Integrand size = 23, antiderivative size = 142 \[ \int \frac {\coth ^3(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=-\frac {(2 a-b) \text {arctanh}\left (\frac {2 a+b \tanh ^2(x)}{2 \sqrt {a} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{4 a^{3/2}}+\frac {\text {arctanh}\left (\frac {2 a+b+(b+2 c) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}}-\frac {\coth ^2(x) \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}{2 a} \] Output:
-1/4*(2*a-b)*arctanh(1/2*(2*a+b*tanh(x)^2)/a^(1/2)/(a+b*tanh(x)^2+c*tanh(x )^4)^(1/2))/a^(3/2)+1/2*arctanh(1/2*(2*a+b+(b+2*c)*tanh(x)^2)/(a+b+c)^(1/2 )/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2))/(a+b+c)^(1/2)-1/2*coth(x)^2*(a+b*tanh (x)^2+c*tanh(x)^4)^(1/2)/a
Time = 0.40 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^3(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=-\frac {(2 a-b) \text {arctanh}\left (\frac {2 a+b \tanh ^2(x)}{2 \sqrt {a} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{4 a^{3/2}}+\frac {\text {arctanh}\left (\frac {2 a+b+(b+2 c) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}}-\frac {\coth ^2(x) \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}{2 a} \] Input:
Integrate[Coth[x]^3/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4],x]
Output:
-1/4*((2*a - b)*ArcTanh[(2*a + b*Tanh[x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tanh[x]^ 2 + c*Tanh[x]^4])])/a^(3/2) + ArcTanh[(2*a + b + (b + 2*c)*Tanh[x]^2)/(2*S qrt[a + b + c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])]/(2*Sqrt[a + b + c]) - (Coth[x]^2*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])/(2*a)
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.35, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 26, 4183, 1578, 1289, 2009}
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.
\(\displaystyle \int \frac {\coth ^3(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i}{\tan (i x)^3 \sqrt {a-b \tan (i x)^2+c \tan (i x)^4}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {1}{\tan (i x)^3 \sqrt {c \tan (i x)^4-b \tan (i x)^2+a}}dx\) |
\(\Big \downarrow \) 4183 |
\(\displaystyle -\int \frac {i \coth ^3(x)}{\left (1-\tanh ^2(x)\right ) \sqrt {c \tanh ^4(x)+b \tanh ^2(x)+a}}d(i \tanh (x))\) |
\(\Big \downarrow \) 1578 |
\(\displaystyle -\frac {1}{2} \int -\frac {\coth ^2(x)}{\left (1-\tanh ^2(x)\right ) \sqrt {-c \tanh ^2(x)-i b \tanh (x)+a}}d\left (-\tanh ^2(x)\right )\) |
\(\Big \downarrow \) 1289 |
\(\displaystyle -\frac {1}{2} \int \left (-\frac {\coth ^2(x)}{\sqrt {-c \tanh ^2(x)-i b \tanh (x)+a}}+\frac {i \coth (x)}{\sqrt {-c \tanh ^2(x)-i b \tanh (x)+a}}+\frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {-c \tanh ^2(x)-i b \tanh (x)+a}}\right )d\left (-\tanh ^2(x)\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {b \text {arctanh}\left (\frac {2 a-i b \tanh (x)}{2 \sqrt {a} \sqrt {a-i b \tanh (x)-c \tanh ^2(x)}}\right )}{2 a^{3/2}}-\frac {\text {arctanh}\left (\frac {2 a-i b \tanh (x)}{2 \sqrt {a} \sqrt {a-i b \tanh (x)-c \tanh ^2(x)}}\right )}{\sqrt {a}}+\frac {\text {arctanh}\left (\frac {2 a-i (b+2 c) \tanh (x)+b}{2 \sqrt {a+b+c} \sqrt {a-i b \tanh (x)-c \tanh ^2(x)}}\right )}{\sqrt {a+b+c}}-\frac {i \coth (x) \sqrt {a-i b \tanh (x)-c \tanh ^2(x)}}{a}\right )\) |
Input:
Int[Coth[x]^3/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4],x]
Output:
(-(ArcTanh[(2*a - I*b*Tanh[x])/(2*Sqrt[a]*Sqrt[a - I*b*Tanh[x] - c*Tanh[x] ^2])]/Sqrt[a]) + (b*ArcTanh[(2*a - I*b*Tanh[x])/(2*Sqrt[a]*Sqrt[a - I*b*Ta nh[x] - c*Tanh[x]^2])])/(2*a^(3/2)) + ArcTanh[(2*a + b - I*(b + 2*c)*Tanh[ x])/(2*Sqrt[a + b + c]*Sqrt[a - I*b*Tanh[x] - c*Tanh[x]^2])]/Sqrt[a + b + c] - (I*Coth[x]*Sqrt[a - I*b*Tanh[x] - c*Tanh[x]^2])/a)/2
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ( IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*( x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n2_.))^(p_), x_Symbol] :> Simp[f/e Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)), x ], x, f*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n 2, 2*n] && NeQ[b^2 - 4*a*c, 0]
\[\int \frac {\coth \left (x \right )^{3}}{\sqrt {a +b \tanh \left (x \right )^{2}+c \tanh \left (x \right )^{4}}}d x\]
Input:
int(coth(x)^3/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x)
Output:
int(coth(x)^3/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 2136 vs. \(2 (118) = 236\).
Time = 1.10 (sec) , antiderivative size = 9168, normalized size of antiderivative = 64.56 \[ \int \frac {\coth ^3(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\text {Too large to display} \] Input:
integrate(coth(x)^3/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="fricas ")
Output:
Too large to include
\[ \int \frac {\coth ^3(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int \frac {\coth ^{3}{\left (x \right )}}{\sqrt {a + b \tanh ^{2}{\left (x \right )} + c \tanh ^{4}{\left (x \right )}}}\, dx \] Input:
integrate(coth(x)**3/(a+b*tanh(x)**2+c*tanh(x)**4)**(1/2),x)
Output:
Integral(coth(x)**3/sqrt(a + b*tanh(x)**2 + c*tanh(x)**4), x)
\[ \int \frac {\coth ^3(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int { \frac {\coth \left (x\right )^{3}}{\sqrt {c \tanh \left (x\right )^{4} + b \tanh \left (x\right )^{2} + a}} \,d x } \] Input:
integrate(coth(x)^3/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="maxima ")
Output:
integrate(coth(x)^3/sqrt(c*tanh(x)^4 + b*tanh(x)^2 + a), x)
Timed out. \[ \int \frac {\coth ^3(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\text {Timed out} \] Input:
integrate(coth(x)^3/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\coth ^3(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int \frac {{\mathrm {coth}\left (x\right )}^3}{\sqrt {c\,{\mathrm {tanh}\left (x\right )}^4+b\,{\mathrm {tanh}\left (x\right )}^2+a}} \,d x \] Input:
int(coth(x)^3/(a + b*tanh(x)^2 + c*tanh(x)^4)^(1/2),x)
Output:
int(coth(x)^3/(a + b*tanh(x)^2 + c*tanh(x)^4)^(1/2), x)
\[ \int \frac {\coth ^3(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int \frac {\coth \left (x \right )^{3}}{\sqrt {\tanh \left (x \right )^{4} c +\tanh \left (x \right )^{2} b +a}}d x \] Input:
int(coth(x)^3/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x)
Output:
int(coth(x)^3/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x)