Integrand size = 12, antiderivative size = 67 \[ \int \left (-1-\coth ^2(x)\right )^{3/2} \, dx=-\frac {5}{2} \arctan \left (\frac {\coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} \coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )+\frac {1}{2} \coth (x) \sqrt {-1-\coth ^2(x)} \]
-5/2*arctan(coth(x)/(-1-coth(x)^2)^(1/2))+2*arctan(coth(x)*2^(1/2)/(-1-cot h(x)^2)^(1/2))*2^(1/2)+1/2*coth(x)*(-1-coth(x)^2)^(1/2)
Time = 0.26 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.76 \[ \int \left (-1-\coth ^2(x)\right )^{3/2} \, dx=-\frac {1}{8} \left (-1-\coth ^2(x)\right )^{3/2} \text {sech}^2(2 x) \left (16 \text {arctanh}\left (\frac {\cosh (x)}{\sqrt {\cosh (2 x)}}\right ) \sqrt {\cosh (2 x)} \sinh ^3(x)+4 \left (\arctan \left (\frac {\cosh (x)}{\sqrt {-\cosh (2 x)}}\right ) \sqrt {-\cosh (2 x)}-4 \sqrt {2} \sqrt {\cosh (2 x)} \log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)}\right )\right ) \sinh ^3(x)+\sinh (4 x)\right ) \]
-1/8*((-1 - Coth[x]^2)^(3/2)*Sech[2*x]^2*(16*ArcTanh[Cosh[x]/Sqrt[Cosh[2*x ]]]*Sqrt[Cosh[2*x]]*Sinh[x]^3 + 4*(ArcTan[Cosh[x]/Sqrt[-Cosh[2*x]]]*Sqrt[- Cosh[2*x]] - 4*Sqrt[2]*Sqrt[Cosh[2*x]]*Log[Sqrt[2]*Cosh[x] + Sqrt[Cosh[2*x ]]])*Sinh[x]^3 + Sinh[4*x]))
Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 4144, 318, 25, 398, 224, 216, 291, 216}
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.
| \(\displaystyle \int \left (-\coth ^2(x)-1\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (-1+\tan \left (\frac {\pi }{2}+i x\right )^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4144 |
| \(\displaystyle \int \frac {\left (-\coth ^2(x)-1\right )^{3/2}}{1-\coth ^2(x)}d\coth (x)\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {1}{2} \coth (x) \sqrt {-\coth ^2(x)-1}-\frac {1}{2} \int -\frac {5 \coth ^2(x)+3}{\sqrt {-\coth ^2(x)-1} \left (1-\coth ^2(x)\right )}d\coth (x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \int \frac {5 \coth ^2(x)+3}{\sqrt {-\coth ^2(x)-1} \left (1-\coth ^2(x)\right )}d\coth (x)+\frac {1}{2} \sqrt {-\coth ^2(x)-1} \coth (x)\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {1}{2} \left (8 \int \frac {1}{\sqrt {-\coth ^2(x)-1} \left (1-\coth ^2(x)\right )}d\coth (x)-5 \int \frac {1}{\sqrt {-\coth ^2(x)-1}}d\coth (x)\right )+\frac {1}{2} \sqrt {-\coth ^2(x)-1} \coth (x)\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {1}{2} \left (8 \int \frac {1}{\sqrt {-\coth ^2(x)-1} \left (1-\coth ^2(x)\right )}d\coth (x)-5 \int \frac {1}{\frac {\coth ^2(x)}{-\coth ^2(x)-1}+1}d\frac {\coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )+\frac {1}{2} \sqrt {-\coth ^2(x)-1} \coth (x)\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{2} \left (8 \int \frac {1}{\sqrt {-\coth ^2(x)-1} \left (1-\coth ^2(x)\right )}d\coth (x)-5 \arctan \left (\frac {\coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )\right )+\frac {1}{2} \sqrt {-\coth ^2(x)-1} \coth (x)\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {1}{2} \left (8 \int \frac {1}{\frac {2 \coth ^2(x)}{-\coth ^2(x)-1}+1}d\frac {\coth (x)}{\sqrt {-\coth ^2(x)-1}}-5 \arctan \left (\frac {\coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )\right )+\frac {1}{2} \sqrt {-\coth ^2(x)-1} \coth (x)\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{2} \left (4 \sqrt {2} \arctan \left (\frac {\sqrt {2} \coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )-5 \arctan \left (\frac {\coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )\right )+\frac {1}{2} \coth (x) \sqrt {-\coth ^2(x)-1}\) |
(-5*ArcTan[Coth[x]/Sqrt[-1 - Coth[x]^2]] + 4*Sqrt[2]*ArcTan[(Sqrt[2]*Coth[ x])/Sqrt[-1 - Coth[x]^2]])/2 + (Coth[x]*Sqrt[-1 - Coth[x]^2])/2
3.1.31.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((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[(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, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Leaf count of result is larger than twice the leaf count of optimal. \(210\) vs. \(2(53)=106\).
Time = 0.11 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.15
| method | result | size |
| derivativedivides | \(-\frac {\left (-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\coth \left (x \right ) \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{4}+\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}-\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )+\frac {\left (-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\coth \left (x \right ) \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{4}-\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}+\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )\) | \(211\) |
| default | \(-\frac {\left (-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\coth \left (x \right ) \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{4}+\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}-\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )+\frac {\left (-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\coth \left (x \right ) \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{4}-\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}+\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )\) | \(211\) |
-1/6*(-(coth(x)-1)^2-2*coth(x))^(3/2)+1/4*coth(x)*(-(coth(x)-1)^2-2*coth(x ))^(1/2)-5/4*arctan(coth(x)/(-(coth(x)-1)^2-2*coth(x))^(1/2))+(-(coth(x)-1 )^2-2*coth(x))^(1/2)-2^(1/2)*arctan(1/4*(-2-2*coth(x))*2^(1/2)/(-(coth(x)- 1)^2-2*coth(x))^(1/2))+1/6*(-(1+coth(x))^2+2*coth(x))^(3/2)+1/4*coth(x)*(- (1+coth(x))^2+2*coth(x))^(1/2)-5/4*arctan(coth(x)/(-(1+coth(x))^2+2*coth(x ))^(1/2))-(-(1+coth(x))^2+2*coth(x))^(1/2)+2^(1/2)*arctan(1/4*(-2+2*coth(x ))*2^(1/2)/(-(1+coth(x))^2+2*coth(x))^(1/2))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 361, normalized size of antiderivative = 5.39 \[ \int \left (-1-\coth ^2(x)\right )^{3/2} \, dx=\frac {2 \, {\left (\sqrt {-2} e^{\left (4 \, x\right )} - 2 \, \sqrt {-2} e^{\left (2 \, x\right )} + \sqrt {-2}\right )} \log \left (2 \, {\left (\sqrt {-2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + 2 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )}\right ) - 2 \, {\left (\sqrt {-2} e^{\left (4 \, x\right )} - 2 \, \sqrt {-2} e^{\left (2 \, x\right )} + \sqrt {-2}\right )} \log \left (-2 \, {\left (\sqrt {-2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} - 2 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-2 \, x\right )}\right ) - 5 \, {\left (i \, e^{\left (4 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} + i\right )} \log \left (-4 \, {\left (i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}\right ) - 5 \, {\left (-i \, e^{\left (4 \, x\right )} + 2 i \, e^{\left (2 \, x\right )} - i\right )} \log \left (-4 \, {\left (-i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}\right ) - 2 \, {\left (\sqrt {-2} e^{\left (4 \, x\right )} - 2 \, \sqrt {-2} e^{\left (2 \, x\right )} + \sqrt {-2}\right )} \log \left (4 \, {\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 2\right )} + \sqrt {-2} e^{\left (4 \, x\right )} + \sqrt {-2} e^{\left (2 \, x\right )} + 2 \, \sqrt {-2}\right )} e^{\left (-4 \, x\right )}\right ) + 2 \, {\left (\sqrt {-2} e^{\left (4 \, x\right )} - 2 \, \sqrt {-2} e^{\left (2 \, x\right )} + \sqrt {-2}\right )} \log \left (4 \, {\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 2\right )} - \sqrt {-2} e^{\left (4 \, x\right )} - \sqrt {-2} e^{\left (2 \, x\right )} - 2 \, \sqrt {-2}\right )} e^{\left (-4 \, x\right )}\right ) + 2 \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 1\right )}}{4 \, {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )}} \]
1/4*(2*(sqrt(-2)*e^(4*x) - 2*sqrt(-2)*e^(2*x) + sqrt(-2))*log(2*(sqrt(-2)* sqrt(-2*e^(4*x) - 2) + 2*e^(2*x) - 2)*e^(-2*x)) - 2*(sqrt(-2)*e^(4*x) - 2* sqrt(-2)*e^(2*x) + sqrt(-2))*log(-2*(sqrt(-2)*sqrt(-2*e^(4*x) - 2) - 2*e^( 2*x) + 2)*e^(-2*x)) - 5*(I*e^(4*x) - 2*I*e^(2*x) + I)*log(-4*(I*sqrt(-2*e^ (4*x) - 2) + e^(2*x) + 1)*e^(-2*x)) - 5*(-I*e^(4*x) + 2*I*e^(2*x) - I)*log (-4*(-I*sqrt(-2*e^(4*x) - 2) + e^(2*x) + 1)*e^(-2*x)) - 2*(sqrt(-2)*e^(4*x ) - 2*sqrt(-2)*e^(2*x) + sqrt(-2))*log(4*(sqrt(-2*e^(4*x) - 2)*(e^(2*x) + 2) + sqrt(-2)*e^(4*x) + sqrt(-2)*e^(2*x) + 2*sqrt(-2))*e^(-4*x)) + 2*(sqrt (-2)*e^(4*x) - 2*sqrt(-2)*e^(2*x) + sqrt(-2))*log(4*(sqrt(-2*e^(4*x) - 2)* (e^(2*x) + 2) - sqrt(-2)*e^(4*x) - sqrt(-2)*e^(2*x) - 2*sqrt(-2))*e^(-4*x) ) + 2*sqrt(-2*e^(4*x) - 2)*(e^(2*x) + 1))/(e^(4*x) - 2*e^(2*x) + 1)
\[ \int \left (-1-\coth ^2(x)\right )^{3/2} \, dx=\int \left (- \coth ^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \]
\[ \int \left (-1-\coth ^2(x)\right )^{3/2} \, dx=\int { {\left (-\coth \left (x\right )^{2} - 1\right )}^{\frac {3}{2}} \,d x } \]
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 285, normalized size of antiderivative = 4.25 \[ \int \left (-1-\coth ^2(x)\right )^{3/2} \, dx=-\frac {1}{4} \, \sqrt {2} {\left (-5 i \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} + 2 \right |}}{2 \, {\left (\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right )}}\right ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) - 4 i \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + 4 i \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + 4 i \, \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + \frac {4 \, {\left (3 i \, {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{3} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + i \, {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + {\left (-i \, \sqrt {e^{\left (4 \, x\right )} + 1} + i \, e^{\left (2 \, x\right )}\right )} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + i \, \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )\right )}}{{\left ({\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} + 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} - 1\right )}^{2}}\right )} \]
-1/4*sqrt(2)*(-5*I*sqrt(2)*log(1/2*abs(-2*sqrt(2) + 2*sqrt(e^(4*x) + 1) - 2*e^(2*x) + 2)/(sqrt(2) + sqrt(e^(4*x) + 1) - e^(2*x) + 1))*sgn(-e^(2*x) + 1) - 4*I*log(sqrt(e^(4*x) + 1) - e^(2*x) + 1)*sgn(-e^(2*x) + 1) + 4*I*log (sqrt(e^(4*x) + 1) - e^(2*x))*sgn(-e^(2*x) + 1) + 4*I*log(-sqrt(e^(4*x) + 1) + e^(2*x) + 1)*sgn(-e^(2*x) + 1) + 4*(3*I*(sqrt(e^(4*x) + 1) - e^(2*x)) ^3*sgn(-e^(2*x) + 1) + I*(sqrt(e^(4*x) + 1) - e^(2*x))^2*sgn(-e^(2*x) + 1) + (-I*sqrt(e^(4*x) + 1) + I*e^(2*x))*sgn(-e^(2*x) + 1) + I*sgn(-e^(2*x) + 1))/((sqrt(e^(4*x) + 1) - e^(2*x))^2 + 2*sqrt(e^(4*x) + 1) - 2*e^(2*x) - 1)^2)
Timed out. \[ \int \left (-1-\coth ^2(x)\right )^{3/2} \, dx=\int {\left (-{\mathrm {coth}\left (x\right )}^2-1\right )}^{3/2} \,d x \]