3.17 \(\int \coth ^2(x) \sqrt{a+b \coth ^2(x)} \, dx\)
Optimal. Leaf size=85 \[ -\frac{1}{2} \coth (x) \sqrt{a+b \coth ^2(x)}-\frac{(a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right )}{2 \sqrt{b}}+\sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right ) \]
[Out]
-((a + 2*b)*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]])/(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
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Rubi [A] time = 0.127882, antiderivative size = 85, normalized size of antiderivative = 1.,
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 =
{3670, 478, 523, 217, 206, 377} \[ -\frac{1}{2} \coth (x) \sqrt{a+b \coth ^2(x)}-\frac{(a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right )}{2 \sqrt{b}}+\sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right ) \]
Antiderivative was successfully verified.
[In]
Int[Coth[x]^2*Sqrt[a + b*Coth[x]^2],x]
[Out]
-((a + 2*b)*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]])/(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 3670
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]))
Rule 478
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 523
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 217
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 377
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]
Rubi steps
\begin{align*} \int \coth ^2(x) \sqrt{a+b \coth ^2(x)} \, dx &=\operatorname{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} \operatorname{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) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\coth (x)\right )+(a+b) \operatorname{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) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\coth (x)}{\sqrt{a+b \coth ^2(x)}}\right )+(a+b) \operatorname{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*}
Mathematica [B] time = 0.784428, size = 191, normalized size = 2.25 \[ -\frac{\sinh (x) \sqrt{\text{csch}^2(x) ((a+b) \cosh (2 x)-a+b)} \left (\sqrt{2} \sqrt{a+b} (a+2 b) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right )+\sqrt{b} \left (\sqrt{a+b} \coth (x) \text{csch}(x) \sqrt{(a+b) \cosh (2 x)-a+b}-2 \sqrt{2} (a+b) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right )\right )\right )}{2 \sqrt{2} \sqrt{b} \sqrt{a+b} \sqrt{(a+b) \cosh (2 x)-a+b}} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[x]^2*Sqrt[a + b*Coth[x]^2],x]
[Out]
-(Sqrt[(-a + b + (a + b)*Cosh[2*x])*Csch[x]^2]*(Sqrt[2]*Sqrt[a + b]*(a + 2*b)*ArcTanh[(Sqrt[2]*Sqrt[b]*Cosh[x]
)/Sqrt[-a + b + (a + b)*Cosh[2*x]]] + Sqrt[b]*(-2*Sqrt[2]*(a + b)*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[-
a + b + (a + b)*Cosh[2*x]]] + Sqrt[a + b]*Sqrt[-a + b + (a + b)*Cosh[2*x]]*Coth[x]*Csch[x]))*Sinh[x])/(2*Sqrt[
2]*Sqrt[b]*Sqrt[a + b]*Sqrt[-a + b + (a + b)*Cosh[2*x]])
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Maple [B] time = 0.046, size = 276, normalized size = 3.3 \begin{align*} -{\frac{{\rm coth} \left (x\right )}{2}\sqrt{a+b \left ({\rm coth} \left (x\right ) \right ) ^{2}}}-{\frac{a}{2}\ln \left ({\rm coth} \left (x\right )\sqrt{b}+\sqrt{a+b \left ({\rm coth} \left (x\right ) \right ) ^{2}} \right ){\frac{1}{\sqrt{b}}}}+{\frac{1}{2}\sqrt{ \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}b-2\, \left ( 1+{\rm coth} \left (x\right ) \right ) b+a+b}}-{\frac{1}{2}\sqrt{b}\ln \left ({( \left ( 1+{\rm coth} \left (x\right ) \right ) b-b){\frac{1}{\sqrt{b}}}}+\sqrt{ \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}b-2\, \left ( 1+{\rm coth} \left (x\right ) \right ) b+a+b} \right ) }-{\frac{1}{2}\sqrt{a+b}\ln \left ({\frac{1}{1+{\rm coth} \left (x\right )} \left ( 2\,a+2\,b-2\, \left ( 1+{\rm coth} \left (x\right ) \right ) b+2\,\sqrt{a+b}\sqrt{ \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}b-2\, \left ( 1+{\rm coth} \left (x\right ) \right ) b+a+b} \right ) } \right ) }-{\frac{1}{2}\sqrt{ \left ({\rm coth} \left (x\right )-1 \right ) ^{2}b+2\, \left ({\rm coth} \left (x\right )-1 \right ) b+a+b}}-{\frac{1}{2}\sqrt{b}\ln \left ({( \left ({\rm coth} \left (x\right )-1 \right ) b+b){\frac{1}{\sqrt{b}}}}+\sqrt{ \left ({\rm coth} \left (x\right )-1 \right ) ^{2}b+2\, \left ({\rm coth} \left (x\right )-1 \right ) b+a+b} \right ) }+{\frac{1}{2}\sqrt{a+b}\ln \left ({\frac{1}{{\rm coth} \left (x\right )-1} \left ( 2\,a+2\,b+2\, \left ({\rm coth} \left (x\right )-1 \right ) b+2\,\sqrt{a+b}\sqrt{ \left ({\rm coth} \left (x\right )-1 \right ) ^{2}b+2\, \left ({\rm coth} \left (x\right )-1 \right ) b+a+b} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)^2*(a+b*coth(x)^2)^(1/2),x)
[Out]
-1/2*coth(x)*(a+b*coth(x)^2)^(1/2)-1/2*a/b^(1/2)*ln(coth(x)*b^(1/2)+(a+b*coth(x)^2)^(1/2))+1/2*((1+coth(x))^2*
b-2*(1+coth(x))*b+a+b)^(1/2)-1/2*b^(1/2)*ln(((1+coth(x))*b-b)/b^(1/2)+((1+coth(x))^2*b-2*(1+coth(x))*b+a+b)^(1
/2))-1/2*(a+b)^(1/2)*ln((2*a+2*b-2*(1+coth(x))*b+2*(a+b)^(1/2)*((1+coth(x))^2*b-2*(1+coth(x))*b+a+b)^(1/2))/(1
+coth(x)))-1/2*((coth(x)-1)^2*b+2*(coth(x)-1)*b+a+b)^(1/2)-1/2*b^(1/2)*ln(((coth(x)-1)*b+b)/b^(1/2)+((coth(x)-
1)^2*b+2*(coth(x)-1)*b+a+b)^(1/2))+1/2*(a+b)^(1/2)*ln((2*a+2*b+2*(coth(x)-1)*b+2*(a+b)^(1/2)*((coth(x)-1)^2*b+
2*(coth(x)-1)*b+a+b)^(1/2))/(coth(x)-1))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \coth \left (x\right )^{2} + a} \coth \left (x\right )^{2}\,{d x} \end{align*}
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] time = 5.27349, size = 14340, normalized size = 168.71 \begin{align*} \text{result too large to display} \end{align*}
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 - 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)) + (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*(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)*arctan(sqrt(2)*(b*cosh(x)^2 + 2*b*cosh
(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 + b^2)*cosh(x)^4 + 4*(a*b + b^2)*cosh(x)*sinh(x)^3 + (a*b + b^2)*sin
h(x)^4 - (a^2 - a*b - 2*b^2)*cosh(x)^2 + (6*(a*b + b^2)*cosh(x)^2 - a^2 + a*b + 2*b^2)*sinh(x)^2 + a^2 + 2*a*b
+ b^2 + 2*(2*(a*b + b^2)*cosh(x)^3 - (a^2 - a*b - 2*b^2)*cosh(x))*sinh(x))) + 2*(b*cosh(x)^4 + 4*b*cosh(x)*si
nh(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)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-a - b)*sqrt(((a + b)*co
sh(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))*sinh(x) + a + b)) - ((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)*log(-((a + 2*b)*cosh(x)^4 + 4*(a + 2*b)*cosh(x)*sin
h(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)) + 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
), -1/2*((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)*sqrt(-a - b)*arctan(sqrt(2)*(b*cosh(x)^2 + 2*b*cosh(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)*si
nh(x) + sinh(x)^2))/((a*b + b^2)*cosh(x)^4 + 4*(a*b + b^2)*cosh(x)*sinh(x)^3 + (a*b + b^2)*sinh(x)^4 - (a^2 -
a*b - 2*b^2)*cosh(x)^2 + (6*(a*b + b^2)*cosh(x)^2 - a^2 + a*b + 2*b^2)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 2*(2*(a
*b + b^2)*cosh(x)^3 - (a^2 - a*b - 2*b^2)*cosh(x))*sinh(x))) + (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)*
arctan(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))/((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)) - ((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))*sinh(x) + a + b)) + 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)*sin
h(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)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \coth ^{2}{\left (x \right )}} \coth ^{2}{\left (x \right )}\, dx \end{align*}
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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
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