3.1.17 \(\int \coth ^2(x) \sqrt {a+b \coth ^2(x)} \, dx\) [17]
Optimal. Leaf size=85 \[ -\frac {(a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{2 \sqrt {b}}+\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-\frac {1}{2} \coth (x) \sqrt {a+b \coth ^2(x)} \]
[Out]
-1/2*(a+2*b)*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)-1/2*coth(x)*(a+b*coth(x)^2)^(1/2)
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Rubi [A]
time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3751, 489, 537,
223, 212, 385} \begin {gather*} -\frac {1}{2} \coth (x) \sqrt {a+b \coth ^2(x)}-\frac {(a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{2 \sqrt {b}}+\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Coth[x]^2*Sqrt[a + b*Coth[x]^2],x]
[Out]
-1/2*((a + 2*b)*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]])/Sqrt[b] + Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*C
oth[x])/Sqrt[a + b*Coth[x]^2]] - (Coth[x]*Sqrt[a + b*Coth[x]^2])/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 489
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(m + n*(p + q) + 1))), x] - Dist[e^n/(b*(m + n*(p +
q) + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b
*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] &&
GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 537
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]
Rule 3751
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((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[(d*ff*(x/c))^m*((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, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))
Rubi steps
\begin {align*} \int \coth ^2(x) \sqrt {a+b \coth ^2(x)} \, dx &=\text {Subst}\left (\int \frac {x^2 \sqrt {a+b x^2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{2} \coth (x) \sqrt {a+b \coth ^2(x)}+\frac {1}{2} \text {Subst}\left (\int \frac {a+(a+2 b) x^2}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{2} \coth (x) \sqrt {a+b \coth ^2(x)}+\frac {1}{2} (-a-2 b) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\coth (x)\right )+(a+b) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{2} \coth (x) \sqrt {a+b \coth ^2(x)}+\frac {1}{2} (-a-2 b) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )+(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 )\\ &=-\frac {(a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{2 \sqrt {b}}+\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-\frac {1}{2} \coth (x) \sqrt {a+b \coth ^2(x)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(191\) vs. \(2(85)=170\).
time = 0.58, size = 191, normalized size = 2.25 \begin {gather*} -\frac {\sqrt {(-a+b+(a+b) \cosh (2 x)) \text {csch}^2(x)} \left (\sqrt {2} \sqrt {a+b} (a+2 b) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \cosh (x)}{\sqrt {-a+b+(a+b) \cosh (2 x)}}\right )+\sqrt {b} \left (-2 \sqrt {2} (a+b) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a+b} \cosh (x)}{\sqrt {-a+b+(a+b) \cosh (2 x)}}\right )+\sqrt {a+b} \sqrt {-a+b+(a+b) \cosh (2 x)} \coth (x) \text {csch}(x)\right )\right ) \sinh (x)}{2 \sqrt {2} \sqrt {b} \sqrt {a+b} \sqrt {-a+b+(a+b) \cosh (2 x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Coth[x]^2*Sqrt[a + b*Coth[x]^2],x]
[Out]
-1/2*(Sqrt[(-a + b + (a + b)*Cosh[2*x])*Csch[x]^2]*(Sqrt[2]*Sqrt[a + b]*(a + 2*b)*ArcTanh[(Sqrt[2]*Sqrt[b]*Cos
h[x])/Sqrt[-a + b + (a + b)*Cosh[2*x]]] + Sqrt[b]*(-2*Sqrt[2]*(a + b)*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sq
rt[-a + b + (a + b)*Cosh[2*x]]] + Sqrt[a + b]*Sqrt[-a + b + (a + b)*Cosh[2*x]]*Coth[x]*Csch[x]))*Sinh[x])/(Sqr
t[2]*Sqrt[b]*Sqrt[a + b]*Sqrt[-a + b + (a + b)*Cosh[2*x]])
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(275\) vs.
\(2(67)=134\).
time = 0.72, size = 276, normalized size = 3.25
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(-\frac {\coth \left (x \right ) \sqrt {a +b \left (\coth ^{2}\left (x \right )\right )}}{2}-\frac {a \ln \left (\sqrt {b}\, \coth \left (x \right )+\sqrt {a +b \left (\coth ^{2}\left (x \right )\right )}\right )}{2 \sqrt {b}}-\frac {\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2}+\frac {\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )}{2}-\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2}\) |
\(276\) |
| default |
\(-\frac {\coth \left (x \right ) \sqrt {a +b \left (\coth ^{2}\left (x \right )\right )}}{2}-\frac {a \ln \left (\sqrt {b}\, \coth \left (x \right )+\sqrt {a +b \left (\coth ^{2}\left (x \right )\right )}\right )}{2 \sqrt {b}}-\frac {\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2}+\frac {\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )}{2}-\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2}\) |
\(276\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)^2*(a+b*coth(x)^2)^(1/2),x,method=_RETURNVERBOSE)
[Out]
-1/2*coth(x)*(a+b*coth(x)^2)^(1/2)-1/2*a/b^(1/2)*ln(b^(1/2)*coth(x)+(a+b*coth(x)^2)^(1/2))-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))/(c
oth(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(coth(x)^2*(a+b*coth(x)^2)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(b*coth(x)^2 + a)*coth(x)^2, x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 907 vs.
\(2 (67) = 134\).
time = 0.61, size = 4877, normalized size = 57.38 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^2*(a+b*coth(x)^2)^(1/2),x, algorithm="fricas")
[Out]
[1/4*((b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 2*b*cosh(x)^2 + 2*(3*b*cosh(x)^2 - b)*sinh(x)^2 + 4
*(b*cosh(x)^3 - b*cosh(x))*sinh(x) + b)*sqrt(a + b)*log(((a*b^2 + b^3)*cosh(x)^8 + 8*(a*b^2 + b^3)*cosh(x)*sin
h(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)*co
sh(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)*c
osh(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*cos
h(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 + (1
5*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)*si
nh(x)^5 + sinh(x)^6)) + ((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 + 4*((a + 2*b)*cosh(x)^3 - (a + 2*b)*cosh(x))*sin
h(x) + a + 2*b)*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)) + (b*cosh(x)
^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 2*b*cosh(x)^2 + 2*(3*b*cosh(x)^2 - b)*sinh(x)^2 + 4*(b*cosh(x)^3 -
b*cosh(x))*sinh(x) + b)*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)) - 2*sqrt
(2)*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + 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)))/(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 2*b*cosh(x
)^2 + 2*(3*b*cosh(x)^2 - b)*sinh(x)^2 + 4*(b*cosh(x)^3 - b*cosh(x))*sinh(x) + b), 1/4*(2*((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 + 4*((a + 2*b)*cosh(x)^3 - (a + 2*b)*cosh(x))*sinh(x) + a + 2*b)*sqrt(-b)*arctan(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))/((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))*sin
h(x) + a + b)) + (b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 2*b*cosh(x)^2 + 2*(3*b*cosh(x)^2 - b)*si
nh(x)^2 + 4*(b*cosh(x)^3 - b*cosh(x))*sinh(x) + b)*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)*cos
h(x)^2 + (15*b^2*cosh(x)^4 + 18*b^2*cosh(x)^2 -...
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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 )}} \coth ^{2}{\left (x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)**2*(a+b*coth(x)**2)**(1/2),x)
[Out]
Integral(sqrt(a + b*coth(x)**2)*coth(x)**2, x)
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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(x)^2*(a+b*coth(x)^2)^(1/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(ex
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {coth}\left (x\right )}^2\,\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(coth(x)^2*(a + b*coth(x)^2)^(1/2),x)
[Out]
int(coth(x)^2*(a + b*coth(x)^2)^(1/2), x)
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