Integrand size = 10, antiderivative size = 200 \[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=\frac {2}{3} a \sqrt {a \cot ^3(x)}+\frac {a \arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {a \arctan \left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {a \sqrt {a \cot ^3(x)} \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {a \sqrt {a \cot ^3(x)} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)} \]
2/3*a*(a*cot(x)^3)^(1/2)-2/7*a*cot(x)^2*(a*cot(x)^3)^(1/2)-1/2*a*arctan(-1 +2^(1/2)*cot(x)^(1/2))*(a*cot(x)^3)^(1/2)/cot(x)^(3/2)*2^(1/2)-1/2*a*arcta n(1+2^(1/2)*cot(x)^(1/2))*(a*cot(x)^3)^(1/2)/cot(x)^(3/2)*2^(1/2)-1/4*a*ln (1+cot(x)-2^(1/2)*cot(x)^(1/2))*(a*cot(x)^3)^(1/2)/cot(x)^(3/2)*2^(1/2)+1/ 4*a*ln(1+cot(x)+2^(1/2)*cot(x)^(1/2))*(a*cot(x)^3)^(1/2)/cot(x)^(3/2)*2^(1 /2)
Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.40 \[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=\frac {a \sqrt {a \cot ^3(x)} \left (-21 \arctan \left (\sqrt [4]{-\cot ^2(x)}\right ) \sqrt [4]{-\cot (x)}+21 \text {arctanh}\left (\sqrt [4]{-\cot ^2(x)}\right ) \sqrt [4]{-\cot (x)}+2 \cot ^{\frac {7}{4}}(x) \left (7-3 \cot ^2(x)\right )\right )}{21 \cot ^{\frac {7}{4}}(x)} \]
(a*Sqrt[a*Cot[x]^3]*(-21*ArcTan[(-Cot[x]^2)^(1/4)]*(-Cot[x])^(1/4) + 21*Ar cTanh[(-Cot[x]^2)^(1/4)]*(-Cot[x])^(1/4) + 2*Cot[x]^(7/4)*(7 - 3*Cot[x]^2) ))/(21*Cot[x]^(7/4))
Time = 0.51 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.74, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.700, Rules used = {3042, 4141, 3042, 3954, 3042, 3954, 3042, 3957, 266, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \left (a \cot ^3(x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (-a \tan \left (x+\frac {\pi }{2}\right )^3\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4141 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \int \cot ^{\frac {9}{2}}(x)dx}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \int \left (-\tan \left (x+\frac {\pi }{2}\right )\right )^{9/2}dx}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (-\int \cot ^{\frac {5}{2}}(x)dx-\frac {2}{7} \cot ^{\frac {7}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (-\int \left (-\tan \left (x+\frac {\pi }{2}\right )\right )^{5/2}dx-\frac {2}{7} \cot ^{\frac {7}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (\int \sqrt {\cot (x)}dx-\frac {2}{7} \cot ^{\frac {7}{2}}(x)+\frac {2}{3} \cot ^{\frac {3}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (\int \sqrt {-\tan \left (x+\frac {\pi }{2}\right )}dx-\frac {2}{7} \cot ^{\frac {7}{2}}(x)+\frac {2}{3} \cot ^{\frac {3}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (-\int \frac {\sqrt {\cot (x)}}{\cot ^2(x)+1}d\cot (x)-\frac {2}{7} \cot ^{\frac {7}{2}}(x)+\frac {2}{3} \cot ^{\frac {3}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (-2 \int \frac {\cot (x)}{\cot ^2(x)+1}d\sqrt {\cot (x)}-\frac {2}{7} \cot ^{\frac {7}{2}}(x)+\frac {2}{3} \cot ^{\frac {3}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (-2 \left (\frac {1}{2} \int \frac {\cot (x)+1}{\cot ^2(x)+1}d\sqrt {\cot (x)}-\frac {1}{2} \int \frac {1-\cot (x)}{\cot ^2(x)+1}d\sqrt {\cot (x)}\right )-\frac {2}{7} \cot ^{\frac {7}{2}}(x)+\frac {2}{3} \cot ^{\frac {3}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (-2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1}d\sqrt {\cot (x)}+\frac {1}{2} \int \frac {1}{\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1}d\sqrt {\cot (x)}\right )-\frac {1}{2} \int \frac {1-\cot (x)}{\cot ^2(x)+1}d\sqrt {\cot (x)}\right )-\frac {2}{7} \cot ^{\frac {7}{2}}(x)+\frac {2}{3} \cot ^{\frac {3}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (-2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\cot (x)-1}d\left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (x)-1}d\left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\cot (x)}{\cot ^2(x)+1}d\sqrt {\cot (x)}\right )-\frac {2}{7} \cot ^{\frac {7}{2}}(x)+\frac {2}{3} \cot ^{\frac {3}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (-2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\cot (x)}{\cot ^2(x)+1}d\sqrt {\cot (x)}\right )-\frac {2}{7} \cot ^{\frac {7}{2}}(x)+\frac {2}{3} \cot ^{\frac {3}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (-2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (x)}}{\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1}d\sqrt {\cot (x)}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1}d\sqrt {\cot (x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2}}\right )\right )-\frac {2}{7} \cot ^{\frac {7}{2}}(x)+\frac {2}{3} \cot ^{\frac {3}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (-2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (x)}}{\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1}d\sqrt {\cot (x)}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1}d\sqrt {\cot (x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2}}\right )\right )-\frac {2}{7} \cot ^{\frac {7}{2}}(x)+\frac {2}{3} \cot ^{\frac {3}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (-2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (x)}}{\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1}d\sqrt {\cot (x)}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (x)}+1}{\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1}d\sqrt {\cot (x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2}}\right )\right )-\frac {2}{7} \cot ^{\frac {7}{2}}(x)+\frac {2}{3} \cot ^{\frac {3}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {a \sqrt {a \cot ^3(x)} \left (-2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2}}\right )\right )-\frac {2}{7} \cot ^{\frac {7}{2}}(x)+\frac {2}{3} \cot ^{\frac {3}{2}}(x)\right )}{\cot ^{\frac {3}{2}}(x)}\) |
(a*Sqrt[a*Cot[x]^3]*((2*Cot[x]^(3/2))/3 - (2*Cot[x]^(7/2))/7 - 2*((-(ArcTa n[1 - Sqrt[2]*Sqrt[Cot[x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[x]]]/Sq rt[2])/2 + (Log[1 - Sqrt[2]*Sqrt[Cot[x]] + Cot[x]]/(2*Sqrt[2]) - Log[1 + S qrt[2]*Sqrt[Cot[x]] + Cot[x]]/(2*Sqrt[2]))/2)))/Cot[x]^(3/2)
3.1.29.3.1 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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 0.09 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(-\frac {\left (a \cot \left (x \right )^{3}\right )^{\frac {3}{2}} \left (24 \left (a \cot \left (x \right )\right )^{\frac {7}{2}} \left (a^{2}\right )^{\frac {1}{4}}+21 a^{4} \sqrt {2}\, \ln \left (-\frac {\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}-a \cot \left (x \right )-\sqrt {a^{2}}}{a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}\right )+42 a^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+42 a^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )-56 a^{2} \left (a \cot \left (x \right )\right )^{\frac {3}{2}} \left (a^{2}\right )^{\frac {1}{4}}\right )}{84 \cot \left (x \right )^{3} \left (a \cot \left (x \right )\right )^{\frac {3}{2}} a^{2} \left (a^{2}\right )^{\frac {1}{4}}}\) | \(189\) |
| default | \(-\frac {\left (a \cot \left (x \right )^{3}\right )^{\frac {3}{2}} \left (24 \left (a \cot \left (x \right )\right )^{\frac {7}{2}} \left (a^{2}\right )^{\frac {1}{4}}+21 a^{4} \sqrt {2}\, \ln \left (-\frac {\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}-a \cot \left (x \right )-\sqrt {a^{2}}}{a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}\right )+42 a^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+42 a^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )-56 a^{2} \left (a \cot \left (x \right )\right )^{\frac {3}{2}} \left (a^{2}\right )^{\frac {1}{4}}\right )}{84 \cot \left (x \right )^{3} \left (a \cot \left (x \right )\right )^{\frac {3}{2}} a^{2} \left (a^{2}\right )^{\frac {1}{4}}}\) | \(189\) |
-1/84*(a*cot(x)^3)^(3/2)*(24*(a*cot(x))^(7/2)*(a^2)^(1/4)+21*a^4*2^(1/2)*l n(-((a^2)^(1/4)*(a*cot(x))^(1/2)*2^(1/2)-a*cot(x)-(a^2)^(1/2))/(a*cot(x)+( a^2)^(1/4)*(a*cot(x))^(1/2)*2^(1/2)+(a^2)^(1/2)))+42*a^4*2^(1/2)*arctan((2 ^(1/2)*(a*cot(x))^(1/2)+(a^2)^(1/4))/(a^2)^(1/4))+42*a^4*2^(1/2)*arctan((2 ^(1/2)*(a*cot(x))^(1/2)-(a^2)^(1/4))/(a^2)^(1/4))-56*a^2*(a*cot(x))^(3/2)* (a^2)^(1/4))/cot(x)^3/(a*cot(x))^(3/2)/a^2/(a^2)^(1/4)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.00 \[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=-\frac {21 \, \left (-a^{6}\right )^{\frac {1}{4}} {\left (\cos \left (2 \, x\right ) - 1\right )} \log \left (\frac {a^{4} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) + \left (-a^{6}\right )^{\frac {3}{4}} {\left (\cos \left (2 \, x\right ) + 1\right )}}{\cos \left (2 \, x\right ) + 1}\right ) - 21 \, \left (-a^{6}\right )^{\frac {1}{4}} {\left (\cos \left (2 \, x\right ) - 1\right )} \log \left (\frac {a^{4} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) - \left (-a^{6}\right )^{\frac {3}{4}} {\left (\cos \left (2 \, x\right ) + 1\right )}}{\cos \left (2 \, x\right ) + 1}\right ) + 21 \, \left (-a^{6}\right )^{\frac {1}{4}} {\left (-i \, \cos \left (2 \, x\right ) + i\right )} \log \left (\frac {a^{4} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) + \left (-a^{6}\right )^{\frac {3}{4}} {\left (i \, \cos \left (2 \, x\right ) + i\right )}}{\cos \left (2 \, x\right ) + 1}\right ) + 21 \, \left (-a^{6}\right )^{\frac {1}{4}} {\left (i \, \cos \left (2 \, x\right ) - i\right )} \log \left (\frac {a^{4} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) + \left (-a^{6}\right )^{\frac {3}{4}} {\left (-i \, \cos \left (2 \, x\right ) - i\right )}}{\cos \left (2 \, x\right ) + 1}\right ) - 8 \, {\left (5 \, a \cos \left (2 \, x\right ) - 2 \, a\right )} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}}}{42 \, {\left (\cos \left (2 \, x\right ) - 1\right )}} \]
-1/42*(21*(-a^6)^(1/4)*(cos(2*x) - 1)*log((a^4*sqrt(-(a*cos(2*x)^2 + 2*a*c os(2*x) + a)/((cos(2*x) - 1)*sin(2*x)))*sin(2*x) + (-a^6)^(3/4)*(cos(2*x) + 1))/(cos(2*x) + 1)) - 21*(-a^6)^(1/4)*(cos(2*x) - 1)*log((a^4*sqrt(-(a*c os(2*x)^2 + 2*a*cos(2*x) + a)/((cos(2*x) - 1)*sin(2*x)))*sin(2*x) - (-a^6) ^(3/4)*(cos(2*x) + 1))/(cos(2*x) + 1)) + 21*(-a^6)^(1/4)*(-I*cos(2*x) + I) *log((a^4*sqrt(-(a*cos(2*x)^2 + 2*a*cos(2*x) + a)/((cos(2*x) - 1)*sin(2*x) ))*sin(2*x) + (-a^6)^(3/4)*(I*cos(2*x) + I))/(cos(2*x) + 1)) + 21*(-a^6)^( 1/4)*(I*cos(2*x) - I)*log((a^4*sqrt(-(a*cos(2*x)^2 + 2*a*cos(2*x) + a)/((c os(2*x) - 1)*sin(2*x)))*sin(2*x) + (-a^6)^(3/4)*(-I*cos(2*x) - I))/(cos(2* x) + 1)) - 8*(5*a*cos(2*x) - 2*a)*sqrt(-(a*cos(2*x)^2 + 2*a*cos(2*x) + a)/ ((cos(2*x) - 1)*sin(2*x))))/(cos(2*x) - 1)
\[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=\int \left (a \cot ^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
Time = 0.42 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.56 \[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=\frac {1}{4} \, {\left (2 \, \sqrt {2} \sqrt {a} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) + 2 \, \sqrt {2} \sqrt {a} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) + \sqrt {2} \sqrt {a} \log \left (\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) - \sqrt {2} \sqrt {a} \log \left (-\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right )\right )} a + \frac {2 \, a^{\frac {3}{2}}}{3 \, \tan \left (x\right )^{\frac {3}{2}}} - \frac {2 \, a^{\frac {3}{2}}}{7 \, \tan \left (x\right )^{\frac {7}{2}}} \]
1/4*(2*sqrt(2)*sqrt(a)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(x)))) + 2* sqrt(2)*sqrt(a)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(x)))) + sqrt(2)* sqrt(a)*log(sqrt(2)*sqrt(tan(x)) + tan(x) + 1) - sqrt(2)*sqrt(a)*log(-sqrt (2)*sqrt(tan(x)) + tan(x) + 1))*a + 2/3*a^(3/2)/tan(x)^(3/2) - 2/7*a^(3/2) /tan(x)^(7/2)
\[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=\int { \left (a \cot \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=\int {\left (a\,{\mathrm {cot}\left (x\right )}^3\right )}^{3/2} \,d x \]