3.16 \(\int \coth ^3(x) \sqrt {a+b \coth ^2(x)} \, dx\)
Optimal. Leaf size=63 \[ -\frac {\left (a+b \coth ^2(x)\right )^{3/2}}{3 b}-\sqrt {a+b \coth ^2(x)}+\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right ) \]
[Out]
-1/3*(a+b*coth(x)^2)^(3/2)/b+arctanh((a+b*coth(x)^2)^(1/2)/(a+b)^(1/2))*(a+b)^(1/2)-(a+b*coth(x)^2)^(1/2)
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Rubi [A] time = 0.12, antiderivative size = 63, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used =
{3670, 446, 80, 50, 63, 208} \[ -\frac {\left (a+b \coth ^2(x)\right )^{3/2}}{3 b}-\sqrt {a+b \coth ^2(x)}+\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right ) \]
Antiderivative was successfully verified.
[In]
Int[Coth[x]^3*Sqrt[a + b*Coth[x]^2],x]
[Out]
Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Coth[x]^2] - (a + b*Coth[x]^2)^(3/2)/(3*b)
Rule 50
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 80
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 446
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
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]))
Rubi steps
\begin {align*} \int \coth ^3(x) \sqrt {a+b \coth ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {x^3 \sqrt {a+b x^2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x \sqrt {a+b x}}{1-x} \, dx,x,\coth ^2(x)\right )\\ &=-\frac {\left (a+b \coth ^2(x)\right )^{3/2}}{3 b}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{1-x} \, dx,x,\coth ^2(x)\right )\\ &=-\sqrt {a+b \coth ^2(x)}-\frac {\left (a+b \coth ^2(x)\right )^{3/2}}{3 b}+\frac {1}{2} (a+b) \operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x}} \, dx,x,\coth ^2(x)\right )\\ &=-\sqrt {a+b \coth ^2(x)}-\frac {\left (a+b \coth ^2(x)\right )^{3/2}}{3 b}+\frac {(a+b) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \coth ^2(x)}\right )}{b}\\ &=\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )-\sqrt {a+b \coth ^2(x)}-\frac {\left (a+b \coth ^2(x)\right )^{3/2}}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 60, normalized size = 0.95 \[ \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )-\frac {\sqrt {a+b \coth ^2(x)} \left (a+b \coth ^2(x)+3 b\right )}{3 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[x]^3*Sqrt[a + b*Coth[x]^2],x]
[Out]
Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - (Sqrt[a + b*Coth[x]^2]*(a + 3*b + b*Coth[x]^2))/(3*b)
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fricas [B] time = 0.57, size = 2367, normalized size = 37.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^3*(a+b*coth(x)^2)^(1/2),x, algorithm="fricas")
[Out]
[1/12*(3*(b*cosh(x)^6 + 6*b*cosh(x)*sinh(x)^5 + b*sinh(x)^6 - 3*b*cosh(x)^4 + 3*(5*b*cosh(x)^2 - b)*sinh(x)^4
+ 4*(5*b*cosh(x)^3 - 3*b*cosh(x))*sinh(x)^3 + 3*b*cosh(x)^2 + 3*(5*b*cosh(x)^4 - 6*b*cosh(x)^2 + b)*sinh(x)^2
+ 6*(b*cosh(x)^5 - 2*b*cosh(x)^3 + b*cosh(x))*sinh(x) - b)*sqrt(a + b)*log(-((a^3 + a^2*b)*cosh(x)^8 + 8*(a^3
+ a^2*b)*cosh(x)*sinh(x)^7 + (a^3 + a^2*b)*sinh(x)^8 - 2*(2*a^3 + a^2*b)*cosh(x)^6 - 2*(2*a^3 + a^2*b - 14*(a^
3 + a^2*b)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a^3 + a^2*b)*cosh(x)^3 - 3*(2*a^3 + a^2*b)*cosh(x))*sinh(x)^5 + (6*a^
3 + 4*a^2*b - a*b^2 + b^3)*cosh(x)^4 + (70*(a^3 + a^2*b)*cosh(x)^4 + 6*a^3 + 4*a^2*b - a*b^2 + b^3 - 30*(2*a^3
+ a^2*b)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a^3 + a^2*b)*cosh(x)^5 - 10*(2*a^3 + a^2*b)*cosh(x)^3 + (6*a^3 + 4*a^2
*b - a*b^2 + b^3)*cosh(x))*sinh(x)^3 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(2*a^3 + 3*a^2*b - b^3)*cosh(x)^2 + 2
*(14*(a^3 + a^2*b)*cosh(x)^6 - 15*(2*a^3 + a^2*b)*cosh(x)^4 - 2*a^3 - 3*a^2*b + b^3 + 3*(6*a^3 + 4*a^2*b - a*b
^2 + b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(a^2*cosh(x)^6 + 6*a^2*cosh(x)*sinh(x)^5 + a^2*sinh(x)^6 - 3*a^2*cosh
(x)^4 + 3*(5*a^2*cosh(x)^2 - a^2)*sinh(x)^4 + 4*(5*a^2*cosh(x)^3 - 3*a^2*cosh(x))*sinh(x)^3 + (3*a^2 + 2*a*b -
b^2)*cosh(x)^2 + (15*a^2*cosh(x)^4 - 18*a^2*cosh(x)^2 + 3*a^2 + 2*a*b - b^2)*sinh(x)^2 - a^2 - 2*a*b - b^2 +
2*(3*a^2*cosh(x)^5 - 6*a^2*cosh(x)^3 + (3*a^2 + 2*a*b - 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^3 + a^2*b)*cosh(x)^7
- 3*(2*a^3 + a^2*b)*cosh(x)^5 + (6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x)^3 - (2*a^3 + 3*a^2*b - b^3)*cosh(x))*s
inh(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)) + 3*(b*cosh(x)^6 + 6*b*cosh(x)*sinh(x)^5 + b*sinh(x)^6 - 3*b*cosh(x)
^4 + 3*(5*b*cosh(x)^2 - b)*sinh(x)^4 + 4*(5*b*cosh(x)^3 - 3*b*cosh(x))*sinh(x)^3 + 3*b*cosh(x)^2 + 3*(5*b*cosh
(x)^4 - 6*b*cosh(x)^2 + b)*sinh(x)^2 + 6*(b*cosh(x)^5 - 2*b*cosh(x)^3 + b*cosh(x))*sinh(x) - b)*sqrt(a + b)*lo
g(((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 + 2*b*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^
2 + b)*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 + b*cosh(x))
*sinh(x) + a + b)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) - 4*sqrt(2)*((a + 4*b)*cosh(x)^4 + 4*(a + 4*b)*
cosh(x)*sinh(x)^3 + (a + 4*b)*sinh(x)^4 - 2*(a + 2*b)*cosh(x)^2 + 2*(3*(a + 4*b)*cosh(x)^2 - a - 2*b)*sinh(x)^
2 + 4*((a + 4*b)*cosh(x)^3 - (a + 2*b)*cosh(x))*sinh(x) + a + 4*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)^6 + 6*b*cosh(x)*sinh(x)^5 + b*sinh(x)^6 - 3
*b*cosh(x)^4 + 3*(5*b*cosh(x)^2 - b)*sinh(x)^4 + 4*(5*b*cosh(x)^3 - 3*b*cosh(x))*sinh(x)^3 + 3*b*cosh(x)^2 + 3
*(5*b*cosh(x)^4 - 6*b*cosh(x)^2 + b)*sinh(x)^2 + 6*(b*cosh(x)^5 - 2*b*cosh(x)^3 + b*cosh(x))*sinh(x) - b), -1/
6*(3*(b*cosh(x)^6 + 6*b*cosh(x)*sinh(x)^5 + b*sinh(x)^6 - 3*b*cosh(x)^4 + 3*(5*b*cosh(x)^2 - b)*sinh(x)^4 + 4*
(5*b*cosh(x)^3 - 3*b*cosh(x))*sinh(x)^3 + 3*b*cosh(x)^2 + 3*(5*b*cosh(x)^4 - 6*b*cosh(x)^2 + b)*sinh(x)^2 + 6*
(b*cosh(x)^5 - 2*b*cosh(x)^3 + b*cosh(x))*sinh(x) - b)*sqrt(-a - b)*arctan(sqrt(2)*(a*cosh(x)^2 + 2*a*cosh(x)*
sinh(x) + a*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^2 + a*b)*cosh(x)^4 + 4*(a^2 + a*b)*cosh(x)*sinh(x)^3 + (a^2 + a*b)*sinh(x)
^4 - (2*a^2 + a*b - b^2)*cosh(x)^2 + (6*(a^2 + a*b)*cosh(x)^2 - 2*a^2 - a*b + b^2)*sinh(x)^2 + a^2 + 2*a*b + b
^2 + 2*(2*(a^2 + a*b)*cosh(x)^3 - (2*a^2 + a*b - b^2)*cosh(x))*sinh(x))) + 3*(b*cosh(x)^6 + 6*b*cosh(x)*sinh(x
)^5 + b*sinh(x)^6 - 3*b*cosh(x)^4 + 3*(5*b*cosh(x)^2 - b)*sinh(x)^4 + 4*(5*b*cosh(x)^3 - 3*b*cosh(x))*sinh(x)^
3 + 3*b*cosh(x)^2 + 3*(5*b*cosh(x)^4 - 6*b*cosh(x)^2 + b)*sinh(x)^2 + 6*(b*cosh(x)^5 - 2*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)*s
inh(x)^2 + 4*((a + b)*cosh(x)^3 - (a - b)*cosh(x))*sinh(x) + a + b)) + 2*sqrt(2)*((a + 4*b)*cosh(x)^4 + 4*(a +
4*b)*cosh(x)*sinh(x)^3 + (a + 4*b)*sinh(x)^4 - 2*(a + 2*b)*cosh(x)^2 + 2*(3*(a + 4*b)*cosh(x)^2 - a - 2*b)*si
nh(x)^2 + 4*((a + 4*b)*cosh(x)^3 - (a + 2*b)*cosh(x))*sinh(x) + a + 4*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)))/(b*cosh(x)^6 + 6*b*cosh(x)*sinh(x)^5 + b*sinh(x)
^6 - 3*b*cosh(x)^4 + 3*(5*b*cosh(x)^2 - b)*sinh(x)^4 + 4*(5*b*cosh(x)^3 - 3*b*cosh(x))*sinh(x)^3 + 3*b*cosh(x)
^2 + 3*(5*b*cosh(x)^4 - 6*b*cosh(x)^2 + b)*sinh(x)^2 + 6*(b*cosh(x)^5 - 2*b*cosh(x)^3 + b*cosh(x))*sinh(x) - b
)]
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^3*(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,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(exp(2*x)-1)]Evaluation time: 2.04Error: Bad Argument Type
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maple [B] time = 0.14, size = 253, normalized size = 4.02 \[ -\frac {\left (a +b \left (\coth ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}{3 b}-\frac {\sqrt {\left (\coth \relax (x )-1\right )^{2} b +2 \left (\coth \relax (x )-1\right ) b +a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {\left (\coth \relax (x )-1\right ) b +b}{\sqrt {b}}+\sqrt {\left (\coth \relax (x )-1\right )^{2} b +2 \left (\coth \relax (x )-1\right ) b +a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 \left (\coth \relax (x )-1\right ) b +2 \sqrt {a +b}\, \sqrt {\left (\coth \relax (x )-1\right )^{2} b +2 \left (\coth \relax (x )-1\right ) b +a +b}}{\coth \relax (x )-1}\right )}{2}-\frac {\sqrt {\left (1+\coth \relax (x )\right )^{2} b -2 \left (1+\coth \relax (x )\right ) b +a +b}}{2}+\frac {\sqrt {b}\, \ln \left (\frac {\left (1+\coth \relax (x )\right ) b -b}{\sqrt {b}}+\sqrt {\left (1+\coth \relax (x )\right )^{2} b -2 \left (1+\coth \relax (x )\right ) b +a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 \left (1+\coth \relax (x )\right ) b +2 \sqrt {a +b}\, \sqrt {\left (1+\coth \relax (x )\right )^{2} b -2 \left (1+\coth \relax (x )\right ) b +a +b}}{1+\coth \relax (x )}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)^3*(a+b*coth(x)^2)^(1/2),x)
[Out]
-1/3*(a+b*coth(x)^2)^(3/2)/b-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))-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)))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \coth \relax (x)^{2} + a} \coth \relax (x)^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^3*(a+b*coth(x)^2)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(b*coth(x)^2 + a)*coth(x)^3, x)
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mupad [B] time = 3.08, size = 66, normalized size = 1.05 \[ -\sqrt {b\,{\mathrm {coth}\relax (x)}^2+a}-\frac {{\left (b\,{\mathrm {coth}\relax (x)}^2+a\right )}^{3/2}}{3\,b}-2\,\mathrm {atan}\left (\frac {2\,\sqrt {b\,{\mathrm {coth}\relax (x)}^2+a}\,\sqrt {-\frac {a}{4}-\frac {b}{4}}}{a+b}\right )\,\sqrt {-\frac {a}{4}-\frac {b}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)^3*(a + b*coth(x)^2)^(1/2),x)
[Out]
- (a + b*coth(x)^2)^(1/2) - (a + b*coth(x)^2)^(3/2)/(3*b) - 2*atan((2*(a + b*coth(x)^2)^(1/2)*(- a/4 - b/4)^(1
/2))/(a + b))*(- a/4 - b/4)^(1/2)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \coth ^{2}{\relax (x )}} \coth ^{3}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)**3*(a+b*coth(x)**2)**(1/2),x)
[Out]
Integral(sqrt(a + b*coth(x)**2)*coth(x)**3, x)
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