Integrand size = 17, antiderivative size = 51 \[ \int \frac {\coth ^2(x)}{\sqrt {a+b \tanh ^2(x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{\sqrt {a+b}}-\frac {\coth (x) \sqrt {a+b \tanh ^2(x)}}{a} \]
arctanh((a+b)^(1/2)*tanh(x)/(a+b*tanh(x)^2)^(1/2))/(a+b)^(1/2)-coth(x)*(a+ b*tanh(x)^2)^(1/2)/a
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.85 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.55 \[ \int \frac {\coth ^2(x)}{\sqrt {a+b \tanh ^2(x)}} \, dx=-\frac {\cosh ^2(x) \coth (x) \left (1+\frac {b \tanh ^2(x)}{a}\right ) \left (-\frac {4 (a+b) \operatorname {Hypergeometric2F1}\left (2,2,\frac {5}{2},-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{3 a^2}+\frac {\arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \left (a+2 b \tanh ^2(x)\right )}{a \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}\right )}{\sqrt {a+b \tanh ^2(x)}} \]
-((Cosh[x]^2*Coth[x]*(1 + (b*Tanh[x]^2)/a)*((-4*(a + b)*Hypergeometric2F1[ 2, 2, 5/2, -(((a + b)*Sinh[x]^2)/a)]*Sinh[x]^2*(a + b*Tanh[x]^2))/(3*a^2) + (ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*(a + 2*b*Tanh[x]^2))/(a*Sqrt[-(( (a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)])))/Sqrt[a + b*Tanh[x] ^2])
Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {3042, 25, 4153, 25, 382, 27, 291, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\coth ^2(x)}{\sqrt {a+b \tanh ^2(x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\tan (i x)^2 \sqrt {a-b \tan (i x)^2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\tan (i x)^2 \sqrt {a-b \tan (i x)^2}}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle -\int -\frac {\coth ^2(x)}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\coth ^2(x)}{\left (1-\tanh ^2(x)\right ) \sqrt {a+b \tanh ^2(x)}}d\tanh (x)\) |
| \(\Big \downarrow \) 382 |
| \(\displaystyle \frac {\int \frac {a}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)}{a}-\frac {\coth (x) \sqrt {a+b \tanh ^2(x)}}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)-\frac {\coth (x) \sqrt {a+b \tanh ^2(x)}}{a}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \int \frac {1}{1-\frac {(a+b) \tanh ^2(x)}{b \tanh ^2(x)+a}}d\frac {\tanh (x)}{\sqrt {b \tanh ^2(x)+a}}-\frac {\coth (x) \sqrt {a+b \tanh ^2(x)}}{a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{\sqrt {a+b}}-\frac {\coth (x) \sqrt {a+b \tanh ^2(x)}}{a}\) |
ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/Sqrt[a + b] - (Coth[x ]*Sqrt[a + b*Tanh[x]^2])/a
3.3.36.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/ (a*c*e*(m + 1))), x] - Simp[1/(a*c*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b* x^2)^p*(c + d*x^2)^q*Simp[(b*c + a*d)*(m + 3) + 2*(b*c*p + a*d*q) + b*d*(m + 2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[ b*c - a*d, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
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]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^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] && Ratio nalQ[n]))
\[\int \frac {\coth \left (x \right )^{2}}{\sqrt {a +b \tanh \left (x \right )^{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (43) = 86\).
Time = 0.32 (sec) , antiderivative size = 1565, normalized size of antiderivative = 30.69 \[ \int \frac {\coth ^2(x)}{\sqrt {a+b \tanh ^2(x)}} \, dx=\text {Too large to display} \]
[1/4*((a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 - a)*sqrt(a + b)*lo g(-((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))*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*...
\[ \int \frac {\coth ^2(x)}{\sqrt {a+b \tanh ^2(x)}} \, dx=\int \frac {\coth ^{2}{\left (x \right )}}{\sqrt {a + b \tanh ^{2}{\left (x \right )}}}\, dx \]
\[ \int \frac {\coth ^2(x)}{\sqrt {a+b \tanh ^2(x)}} \, dx=\int { \frac {\coth \left (x\right )^{2}}{\sqrt {b \tanh \left (x\right )^{2} + a}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (43) = 86\).
Time = 0.48 (sec) , antiderivative size = 343, normalized size of antiderivative = 6.73 \[ \int \frac {\coth ^2(x)}{\sqrt {a+b \tanh ^2(x)}} \, dx=-\frac {\log \left ({\left | -{\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} {\left (a + b\right )} - \sqrt {a + b} {\left (a - b\right )} \right |}\right )}{2 \, \sqrt {a + b}} - \frac {\log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} + \sqrt {a + b} \right |}\right )}{2 \, \sqrt {a + b}} + \frac {\log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b} \right |}\right )}{2 \, \sqrt {a + b}} + \frac {4 \, {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} + \sqrt {a + b}\right )}}{{\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )}^{2} - 2 \, {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} \sqrt {a + b} - 3 \, a + b} \]
-1/2*log(abs(-(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2 *x) - 2*b*e^(2*x) + a + b))*(a + b) - sqrt(a + b)*(a - b)))/sqrt(a + b) - 1/2*log(abs(-sqrt(a + b)*e^(2*x) + sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x ) - 2*b*e^(2*x) + a + b) + sqrt(a + b)))/sqrt(a + b) + 1/2*log(abs(-sqrt(a + b)*e^(2*x) + sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b) - sqrt(a + b)))/sqrt(a + b) + 4*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x ) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b) + sqrt(a + b))/((sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^2 - 2*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2* x) - 2*b*e^(2*x) + a + b))*sqrt(a + b) - 3*a + b)
Timed out. \[ \int \frac {\coth ^2(x)}{\sqrt {a+b \tanh ^2(x)}} \, dx=\int \frac {{\mathrm {coth}\left (x\right )}^2}{\sqrt {b\,{\mathrm {tanh}\left (x\right )}^2+a}} \,d x \]