Integrand size = 17, antiderivative size = 88 \[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {(2 a-b) \coth (x)}{3 a (a+b)^2 \sqrt {a+b \coth ^2(x)}} \]
arctanh(coth(x)*(a+b)^(1/2)/(a+b*coth(x)^2)^(1/2))/(a+b)^(5/2)-1/3*coth(x) /(a+b)/(a+b*coth(x)^2)^(3/2)-1/3*(2*a-b)*coth(x)/a/(a+b)^2/(a+b*coth(x)^2) ^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 7.07 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.44 \[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=-\frac {\cosh ^2(x) \coth (x) \left (\frac {4 (a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^2}-\frac {\left (-5 a-2 b \coth ^2(x)\right ) \left (3 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \left (a+b \coth ^2(x)\right )^2-a \left (-4 b \coth ^2(x)+a \left (-3-\coth ^2(x)\right )\right ) \text {csch}^2(x) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}\right ) \tanh ^4(x)}{3 a (a+b)^2 \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}\right )}{3 a^2 \sqrt {a+b \coth ^2(x)} \left (1+\frac {b \coth ^2(x)}{a}\right )} \]
-1/3*(Cosh[x]^2*Coth[x]*((4*(a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Hypergeome tric2F1[2, 2, 9/2, ((a + b)*Cosh[x]^2)/a])/(35*a^2) - ((-5*a - 2*b*Coth[x] ^2)*(3*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*(a + b*Coth[x]^2)^2 - a*(-4*b*C oth[x]^2 + a*(-3 - Coth[x]^2))*Csch[x]^2*Sqrt[-(((a + b)*Cosh[x]^2*(a + b* Coth[x]^2)*Sinh[x]^2)/a^2)])*Tanh[x]^4)/(3*a*(a + b)^2*Sqrt[-(((a + b)*Cos h[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)])))/(a^2*Sqrt[a + b*Coth[x]^2]*(1 + (b*Coth[x]^2)/a))
Time = 0.33 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3042, 25, 4153, 25, 373, 402, 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.
\(\displaystyle \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\tan \left (\frac {\pi }{2}+i x\right )^2}{\left (a-b \tan \left (\frac {\pi }{2}+i x\right )^2\right )^{5/2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\tan \left (i x+\frac {\pi }{2}\right )^2}{\left (a-b \tan \left (i x+\frac {\pi }{2}\right )^2\right )^{5/2}}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle -\int -\frac {\coth ^2(x)}{\left (1-\coth ^2(x)\right ) \left (b \coth ^2(x)+a\right )^{5/2}}d\coth (x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\coth ^2(x)}{\left (1-\coth ^2(x)\right ) \left (a+b \coth ^2(x)\right )^{5/2}}d\coth (x)\) |
\(\Big \downarrow \) 373 |
\(\displaystyle \frac {\int \frac {2 \coth ^2(x)+1}{\left (1-\coth ^2(x)\right ) \left (b \coth ^2(x)+a\right )^{3/2}}d\coth (x)}{3 (a+b)}-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {-\frac {\int -\frac {3 a}{\left (1-\coth ^2(x)\right ) \sqrt {b \coth ^2(x)+a}}d\coth (x)}{a (a+b)}-\frac {(2 a-b) \coth (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}}{3 (a+b)}-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {3 \int \frac {1}{\left (1-\coth ^2(x)\right ) \sqrt {b \coth ^2(x)+a}}d\coth (x)}{a+b}-\frac {(2 a-b) \coth (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}}{3 (a+b)}-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {\frac {3 \int \frac {1}{1-\frac {(a+b) \coth ^2(x)}{b \coth ^2(x)+a}}d\frac {\coth (x)}{\sqrt {b \coth ^2(x)+a}}}{a+b}-\frac {(2 a-b) \coth (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}}{3 (a+b)}-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {3 \text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{3/2}}-\frac {(2 a-b) \coth (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}}{3 (a+b)}-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}\) |
-1/3*Coth[x]/((a + b)*(a + b*Coth[x]^2)^(3/2)) + ((3*ArcTanh[(Sqrt[a + b]* Coth[x])/Sqrt[a + b*Coth[x]^2]])/(a + b)^(3/2) - ((2*a - b)*Coth[x])/(a*(a + b)*Sqrt[a + b*Coth[x]^2]))/(3*(a + b))
3.1.44.3.1 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*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
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]))
Leaf count of result is larger than twice the leaf count of optimal. \(453\) vs. \(2(74)=148\).
Time = 0.10 (sec) , antiderivative size = 454, normalized size of antiderivative = 5.16
method | result | size |
derivativedivides | \(-\frac {\coth \left (x \right )}{3 a \left (a +b \coth \left (x \right )^{2}\right )^{\frac {3}{2}}}-\frac {2 \coth \left (x \right )}{3 a^{2} \sqrt {a +b \coth \left (x \right )^{2}}}-\frac {1}{6 \left (a +b \right ) \left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{6 \left (a +b \right ) a \left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}-\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {b \coth \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {\ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}+\frac {1}{6 \left (a +b \right ) \left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{6 \left (a +b \right ) a \left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}+\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}+\frac {b \coth \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}-\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}\) | \(454\) |
default | \(-\frac {\coth \left (x \right )}{3 a \left (a +b \coth \left (x \right )^{2}\right )^{\frac {3}{2}}}-\frac {2 \coth \left (x \right )}{3 a^{2} \sqrt {a +b \coth \left (x \right )^{2}}}-\frac {1}{6 \left (a +b \right ) \left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{6 \left (a +b \right ) a \left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}-\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {b \coth \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {\ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}+\frac {1}{6 \left (a +b \right ) \left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{6 \left (a +b \right ) a \left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}+\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}+\frac {b \coth \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}-\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}\) | \(454\) |
-1/3*coth(x)/a/(a+b*coth(x)^2)^(3/2)-2/3/a^2*coth(x)/(a+b*coth(x)^2)^(1/2) -1/6/(a+b)/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(3/2)+1/6*b/(a+b)/a/(b*(c oth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(3/2)*coth(x)+1/3*b/(a+b)/a^2/(b*(coth(x) -1)^2+2*b*(coth(x)-1)+a+b)^(1/2)*coth(x)-1/2/(a+b)^2/(b*(coth(x)-1)^2+2*b* (coth(x)-1)+a+b)^(1/2)+1/2/(a+b)^2/a/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b) ^(1/2)*b*coth(x)+1/2/(a+b)^(5/2)*ln((2*a+2*b+2*b*(coth(x)-1)+2*(a+b)^(1/2) *(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2))/(coth(x)-1))+1/6/(a+b)/(b*(1 +coth(x))^2-2*b*(1+coth(x))+a+b)^(3/2)+1/6*b/(a+b)/a/(b*(1+coth(x))^2-2*b* (1+coth(x))+a+b)^(3/2)*coth(x)+1/3*b/(a+b)/a^2/(b*(1+coth(x))^2-2*b*(1+cot h(x))+a+b)^(1/2)*coth(x)+1/2/(a+b)^2/(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b) ^(1/2)+1/2/(a+b)^2/a/(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2)*b*coth(x) -1/2/(a+b)^(5/2)*ln((2*a+2*b-2*b*(1+coth(x))+2*(a+b)^(1/2)*(b*(1+coth(x))^ 2-2*b*(1+coth(x))+a+b)^(1/2))/(1+coth(x)))
Leaf count of result is larger than twice the leaf count of optimal. 3016 vs. \(2 (74) = 148\).
Time = 0.68 (sec) , antiderivative size = 6591, normalized size of antiderivative = 74.90 \[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\int \frac {\coth ^{2}{\left (x \right )}}{\left (a + b \coth ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\int { \frac {\coth \left (x\right )^{2}}{{\left (b \coth \left (x\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 952 vs. \(2 (74) = 148\).
Time = 0.55 (sec) , antiderivative size = 952, normalized size of antiderivative = 10.82 \[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
-1/3*((((3*a^7*b^2*sgn(e^(2*x) - 1) + 14*a^6*b^3*sgn(e^(2*x) - 1) + 25*a^5 *b^4*sgn(e^(2*x) - 1) + 20*a^4*b^5*sgn(e^(2*x) - 1) + 5*a^3*b^6*sgn(e^(2*x ) - 1) - 2*a^2*b^7*sgn(e^(2*x) - 1) - a*b^8*sgn(e^(2*x) - 1))*e^(2*x)/(a^8 *b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6*a^3*b^7 + a^2* b^8) - 3*(a^7*b^2*sgn(e^(2*x) - 1) + 2*a^6*b^3*sgn(e^(2*x) - 1) - a^5*b^4* sgn(e^(2*x) - 1) - 4*a^4*b^5*sgn(e^(2*x) - 1) - a^3*b^6*sgn(e^(2*x) - 1) + 2*a^2*b^7*sgn(e^(2*x) - 1) + a*b^8*sgn(e^(2*x) - 1))/(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6*a^3*b^7 + a^2*b^8))*e^(2*x) - 3*(a^7*b^2*sgn(e^(2*x) - 1) + 2*a^6*b^3*sgn(e^(2*x) - 1) - a^5*b^4*sgn(e^( 2*x) - 1) - 4*a^4*b^5*sgn(e^(2*x) - 1) - a^3*b^6*sgn(e^(2*x) - 1) + 2*a^2* b^7*sgn(e^(2*x) - 1) + a*b^8*sgn(e^(2*x) - 1))/(a^8*b^2 + 6*a^7*b^3 + 15*a ^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6*a^3*b^7 + a^2*b^8))*e^(2*x) + (3*a^7* b^2*sgn(e^(2*x) - 1) + 14*a^6*b^3*sgn(e^(2*x) - 1) + 25*a^5*b^4*sgn(e^(2*x ) - 1) + 20*a^4*b^5*sgn(e^(2*x) - 1) + 5*a^3*b^6*sgn(e^(2*x) - 1) - 2*a^2* b^7*sgn(e^(2*x) - 1) - a*b^8*sgn(e^(2*x) - 1))/(a^8*b^2 + 6*a^7*b^3 + 15*a ^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6*a^3*b^7 + a^2*b^8))/(a*e^(4*x) + b*e^ (4*x) - 2*a*e^(2*x) + 2*b*e^(2*x) + a + b)^(3/2) - 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) - a + b))/((a^2 + 2*a*b + b^2)*sqrt(a + b)*sgn(e^(2*x) - 1) ) + 1/2*log(abs((sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) - 2*a...
Timed out. \[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {coth}\left (x\right )}^2}{{\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{5/2}} \,d x \]