3.1.19 \(\int \sqrt {a+b \coth ^2(x)} \, dx\) [19]
Optimal. Leaf size=60 \[ -\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )+\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right ) \]
[Out]
-arctanh(coth(x)*b^(1/2)/(a+b*coth(x)^2)^(1/2))*b^(1/2)+arctanh(coth(x)*(a+b)^(1/2)/(a+b*coth(x)^2)^(1/2))*(a+
b)^(1/2)
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Rubi [A]
time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3742, 399, 223,
212, 385} \begin {gather*} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[a + b*Coth[x]^2],x]
[Out]
-(Sqrt[b]*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]) + Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a
+ b*Coth[x]^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 223
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
Rule 385
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 399
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]
Rule 3742
Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[c*(ff/f), Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Rubi steps
\begin {align*} \int \sqrt {a+b \coth ^2(x)} \, dx &=\text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=-\left ((-a-b) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )\right )-b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\coth (x)\right )\\ &=-\left ((-a-b) \text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )\right )-b \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )\\ &=-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )+\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 82, normalized size = 1.37 \begin {gather*} \sqrt {-a-b} \text {ArcTan}\left (\frac {\coth (x) \sqrt {a+b \coth ^2(x)}-\sqrt {b} \text {csch}^2(x)}{\sqrt {-a-b}}\right )+\sqrt {b} \log \left (-\sqrt {b} \coth (x)+\sqrt {a+b \coth ^2(x)}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[a + b*Coth[x]^2],x]
[Out]
Sqrt[-a - b]*ArcTan[(Coth[x]*Sqrt[a + b*Coth[x]^2] - Sqrt[b]*Csch[x]^2)/Sqrt[-a - b]] + Sqrt[b]*Log[-(Sqrt[b]*
Coth[x]) + Sqrt[a + b*Coth[x]^2]]
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(237\) vs.
\(2(48)=96\).
time = 0.84, size = 238, normalized size = 3.97
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(-\frac {\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2}+\frac {\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )}{2}-\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2}\) |
\(238\) |
| default |
\(-\frac {\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2}+\frac {\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )}{2}-\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2}\) |
\(238\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*coth(x)^2)^(1/2),x,method=_RETURNVERBOSE)
[Out]
-1/2*(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2)-1/2*b^(1/2)*ln((b*(coth(x)-1)+b)/b^(1/2)+(b*(coth(x)-1)^2+2*b
*(coth(x)-1)+a+b)^(1/2))+1/2*(a+b)^(1/2)*ln((2*a+2*b+2*b*(coth(x)-1)+2*(a+b)^(1/2)*(b*(coth(x)-1)^2+2*b*(coth(
x)-1)+a+b)^(1/2))/(coth(x)-1))+1/2*(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2)-1/2*b^(1/2)*ln((b*(1+coth(x))-b
)/b^(1/2)+(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2))-1/2*(a+b)^(1/2)*ln((2*a+2*b-2*b*(1+coth(x))+2*(a+b)^(1/
2)*(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2))/(1+coth(x)))
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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((a+b*coth(x)^2)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(b*coth(x)^2 + a), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 552 vs.
\(2 (48) = 96\).
time = 0.47, size = 3455, normalized size = 57.58 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*coth(x)^2)^(1/2),x, algorithm="fricas")
[Out]
[1/4*sqrt(a + b)*log(((a*b^2 + b^3)*cosh(x)^8 + 8*(a*b^2 + b^3)*cosh(x)*sinh(x)^7 + (a*b^2 + b^3)*sinh(x)^8 +
2*(a*b^2 + 2*b^3)*cosh(x)^6 + 2*(a*b^2 + 2*b^3 + 14*(a*b^2 + b^3)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a*b^2 + b^3)*c
osh(x)^3 + 3*(a*b^2 + 2*b^3)*cosh(x))*sinh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^4 + (70*(a*b^2 + b^3
)*cosh(x)^4 + a^3 - a^2*b + 4*a*b^2 + 6*b^3 + 30*(a*b^2 + 2*b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a*b^2 + b^3)*co
sh(x)^5 + 10*(a*b^2 + 2*b^3)*cosh(x)^3 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x))*sinh(x)^3 + a^3 + 3*a^2*b +
3*a*b^2 + b^3 - 2*(a^3 - 3*a*b^2 - 2*b^3)*cosh(x)^2 + 2*(14*(a*b^2 + b^3)*cosh(x)^6 + 15*(a*b^2 + 2*b^3)*cosh(
x)^4 - a^3 + 3*a*b^2 + 2*b^3 + 3*(a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(b^2*cosh(x)^6
+ 6*b^2*cosh(x)*sinh(x)^5 + b^2*sinh(x)^6 + 3*b^2*cosh(x)^4 + 3*(5*b^2*cosh(x)^2 + b^2)*sinh(x)^4 + 4*(5*b^2*
cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x)^2 + (15*b^2*cosh(x)^4 + 18*b^2*cosh(x)^2
- a^2 + 2*a*b + 3*b^2)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 2*(3*b^2*cosh(x)^5 + 6*b^2*cosh(x)^3 - (a^2 - 2*a*b - 3
*b^2)*cosh(x))*sinh(x))*sqrt(a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x
)*sinh(x) + sinh(x)^2)) + 4*(2*(a*b^2 + b^3)*cosh(x)^7 + 3*(a*b^2 + 2*b^3)*cosh(x)^5 + (a^3 - a^2*b + 4*a*b^2
+ 6*b^3)*cosh(x)^3 - (a^3 - 3*a*b^2 - 2*b^3)*cosh(x))*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4
*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + 1/2*sqrt(b)
*log(-((a + 2*b)*cosh(x)^4 + 4*(a + 2*b)*cosh(x)*sinh(x)^3 + (a + 2*b)*sinh(x)^4 - 2*(a - 2*b)*cosh(x)^2 + 2*(
3*(a + 2*b)*cosh(x)^2 - a + 2*b)*sinh(x)^2 - 2*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(b)
*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*((a + 2
*b)*cosh(x)^3 - (a - 2*b)*cosh(x))*sinh(x) + a + 2*b)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh
(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)) + 1/4*sqrt(a + b)*log(-((a + b)*cos
h(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 - 2*a*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 - a)*sinh(x)
^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sin
h(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*((a + b)*cosh(x)^3 - a*cosh(x))*sinh(x) + a +
b)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)), sqrt(-b)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sin
h(x)^2 + 1)*sqrt(-b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sin
h(x)^2))/((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 - 2*(a - b)*cosh(x)^2 + 2*(3*(a
+ b)*cosh(x)^2 - a + b)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 - (a - b)*cosh(x))*sinh(x) + a + b)) + 1/4*sqrt(a + b
)*log(((a*b^2 + b^3)*cosh(x)^8 + 8*(a*b^2 + b^3)*cosh(x)*sinh(x)^7 + (a*b^2 + b^3)*sinh(x)^8 + 2*(a*b^2 + 2*b^
3)*cosh(x)^6 + 2*(a*b^2 + 2*b^3 + 14*(a*b^2 + b^3)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a*b^2 + b^3)*cosh(x)^3 + 3*(a
*b^2 + 2*b^3)*cosh(x))*sinh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^4 + (70*(a*b^2 + b^3)*cosh(x)^4 + a
^3 - a^2*b + 4*a*b^2 + 6*b^3 + 30*(a*b^2 + 2*b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a*b^2 + b^3)*cosh(x)^5 + 10*(a
*b^2 + 2*b^3)*cosh(x)^3 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x))*sinh(x)^3 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 -
2*(a^3 - 3*a*b^2 - 2*b^3)*cosh(x)^2 + 2*(14*(a*b^2 + b^3)*cosh(x)^6 + 15*(a*b^2 + 2*b^3)*cosh(x)^4 - a^3 + 3*
a*b^2 + 2*b^3 + 3*(a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(b^2*cosh(x)^6 + 6*b^2*cosh(x
)*sinh(x)^5 + b^2*sinh(x)^6 + 3*b^2*cosh(x)^4 + 3*(5*b^2*cosh(x)^2 + b^2)*sinh(x)^4 + 4*(5*b^2*cosh(x)^3 + 3*b
^2*cosh(x))*sinh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x)^2 + (15*b^2*cosh(x)^4 + 18*b^2*cosh(x)^2 - a^2 + 2*a*b +
3*b^2)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 2*(3*b^2*cosh(x)^5 + 6*b^2*cosh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x))*
sinh(x))*sqrt(a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sin
h(x)^2)) + 4*(2*(a*b^2 + b^3)*cosh(x)^7 + 3*(a*b^2 + 2*b^3)*cosh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x
)^3 - (a^3 - 3*a*b^2 - 2*b^3)*cosh(x))*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20
*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + 1/4*sqrt(a + b)*log(-((a +
b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 - 2*a*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 - a)*
sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(a + b)*sqrt(((a + b)*cosh(x)^2 + (a +
b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*((a + b)*cosh(x)^3 - a*cosh(x))*sinh(x
) + a + b)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)), -1/2*sqrt(-a - b)*arctan(sqrt(2)*(b*cosh(x)^2 + 2*b*c
osh(x)*sinh(x) + b*sinh(x)^2 + a + b)*sqrt(-a - b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(
x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/((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 )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*coth(x)**2)**(1/2),x)
[Out]
Integral(sqrt(a + b*coth(x)**2), x)
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 262 vs.
\(2 (48) = 96\).
time = 0.58, size = 262, normalized size = 4.37 \begin {gather*} \frac {1}{2} \, {\left (\frac {4 \, b \arctan \left (-\frac {\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b}}{2 \, \sqrt {-b}}\right )}{\sqrt {-b}} - \sqrt {a + b} \log \left ({\left | {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} {\left (a + b\right )} - \sqrt {a + b} {\left (a - b\right )} \right |}\right ) + \sqrt {a + b} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + 2 \, b e^{\left (2 \, x\right )} + a + b} + \sqrt {a + b} \right |}\right ) - \sqrt {a + b} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b} \right |}\right )\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*coth(x)^2)^(1/2),x, algorithm="giac")
[Out]
1/2*(4*b*arctan(-1/2*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) - 2*a*e^(2*x) + 2*b*e^(2*x) + a + b) -
sqrt(a + b))/sqrt(-b))/sqrt(-b) - sqrt(a + b)*log(abs((sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) - 2*a*
e^(2*x) + 2*b*e^(2*x) + a + b))*(a + b) - sqrt(a + b)*(a - b))) + sqrt(a + b)*log(abs(-sqrt(a + b)*e^(2*x) + s
qrt(a*e^(4*x) + b*e^(4*x) - 2*a*e^(2*x) + 2*b*e^(2*x) + a + b) + sqrt(a + b))) - sqrt(a + b)*log(abs(-sqrt(a +
b)*e^(2*x) + sqrt(a*e^(4*x) + b*e^(4*x) - 2*a*e^(2*x) + 2*b*e^(2*x) + a + b) - sqrt(a + b))))*sgn(e^(2*x) - 1
)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {b\,{\mathrm {coth}\left (x\right )}^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*coth(x)^2)^(1/2),x)
[Out]
int((a + b*coth(x)^2)^(1/2), x)
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