2))/a)^(1/2)))-1/8*(-b-c)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*coth(x)^2)
^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*coth(x)^2)^(1/2)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2)*EllipticF(1/2*coth(x)*
2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-1/2*(a+b+c)*(-1/2/(a+
b+c)^(1/2)*arctanh(1/2*(b*coth(x)^2+2*c*coth(x)^2+2*a+b)/(a+b+c)^(1/2)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2))+2^(1
/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(1-1/2*(-b+(-4*a*c+b^2)^(1/2))/a*coth(x)^2)^(1/2)*(1+1/2*(b+(-4*a*c+b^2)
^(1/2))/a*coth(x)^2)^(1/2)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2)*EllipticPi(1/2*coth(x)*2^(1/2)*((-b+(-4*a*c+b^2)^
(1/2))/a)^(1/2),2/(-b+(-4*a*c+b^2)^(1/2))*a,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1
/2))/a)^(1/2)))
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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(x)*(a+b*coth(x)^2+c*coth(x)^4)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(c*coth(x)^4 + b*coth(x)^2 + a)*coth(x), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1840 vs.
\(2 (108) = 216\).
time = 1.44, size = 7964, normalized size = 60.33 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)*(a+b*coth(x)^2+c*coth(x)^4)^(1/2),x, algorithm="fricas")
[Out]
[1/8*(((b + 2*c)*cosh(x)^4 + 4*(b + 2*c)*cosh(x)*sinh(x)^3 + (b + 2*c)*sinh(x)^4 - 2*(b + 2*c)*cosh(x)^2 + 2*(
3*(b + 2*c)*cosh(x)^2 - b - 2*c)*sinh(x)^2 + 4*((b + 2*c)*cosh(x)^3 - (b + 2*c)*cosh(x))*sinh(x) + b + 2*c)*sq
rt(c)*log(((b^2 + 4*(a + 2*b)*c + 8*c^2)*cosh(x)^8 + 8*(b^2 + 4*(a + 2*b)*c + 8*c^2)*cosh(x)*sinh(x)^7 + (b^2
+ 4*(a + 2*b)*c + 8*c^2)*sinh(x)^8 - 4*(b^2 + 4*a*c - 8*c^2)*cosh(x)^6 + 4*(7*(b^2 + 4*(a + 2*b)*c + 8*c^2)*co
sh(x)^2 - b^2 - 4*a*c + 8*c^2)*sinh(x)^6 + 8*(7*(b^2 + 4*(a + 2*b)*c + 8*c^2)*cosh(x)^3 - 3*(b^2 + 4*a*c - 8*c
^2)*cosh(x))*sinh(x)^5 + 2*(3*b^2 + 4*(3*a - 2*b)*c + 24*c^2)*cosh(x)^4 + 2*(35*(b^2 + 4*(a + 2*b)*c + 8*c^2)*
cosh(x)^4 - 30*(b^2 + 4*a*c - 8*c^2)*cosh(x)^2 + 3*b^2 + 4*(3*a - 2*b)*c + 24*c^2)*sinh(x)^4 + 8*(7*(b^2 + 4*(
a + 2*b)*c + 8*c^2)*cosh(x)^5 - 10*(b^2 + 4*a*c - 8*c^2)*cosh(x)^3 + (3*b^2 + 4*(3*a - 2*b)*c + 24*c^2)*cosh(x
))*sinh(x)^3 - 4*(b^2 + 4*a*c - 8*c^2)*cosh(x)^2 + 4*(7*(b^2 + 4*(a + 2*b)*c + 8*c^2)*cosh(x)^6 - 15*(b^2 + 4*
a*c - 8*c^2)*cosh(x)^4 + 3*(3*b^2 + 4*(3*a - 2*b)*c + 24*c^2)*cosh(x)^2 - b^2 - 4*a*c + 8*c^2)*sinh(x)^2 - 4*s
qrt(2)*((b + 2*c)*cosh(x)^4 + 4*(b + 2*c)*cosh(x)*sinh(x)^3 + (b + 2*c)*sinh(x)^4 - 2*(b - 2*c)*cosh(x)^2 + 2*
(3*(b + 2*c)*cosh(x)^2 - b + 2*c)*sinh(x)^2 + 4*((b + 2*c)*cosh(x)^3 - (b - 2*c)*cosh(x))*sinh(x) + b + 2*c)*s
qrt(c)*sqrt(((a + b + c)*cosh(x)^4 + (a + b + c)*sinh(x)^4 - 4*(a - c)*cosh(x)^2 + 2*(3*(a + b + c)*cosh(x)^2
- 2*a + 2*c)*sinh(x)^2 + 3*a - b + 3*c)/(cosh(x)^4 - 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 - 4*cosh(x)*s
inh(x)^3 + sinh(x)^4)) + b^2 + 4*(a + 2*b)*c + 8*c^2 + 8*((b^2 + 4*(a + 2*b)*c + 8*c^2)*cosh(x)^7 - 3*(b^2 + 4
*a*c - 8*c^2)*cosh(x)^5 + (3*b^2 + 4*(3*a - 2*b)*c + 24*c^2)*cosh(x)^3 - (b^2 + 4*a*c - 8*c^2)*cosh(x))*sinh(x
))/(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 - 1)*sinh(x)^6 - 4*cosh(x)^6 + 8*(7*cosh(x)^3
- 3*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 - 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 - 10*co
sh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 - 15*cosh(x)^4 + 9*cosh(x)^2 - 1)*sinh(x)^2 - 4*cosh(x)^2 + 8*
(cosh(x)^7 - 3*cosh(x)^5 + 3*cosh(x)^3 - cosh(x))*sinh(x) + 1)) + 2*(c*cosh(x)^4 + 4*c*cosh(x)*sinh(x)^3 + c*s
inh(x)^4 - 2*c*cosh(x)^2 + 2*(3*c*cosh(x)^2 - c)*sinh(x)^2 + 4*(c*cosh(x)^3 - c*cosh(x))*sinh(x) + c)*sqrt(a +
b + c)*log(((a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^8 + 8*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cos
h(x)*sinh(x)^7 + (a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*sinh(x)^8 - 4*(a^2 + a*b - b*c - c^2)*cosh(x)^6 + 4*(
7*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^2 - a^2 - a*b + b*c + c^2)*sinh(x)^6 + 8*(7*(a^2 + 2*a*b + b
^2 + 2*(a + b)*c + c^2)*cosh(x)^3 - 3*(a^2 + a*b - b*c - c^2)*cosh(x))*sinh(x)^5 + 2*(3*a^2 + 2*a*b + 2*(a + b
)*c + 3*c^2)*cosh(x)^4 + 2*(35*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^4 - 30*(a^2 + a*b - b*c - c^2)*
cosh(x)^2 + 3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*sinh(x)^4 + 8*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh
(x)^5 - 10*(a^2 + a*b - b*c - c^2)*cosh(x)^3 + (3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*cosh(x))*sinh(x)^3 - 4*(a
^2 + a*b - b*c - c^2)*cosh(x)^2 + 4*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^6 - 15*(a^2 + a*b - b*c
- c^2)*cosh(x)^4 + 3*(3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*cosh(x)^2 - a^2 - a*b + b*c + c^2)*sinh(x)^2 + sqr
t(2)*((a + b + c)*cosh(x)^4 + 4*(a + b + c)*cosh(x)*sinh(x)^3 + (a + b + c)*sinh(x)^4 - 2*(a - c)*cosh(x)^2 +
2*(3*(a + b + c)*cosh(x)^2 - a + c)*sinh(x)^2 + 4*((a + b + c)*cosh(x)^3 - (a - c)*cosh(x))*sinh(x) + a + b +
c)*sqrt(a + b + c)*sqrt(((a + b + c)*cosh(x)^4 + (a + b + c)*sinh(x)^4 - 4*(a - c)*cosh(x)^2 + 2*(3*(a + b + c
)*cosh(x)^2 - 2*a + 2*c)*sinh(x)^2 + 3*a - b + 3*c)/(cosh(x)^4 - 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 -
4*cosh(x)*sinh(x)^3 + sinh(x)^4)) + a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2 + 8*((a^2 + 2*a*b + b^2 + 2*(a + b)
*c + c^2)*cosh(x)^7 - 3*(a^2 + a*b - b*c - c^2)*cosh(x)^5 + (3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*cosh(x)^3 -
(a^2 + a*b - b*c - c^2)*cosh(x))*sinh(x))/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)
*sinh(x)^3 + sinh(x)^4)) - 4*sqrt(2)*c*sqrt(((a + b + c)*cosh(x)^4 + (a + b + c)*sinh(x)^4 - 4*(a - c)*cosh(x)
^2 + 2*(3*(a + b + c)*cosh(x)^2 - 2*a + 2*c)*sinh(x)^2 + 3*a - b + 3*c)/(cosh(x)^4 - 4*cosh(x)^3*sinh(x) + 6*c
osh(x)^2*sinh(x)^2 - 4*cosh(x)*sinh(x)^3 + sinh(x)^4)))/(c*cosh(x)^4 + 4*c*cosh(x)*sinh(x)^3 + c*sinh(x)^4 - 2
*c*cosh(x)^2 + 2*(3*c*cosh(x)^2 - c)*sinh(x)^2 + 4*(c*cosh(x)^3 - c*cosh(x))*sinh(x) + c), -1/8*(4*(c*cosh(x)^
4 + 4*c*cosh(x)*sinh(x)^3 + c*sinh(x)^4 - 2*c*cosh(x)^2 + 2*(3*c*cosh(x)^2 - c)*sinh(x)^2 + 4*(c*cosh(x)^3 - c
*cosh(x))*sinh(x) + c)*sqrt(-a - b - c)*arctan(sqrt(2)*((a + b + c)*cosh(x)^4 + 4*(a + b + c)*cosh(x)*sinh(x)^
3 + (a + b + c)*sinh(x)^4 - 2*(a - c)*cosh(x)^2 + 2*(3*(a + b + c)*cosh(x)^2 - a + c)*sinh(x)^2 + 4*((a + b +
c)*cosh(x)^3 - (a - c)*cosh(x))*sinh(x) + a + b + c)*sqrt(-a - b - c)*sqrt(((a + b + c)*cosh(x)^4 + (a + b + c
)*sinh(x)^4 - 4*(a - c)*cosh(x)^2 + 2*(3*(a + b...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \coth ^{2}{\left (x \right )} + c \coth ^{4}{\left (x \right )}} \coth {\left (x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)*(a+b*coth(x)**2+c*coth(x)**4)**(1/2),x)
[Out]
Integral(sqrt(a + b*coth(x)**2 + c*coth(x)**4)*coth(x), x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)*(a+b*coth(x)^2+c*coth(x)^4)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(c*coth(x)^4 + b*coth(x)^2 + a)*coth(x), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {coth}\left (x\right )\,\sqrt {c\,{\mathrm {coth}\left (x\right )}^4+b\,{\mathrm {coth}\left (x\right )}^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)*(a + b*coth(x)^2 + c*coth(x)^4)^(1/2),x)
[Out]
int(coth(x)*(a + b*coth(x)^2 + c*coth(x)^4)^(1/2), x)
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