\(\int \frac {\tanh ^2(x)}{\sqrt {a+b \coth ^2(x)}} \, dx\) [37]

Optimal result

Integrand size = 17, antiderivative size = 51 \[ \int \frac {\tanh ^2(x)}{\sqrt {a+b \coth ^2(x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{\sqrt {a+b}}-\frac {\sqrt {a+b \coth ^2(x)} \tanh (x)}{a} \]

[Out]

arctanh(coth(x)*(a+b)^(1/2)/(a+b*coth(x)^2)^(1/2))/(a+b)^(1/2)-(a+b*coth(x)^2)^(1/2)*tanh(x)/a

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3751, 491, 12, 385, 212} \[ \int \frac {\tanh ^2(x)}{\sqrt {a+b \coth ^2(x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{\sqrt {a+b}}-\frac {\tanh (x) \sqrt {a+b \coth ^2(x)}}{a} \]

[In]

Int[Tanh[x]^2/Sqrt[a + b*Coth[x]^2],x]

[Out]

ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/Sqrt[a + b] - (Sqrt[a + b*Coth[x]^2]*Tanh[x])/a

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 491

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, 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*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right ) \\ & = -\frac {\sqrt {a+b \coth ^2(x)} \tanh (x)}{a}+\frac {\text {Subst}\left (\int \frac {a}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{a} \\ & = -\frac {\sqrt {a+b \coth ^2(x)} \tanh (x)}{a}+\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right ) \\ & = -\frac {\sqrt {a+b \coth ^2(x)} \tanh (x)}{a}+\text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right ) \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{\sqrt {a+b}}-\frac {\sqrt {a+b \coth ^2(x)} \tanh (x)}{a} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.77 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.49 \[ \int \frac {\tanh ^2(x)}{\sqrt {a+b \coth ^2(x)}} \, dx=\frac {\left (1+\frac {b \coth ^2(x)}{a}\right ) \sinh ^2(x) \left (\frac {4 (a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \operatorname {Hypergeometric2F1}\left (2,2,\frac {5}{2},\frac {(a+b) \cosh ^2(x)}{a}\right )}{3 a^2}+\frac {\arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \left (a+2 b \coth ^2(x)\right )}{a \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}\right ) \tanh (x)}{\sqrt {a+b \coth ^2(x)}} \]

[In]

Integrate[Tanh[x]^2/Sqrt[a + b*Coth[x]^2],x]

[Out]

((1 + (b*Coth[x]^2)/a)*Sinh[x]^2*((4*(a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Hypergeometric2F1[2, 2, 5/2, ((a + b)
*Cosh[x]^2)/a])/(3*a^2) + (ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*(a + 2*b*Coth[x]^2))/(a*Sqrt[-(((a + b)*Cosh[x]
^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)]))*Tanh[x])/Sqrt[a + b*Coth[x]^2]

Maple [F]

\[\int \frac {\tanh \left (x \right )^{2}}{\sqrt {a +b \coth \left (x \right )^{2}}}d x\]

[In]

int(tanh(x)^2/(a+b*coth(x)^2)^(1/2),x)

[Out]

int(tanh(x)^2/(a+b*coth(x)^2)^(1/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (43) = 86\).

Time = 0.33 (sec) , antiderivative size = 1621, normalized size of antiderivative = 31.78 \[ \int \frac {\tanh ^2(x)}{\sqrt {a+b \coth ^2(x)}} \, dx=\text {Too large to display} \]

[In]

integrate(tanh(x)^2/(a+b*coth(x)^2)^(1/2),x, algorithm="fricas")

[Out]

[1/4*((a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 + a)*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) + 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))*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)) + (a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 + a)*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)) - 4*sqrt(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^2 + a*b)*cosh(x)^2 + 2*(a^2 + a
*b)*cosh(x)*sinh(x) + (a^2 + a*b)*sinh(x)^2 + a^2 + a*b), -1/2*((a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)
^2 + a)*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)*sqr
t(((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)*co
sh(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))) + (a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 + a)*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)*si
nh(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)*cos
h(x))*sinh(x) + a + b)) + 2*sqrt(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^2 + a*b)*cosh(x)^2 + 2*(a^2 + a*b)*cosh(x)*sinh(x) + (a^2 + a*b)*sinh(x)^
2 + a^2 + a*b)]

Sympy [F]

\[ \int \frac {\tanh ^2(x)}{\sqrt {a+b \coth ^2(x)}} \, dx=\int \frac {\tanh ^{2}{\left (x \right )}}{\sqrt {a + b \coth ^{2}{\left (x \right )}}}\, dx \]

[In]

integrate(tanh(x)**2/(a+b*coth(x)**2)**(1/2),x)

[Out]

Integral(tanh(x)**2/sqrt(a + b*coth(x)**2), x)

Maxima [F]

\[ \int \frac {\tanh ^2(x)}{\sqrt {a+b \coth ^2(x)}} \, dx=\int { \frac {\tanh \left (x\right )^{2}}{\sqrt {b \coth \left (x\right )^{2} + a}} \,d x } \]

[In]

integrate(tanh(x)^2/(a+b*coth(x)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(tanh(x)^2/sqrt(b*coth(x)^2 + a), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\tanh ^2(x)}{\sqrt {a+b \coth ^2(x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(tanh(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:Error: Bad Argument Type

Mupad [F(-1)]

Timed out. \[ \int \frac {\tanh ^2(x)}{\sqrt {a+b \coth ^2(x)}} \, dx=\int \frac {{\mathrm {tanh}\left (x\right )}^2}{\sqrt {b\,{\mathrm {coth}\left (x\right )}^2+a}} \,d x \]

[In]

int(tanh(x)^2/(a + b*coth(x)^2)^(1/2),x)

[Out]

int(tanh(x)^2/(a + b*coth(x)^2)^(1/2), x)