Integrand size = 23, antiderivative size = 135 \[ \int \frac {\coth ^5(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\frac {(b-2 c) \text {arctanh}\left (\frac {b+2 c \coth ^2(x)}{2 \sqrt {c} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{4 c^{3/2}}+\frac {\text {arctanh}\left (\frac {2 a+b+(b+2 c) \coth ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{2 \sqrt {a+b+c}}-\frac {\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}{2 c} \]
1/4*(b-2*c)*arctanh(1/2*(b+2*c*coth(x)^2)/c^(1/2)/(a+b*coth(x)^2+c*coth(x) ^4)^(1/2))/c^(3/2)+1/2*arctanh(1/2*(2*a+b+(b+2*c)*coth(x)^2)/(a+b+c)^(1/2) /(a+b*coth(x)^2+c*coth(x)^4)^(1/2))/(a+b+c)^(1/2)-1/2*(a+b*coth(x)^2+c*cot h(x)^4)^(1/2)/c
Time = 2.30 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.47 \[ \int \frac {\coth ^5(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\frac {\sqrt {a+b \coth ^2(x)+c \coth ^4(x)} \tanh ^2(x) \left ((b-2 c) (a+b+c) \text {arctanh}\left (\frac {2 c+b \tanh ^2(x)}{2 \sqrt {c} \sqrt {c+b \tanh ^2(x)+a \tanh ^4(x)}}\right )+2 c^{3/2} \sqrt {a+b+c} \text {arctanh}\left (\frac {b+2 c+(2 a+b) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {c+b \tanh ^2(x)+a \tanh ^4(x)}}\right )-2 \sqrt {c} (a+b+c) \coth ^2(x) \sqrt {c+b \tanh ^2(x)+a \tanh ^4(x)}\right )}{4 c^{3/2} (a+b+c) \sqrt {c+b \tanh ^2(x)+a \tanh ^4(x)}} \]
(Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4]*Tanh[x]^2*((b - 2*c)*(a + b + c)*ArcT anh[(2*c + b*Tanh[x]^2)/(2*Sqrt[c]*Sqrt[c + b*Tanh[x]^2 + a*Tanh[x]^4])] + 2*c^(3/2)*Sqrt[a + b + c]*ArcTanh[(b + 2*c + (2*a + b)*Tanh[x]^2)/(2*Sqrt [a + b + c]*Sqrt[c + b*Tanh[x]^2 + a*Tanh[x]^4])] - 2*Sqrt[c]*(a + b + c)* Coth[x]^2*Sqrt[c + b*Tanh[x]^2 + a*Tanh[x]^4]))/(4*c^(3/2)*(a + b + c)*Sqr t[c + b*Tanh[x]^2 + a*Tanh[x]^4])
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.68, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 26, 4184, 1578, 1267, 27, 1269, 1092, 219, 1154, 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.
\(\displaystyle \int \frac {\coth ^5(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \cot (i x)^5}{\sqrt {a-b \cot (i x)^2+c \cot (i x)^4}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\cot (i x)^5}{\sqrt {c \cot (i x)^4-b \cot (i x)^2+a}}dx\) |
\(\Big \downarrow \) 4184 |
\(\displaystyle -\int -\frac {i \coth ^5(x)}{\left (1-\coth ^2(x)\right ) \sqrt {c \coth ^4(x)+b \coth ^2(x)+a}}d(-i \coth (x))\) |
\(\Big \downarrow \) 1578 |
\(\displaystyle -\frac {1}{2} \int -\frac {\coth ^2(x)}{\left (1-\coth ^2(x)\right ) \sqrt {-c \coth ^2(x)+i b \coth (x)+a}}d\left (-\coth ^2(x)\right )\) |
\(\Big \downarrow \) 1267 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {b-(b-2 c) \coth ^2(x)}{2 \left (1-\coth ^2(x)\right ) \sqrt {-c \coth ^2(x)+i b \coth (x)+a}}d\left (-\coth ^2(x)\right )}{c}-\frac {\sqrt {a+i b \coth (x)-c \coth ^2(x)}}{c}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {b-(b-2 c) \coth ^2(x)}{\left (1-\coth ^2(x)\right ) \sqrt {-c \coth ^2(x)+i b \coth (x)+a}}d\left (-\coth ^2(x)\right )}{2 c}-\frac {\sqrt {a+i b \coth (x)-c \coth ^2(x)}}{c}\right )\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {1}{2} \left (-\frac {(b-2 c) \int \frac {1}{\sqrt {-c \coth ^2(x)+i b \coth (x)+a}}d\left (-\coth ^2(x)\right )+2 c \int \frac {1}{\left (1-\coth ^2(x)\right ) \sqrt {-c \coth ^2(x)+i b \coth (x)+a}}d\left (-\coth ^2(x)\right )}{2 c}-\frac {\sqrt {a+i b \coth (x)-c \coth ^2(x)}}{c}\right )\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 (b-2 c) \int \frac {1}{\coth ^2(x)+4 c}d\left (-\frac {b+2 i c \coth (x)}{\sqrt {-c \coth ^2(x)+i b \coth (x)+a}}\right )+2 c \int \frac {1}{\left (1-\coth ^2(x)\right ) \sqrt {-c \coth ^2(x)+i b \coth (x)+a}}d\left (-\coth ^2(x)\right )}{2 c}-\frac {\sqrt {a+i b \coth (x)-c \coth ^2(x)}}{c}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 c \int \frac {1}{\left (1-\coth ^2(x)\right ) \sqrt {-c \coth ^2(x)+i b \coth (x)+a}}d\left (-\coth ^2(x)\right )-\frac {i (b-2 c) \arctan \left (\frac {\coth (x)}{2 \sqrt {c}}\right )}{\sqrt {c}}}{2 c}-\frac {\sqrt {a+i b \coth (x)-c \coth ^2(x)}}{c}\right )\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {1}{2} \left (-\frac {-4 c \int \frac {1}{\coth ^2(x)+4 (a+b+c)}d\frac {2 a+b+i (b+2 c) \coth (x)}{\sqrt {-c \coth ^2(x)+i b \coth (x)+a}}-\frac {i (b-2 c) \arctan \left (\frac {\coth (x)}{2 \sqrt {c}}\right )}{\sqrt {c}}}{2 c}-\frac {\sqrt {a+i b \coth (x)-c \coth ^2(x)}}{c}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (-\frac {\frac {2 i c \arctan \left (\frac {\coth (x)}{2 \sqrt {a+b+c}}\right )}{\sqrt {a+b+c}}-\frac {i (b-2 c) \arctan \left (\frac {\coth (x)}{2 \sqrt {c}}\right )}{\sqrt {c}}}{2 c}-\frac {\sqrt {a+i b \coth (x)-c \coth ^2(x)}}{c}\right )\) |
(-1/2*(((-I)*(b - 2*c)*ArcTan[Coth[x]/(2*Sqrt[c])])/Sqrt[c] + ((2*I)*c*Arc Tan[Coth[x]/(2*Sqrt[a + b + c])])/Sqrt[a + b + c])/c - Sqrt[a + I*b*Coth[x ] - c*Coth[x]^2]/c)/2
3.3.5.3.1 Defintions of rubi rules used
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[(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_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b *x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m + n + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d + e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g , m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 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[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*( f_.))^(n_.) + (c_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(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*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[ n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Time = 0.86 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.10
method | result | size |
derivativedivides | \(-\frac {\ln \left (\frac {\frac {b}{2}+c \coth \left (x \right )^{2}}{\sqrt {c}}+\sqrt {a +b \coth \left (x \right )^{2}+c \coth \left (x \right )^{4}}\right )}{2 \sqrt {c}}-\frac {\sqrt {a +b \coth \left (x \right )^{2}+c \coth \left (x \right )^{4}}}{2 c}+\frac {b \ln \left (\frac {\frac {b}{2}+c \coth \left (x \right )^{2}}{\sqrt {c}}+\sqrt {a +b \coth \left (x \right )^{2}+c \coth \left (x \right )^{4}}\right )}{4 c^{\frac {3}{2}}}+\frac {\operatorname {arctanh}\left (\frac {b \coth \left (x \right )^{2}+2 c \coth \left (x \right )^{2}+2 a +b}{2 \sqrt {a +b +c}\, \sqrt {a +b \coth \left (x \right )^{2}+c \coth \left (x \right )^{4}}}\right )}{2 \sqrt {a +b +c}}\) | \(149\) |
default | \(-\frac {\ln \left (\frac {\frac {b}{2}+c \coth \left (x \right )^{2}}{\sqrt {c}}+\sqrt {a +b \coth \left (x \right )^{2}+c \coth \left (x \right )^{4}}\right )}{2 \sqrt {c}}-\frac {\sqrt {a +b \coth \left (x \right )^{2}+c \coth \left (x \right )^{4}}}{2 c}+\frac {b \ln \left (\frac {\frac {b}{2}+c \coth \left (x \right )^{2}}{\sqrt {c}}+\sqrt {a +b \coth \left (x \right )^{2}+c \coth \left (x \right )^{4}}\right )}{4 c^{\frac {3}{2}}}+\frac {\operatorname {arctanh}\left (\frac {b \coth \left (x \right )^{2}+2 c \coth \left (x \right )^{2}+2 a +b}{2 \sqrt {a +b +c}\, \sqrt {a +b \coth \left (x \right )^{2}+c \coth \left (x \right )^{4}}}\right )}{2 \sqrt {a +b +c}}\) | \(149\) |
-1/2*ln((1/2*b+c*coth(x)^2)/c^(1/2)+(a+b*coth(x)^2+c*coth(x)^4)^(1/2))/c^( 1/2)-1/2*(a+b*coth(x)^2+c*coth(x)^4)^(1/2)/c+1/4*b/c^(3/2)*ln((1/2*b+c*cot h(x)^2)/c^(1/2)+(a+b*coth(x)^2+c*coth(x)^4)^(1/2))+1/2/(a+b+c)^(1/2)*arcta nh(1/2*(b*coth(x)^2+2*c*coth(x)^2+2*a+b)/(a+b+c)^(1/2)/(a+b*coth(x)^2+c*co th(x)^4)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 2086 vs. \(2 (111) = 222\).
Time = 1.11 (sec) , antiderivative size = 8951, normalized size of antiderivative = 66.30 \[ \int \frac {\coth ^5(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\text {Too large to display} \]
\[ \int \frac {\coth ^5(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\int \frac {\coth ^{5}{\left (x \right )}}{\sqrt {a + b \coth ^{2}{\left (x \right )} + c \coth ^{4}{\left (x \right )}}}\, dx \]
\[ \int \frac {\coth ^5(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\int { \frac {\coth \left (x\right )^{5}}{\sqrt {c \coth \left (x\right )^{4} + b \coth \left (x\right )^{2} + a}} \,d x } \]
\[ \int \frac {\coth ^5(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\int { \frac {\coth \left (x\right )^{5}}{\sqrt {c \coth \left (x\right )^{4} + b \coth \left (x\right )^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {\coth ^5(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\int \frac {{\mathrm {coth}\left (x\right )}^5}{\sqrt {c\,{\mathrm {coth}\left (x\right )}^4+b\,{\mathrm {coth}\left (x\right )}^2+a}} \,d x \]