3.39 \(\int \frac {\coth ^2(x)}{(a+b \coth ^2(x))^{3/2}} \, dx\)
Optimal. Leaf size=53 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{3/2}}-\frac {\coth (x)}{(a+b) \sqrt {a+b \coth ^2(x)}} \]
[Out]
arctanh(coth(x)*(a+b)^(1/2)/(a+b*coth(x)^2)^(1/2))/(a+b)^(3/2)-coth(x)/(a+b)/(a+b*coth(x)^2)^(1/2)
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Rubi [A] time = 0.10, antiderivative size = 53, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used =
{3670, 471, 377, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{3/2}}-\frac {\coth (x)}{(a+b) \sqrt {a+b \coth ^2(x)}} \]
Antiderivative was successfully verified.
[In]
Int[Coth[x]^2/(a + b*Coth[x]^2)^(3/2),x]
[Out]
ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(3/2) - Coth[x]/((a + b)*Sqrt[a + b*Coth[x]^2])
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]
Rule 471
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 + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
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 \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx &=\operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\coth (x)\right )\\ &=-\frac {\coth (x)}{(a+b) \sqrt {a+b \coth ^2(x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{a+b}\\ &=-\frac {\coth (x)}{(a+b) \sqrt {a+b \coth ^2(x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{a+b}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{3/2}}-\frac {\coth (x)}{(a+b) \sqrt {a+b \coth ^2(x)}}\\ \end {align*}
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Mathematica [B] time = 1.00, size = 109, normalized size = 2.06 \[ \frac {\frac {\text {csch}(x) \text {sech}(x) ((a+b) \cosh (2 x)-a+b) \sqrt {\frac {(a+b) \coth ^2(x)}{a}} \tanh ^{-1}\left (\frac {\sqrt {\frac {(a+b) \coth ^2(x)}{a}}}{\sqrt {\frac {b \coth ^2(x)}{a}+1}}\right )}{\sqrt {\frac {b \coth ^2(x)}{a}+1}}-2 (a+b) \coth (x)}{2 (a+b)^2 \sqrt {a+b \coth ^2(x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[x]^2/(a + b*Coth[x]^2)^(3/2),x]
[Out]
(-2*(a + b)*Coth[x] + (ArcTanh[Sqrt[((a + b)*Coth[x]^2)/a]/Sqrt[1 + (b*Coth[x]^2)/a]]*(-a + b + (a + b)*Cosh[2
*x])*Sqrt[((a + b)*Coth[x]^2)/a]*Csch[x]*Sech[x])/Sqrt[1 + (b*Coth[x]^2)/a])/(2*(a + b)^2*Sqrt[a + b*Coth[x]^2
])
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fricas [B] time = 0.61, size = 2279, normalized size = 43.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^2/(a+b*coth(x)^2)^(3/2),x, algorithm="fricas")
[Out]
[1/4*(((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(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)*si
nh(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))*s
qrt(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)) + ((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(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*co
sh(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)) - 4*sqrt(2)*((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + 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^3 + 3*a^2*b +
3*a*b^2 + b^3)*cosh(x)^4 + 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)*sinh(x)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b
^3)*sinh(x)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(a^3 + a^2*b - a*b^2 - b^3)*cosh(x)^2 - 2*(a^3 + a^2*b - a*b
^2 - b^3 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^2)*sinh(x)^2 + 4*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)
^3 - (a^3 + a^2*b - a*b^2 - b^3)*cosh(x))*sinh(x)), -1/2*(((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(-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) + sin
h(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)*c
osh(x)^3 - (a^2 - a*b - 2*b^2)*cosh(x))*sinh(x))) + ((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(-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)) + 2*sqrt(2)*((a + b)*cosh(x)^2 +
2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + 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^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^4 + 4*(a^3 + 3*a^2*b + 3
*a*b^2 + b^3)*cosh(x)*sinh(x)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(x)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 -
2*(a^3 + a^2*b - a*b^2 - b^3)*cosh(x)^2 - 2*(a^3 + a^2*b - a*b^2 - b^3 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos
h(x)^2)*sinh(x)^2 + 4*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^3 - (a^3 + a^2*b - a*b^2 - b^3)*cosh(x))*sinh(x
))]
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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)^2/(a+b*coth(x)^2)^(3/2),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(t_nostep-1)]Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacin
g 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution var
iable should perhaps be purged.Error: Bad Argument Type
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maple [B] time = 0.10, size = 289, normalized size = 5.45 \[ -\frac {\coth \relax (x )}{a \sqrt {a +b \left (\coth ^{2}\relax (x )\right )}}-\frac {1}{2 \left (a +b \right ) \sqrt {\left (\coth \relax (x )-1\right )^{2} b +2 \left (\coth \relax (x )-1\right ) b +a +b}}+\frac {b \left (2 \left (\coth \relax (x )-1\right ) b +2 b \right )}{\left (a +b \right ) \left (4 b \left (a +b \right )-4 b^{2}\right ) \sqrt {\left (\coth \relax (x )-1\right )^{2} b +2 \left (\coth \relax (x )-1\right ) b +a +b}}+\frac {\ln \left (\frac {2 a +2 b +2 \left (\coth \relax (x )-1\right ) b +2 \sqrt {a +b}\, \sqrt {\left (\coth \relax (x )-1\right )^{2} b +2 \left (\coth \relax (x )-1\right ) b +a +b}}{\coth \relax (x )-1}\right )}{2 \left (a +b \right )^{\frac {3}{2}}}+\frac {1}{2 \left (a +b \right ) \sqrt {\left (1+\coth \relax (x )\right )^{2} b -2 \left (1+\coth \relax (x )\right ) b +a +b}}+\frac {b \left (2 \left (1+\coth \relax (x )\right ) b -2 b \right )}{\left (a +b \right ) \left (4 b \left (a +b \right )-4 b^{2}\right ) \sqrt {\left (1+\coth \relax (x )\right )^{2} b -2 \left (1+\coth \relax (x )\right ) b +a +b}}-\frac {\ln \left (\frac {2 a +2 b -2 \left (1+\coth \relax (x )\right ) b +2 \sqrt {a +b}\, \sqrt {\left (1+\coth \relax (x )\right )^{2} b -2 \left (1+\coth \relax (x )\right ) b +a +b}}{1+\coth \relax (x )}\right )}{2 \left (a +b \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)^2/(a+b*coth(x)^2)^(3/2),x)
[Out]
-coth(x)/a/(a+b*coth(x)^2)^(1/2)-1/2/(a+b)/((coth(x)-1)^2*b+2*(coth(x)-1)*b+a+b)^(1/2)+b/(a+b)*(2*(coth(x)-1)*
b+2*b)/(4*b*(a+b)-4*b^2)/((coth(x)-1)^2*b+2*(coth(x)-1)*b+a+b)^(1/2)+1/2/(a+b)^(3/2)*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/(a+b)/((1+coth(x))^2*b-2*(1+cot
h(x))*b+a+b)^(1/2)+b/(a+b)*(2*(1+coth(x))*b-2*b)/(4*b*(a+b)-4*b^2)/((1+coth(x))^2*b-2*(1+coth(x))*b+a+b)^(1/2)
-1/2/(a+b)^(3/2)*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+cot
h(x)))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth \relax (x)^{2}}{{\left (b \coth \relax (x)^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^2/(a+b*coth(x)^2)^(3/2),x, algorithm="maxima")
[Out]
integrate(coth(x)^2/(b*coth(x)^2 + a)^(3/2), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {coth}\relax (x)}^2}{{\left (b\,{\mathrm {coth}\relax (x)}^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)^2/(a + b*coth(x)^2)^(3/2),x)
[Out]
int(coth(x)^2/(a + b*coth(x)^2)^(3/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{2}{\relax (x )}}{\left (a + b \coth ^{2}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)**2/(a+b*coth(x)**2)**(3/2),x)
[Out]
Integral(coth(x)**2/(a + b*coth(x)**2)**(3/2), x)
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