Integrand size = 23, antiderivative size = 76 \[ \int \cot (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=-\frac {35}{16} \sqrt {\cot ^2(x)}+\frac {35}{48} \cos ^2(x) \sqrt {\cot ^2(x)}+\frac {7}{24} \cos ^4(x) \sqrt {\cot ^2(x)}+\frac {1}{6} \cos ^6(x) \sqrt {\cot ^2(x)}-\frac {35}{16} x \sqrt {\cot ^2(x)} \tan (x) \]
-35/16*(cot(x)^2)^(1/2)+35/48*cos(x)^2*(cot(x)^2)^(1/2)+7/24*cos(x)^4*(cot (x)^2)^(1/2)+1/6*cos(x)^6*(cot(x)^2)^(1/2)-35/16*x*(cot(x)^2)^(1/2)*tan(x)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.03 \[ \int \cot (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=\frac {1}{288} \cot ^8(x) \sqrt {\cot ^2(x)} \left (-27+36 \cos (2 x)-9 \cos (4 x)+40 \operatorname {Hypergeometric2F1}\left (2,\frac {9}{2},\frac {11}{2},-\cot ^2(x)\right )+32 \operatorname {Hypergeometric2F1}\left (3,\frac {9}{2},\frac {11}{2},-\cot ^2(x)\right )-32 \operatorname {Hypergeometric2F1}\left (4,\frac {9}{2},\frac {11}{2},-\cot ^2(x)\right )\right ) \]
(Cot[x]^8*Sqrt[Cot[x]^2]*(-27 + 36*Cos[2*x] - 9*Cos[4*x] + 40*Hypergeometr ic2F1[2, 9/2, 11/2, -Cot[x]^2] + 32*Hypergeometric2F1[3, 9/2, 11/2, -Cot[x ]^2] - 32*Hypergeometric2F1[4, 9/2, 11/2, -Cot[x]^2]))/288
Time = 0.40 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.25, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 3654, 3042, 4860, 1016, 798, 51, 51, 51, 60, 73, 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.
| \(\displaystyle \int \left (1-\sin ^2(x)\right )^3 \cot (x) \sqrt {\csc ^2(x)-1} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (1-\sin (x)^2\right )^3 \cot (x) \sqrt {\csc (x)^2-1}dx\) |
| \(\Big \downarrow \) 3654 |
| \(\displaystyle \int \cos ^6(x) \cot (x) \sqrt {\csc ^2(x)-1}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (x)^6 \cot (x) \sqrt {\csc (x)^2-1}dx\) |
| \(\Big \downarrow \) 4860 |
| \(\displaystyle \int \left (1-\sin ^2(x)\right )^3 \csc (x) \sqrt {\csc ^2(x)-1}d\sin (x)\) |
| \(\Big \downarrow \) 1016 |
| \(\displaystyle \int \sin ^5(x) \left (\csc ^2(x)-1\right )^{7/2}d\sin (x)\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle -\frac {1}{2} \int \csc ^4(x) \left (\csc ^2(x)-1\right )^{7/2}d\csc ^2(x)\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \csc ^3(x) \left (\csc ^2(x)-1\right )^{7/2}-\frac {7}{6} \int \csc ^3(x) \left (\csc ^2(x)-1\right )^{5/2}d\csc ^2(x)\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \csc ^3(x) \left (\csc ^2(x)-1\right )^{7/2}-\frac {7}{6} \left (\frac {5}{4} \int \csc ^2(x) \left (\csc ^2(x)-1\right )^{3/2}d\csc ^2(x)-\frac {1}{2} \csc ^2(x) \left (\csc ^2(x)-1\right )^{5/2}\right )\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \csc ^3(x) \left (\csc ^2(x)-1\right )^{7/2}-\frac {7}{6} \left (\frac {5}{4} \left (\frac {3}{2} \int \csc (x) \sqrt {\csc ^2(x)-1}d\csc ^2(x)-\csc (x) \left (\csc ^2(x)-1\right )^{3/2}\right )-\frac {1}{2} \csc ^2(x) \left (\csc ^2(x)-1\right )^{5/2}\right )\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \csc ^3(x) \left (\csc ^2(x)-1\right )^{7/2}-\frac {7}{6} \left (\frac {5}{4} \left (\frac {3}{2} \left (2 \sqrt {\csc ^2(x)-1}-\int \frac {\csc (x)}{\sqrt {\csc ^2(x)-1}}d\csc ^2(x)\right )-\csc (x) \left (\csc ^2(x)-1\right )^{3/2}\right )-\frac {1}{2} \csc ^2(x) \left (\csc ^2(x)-1\right )^{5/2}\right )\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \csc ^3(x) \left (\csc ^2(x)-1\right )^{7/2}-\frac {7}{6} \left (\frac {5}{4} \left (\frac {3}{2} \left (2 \sqrt {\csc ^2(x)-1}-2 \int \frac {1}{\csc ^4(x)+1}d\sqrt {\csc ^2(x)-1}\right )-\csc (x) \left (\csc ^2(x)-1\right )^{3/2}\right )-\frac {1}{2} \csc ^2(x) \left (\csc ^2(x)-1\right )^{5/2}\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \csc ^3(x) \left (\csc ^2(x)-1\right )^{7/2}-\frac {7}{6} \left (\frac {5}{4} \left (\frac {3}{2} \left (2 \sqrt {\csc ^2(x)-1}-2 \arctan \left (\sqrt {\csc ^2(x)-1}\right )\right )-\csc (x) \left (\csc ^2(x)-1\right )^{3/2}\right )-\frac {1}{2} \csc ^2(x) \left (\csc ^2(x)-1\right )^{5/2}\right )\right )\) |
((Csc[x]^3*(-1 + Csc[x]^2)^(7/2))/3 - (7*(-1/2*(Csc[x]^2*(-1 + Csc[x]^2)^( 5/2)) + (5*(-(Csc[x]*(-1 + Csc[x]^2)^(3/2)) + (3*(-2*ArcTan[Sqrt[-1 + Csc[ x]^2]] + 2*Sqrt[-1 + Csc[x]^2]))/2))/4))/6)/2
3.9.66.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^( p_.), x_Symbol] :> Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ [{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || !I ntegerQ[p])
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[ a^p Int[ActivateTrig[u*cos[e + f*x]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]
Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFacto rs[Sin[c*(a + b*x)], x]}, Simp[1/(b*c) Subst[Int[SubstFor[1/x, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a + b *x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cot] || EqQ[F, cot])
Time = 1.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.49
| method | result | size |
| default | \(\frac {\sqrt {\cot \left (x \right )^{2}}\, \left (8 \cos \left (x \right )^{6}+14 \cos \left (x \right )^{4}+35 \cos \left (x \right )^{2}-105 x \tan \left (x \right )-105\right ) \sqrt {4}}{96}\) | \(37\) |
| risch | \(\frac {35 i \sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) x}{16 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {\sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, {\mathrm e}^{8 i x}}{384 \,{\mathrm e}^{2 i x}+384}-\frac {47 \sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, {\mathrm e}^{2 i x}}{128 \left ({\mathrm e}^{2 i x}+1\right )}-\frac {47 \sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left (1-{\mathrm e}^{-2 i x}\right )}{128 \left ({\mathrm e}^{2 i x}+1\right )}-\frac {5 \sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, {\mathrm e}^{-2 i x}}{128 \left ({\mathrm e}^{2 i x}+1\right )}-\frac {2 \sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}{{\mathrm e}^{2 i x}+1}+\frac {5 \sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \cos \left (6 x \right )}{128 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {13 i \sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \sin \left (6 x \right )}{384 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {35 \sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \cos \left (4 x \right )}{96 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {7 i \sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \sin \left (4 x \right )}{24 \left ({\mathrm e}^{2 i x}+1\right )}\) | \(383\) |
Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.45 \[ \int \cot (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=-\frac {8 \, \cos \left (x\right )^{7} + 14 \, \cos \left (x\right )^{5} + 35 \, \cos \left (x\right )^{3} - 105 \, x \sin \left (x\right ) - 105 \, \cos \left (x\right )}{48 \, \sin \left (x\right )} \]
Timed out. \[ \int \cot (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (56) = 112\).
Time = 0.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.79 \[ \int \cot (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=-\frac {3}{2} \, \sqrt {\frac {1}{\sin \left (x\right )^{2}} - 1} \sin \left (x\right )^{2} - \sqrt {\frac {1}{\sin \left (x\right )^{2}} - 1} + \frac {3 \, {\left (\frac {1}{\sin \left (x\right )^{2}} - 1\right )}^{\frac {5}{2}} + 8 \, {\left (\frac {1}{\sin \left (x\right )^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {\frac {1}{\sin \left (x\right )^{2}} - 1}}{48 \, {\left ({\left (\frac {1}{\sin \left (x\right )^{2}} - 1\right )}^{3} + 3 \, {\left (\frac {1}{\sin \left (x\right )^{2}} - 1\right )}^{2} + \frac {3}{\sin \left (x\right )^{2}} - 2\right )}} - \frac {3 \, {\left ({\left (\frac {1}{\sin \left (x\right )^{2}} - 1\right )}^{\frac {3}{2}} - \sqrt {\frac {1}{\sin \left (x\right )^{2}} - 1}\right )}}{8 \, {\left ({\left (\frac {1}{\sin \left (x\right )^{2}} - 1\right )}^{2} + \frac {2}{\sin \left (x\right )^{2}} - 1\right )}} + \frac {35}{16} \, \arctan \left (\sqrt {\frac {1}{\sin \left (x\right )^{2}} - 1}\right ) \]
-3/2*sqrt(1/sin(x)^2 - 1)*sin(x)^2 - sqrt(1/sin(x)^2 - 1) + 1/48*(3*(1/sin (x)^2 - 1)^(5/2) + 8*(1/sin(x)^2 - 1)^(3/2) - 3*sqrt(1/sin(x)^2 - 1))/((1/ sin(x)^2 - 1)^3 + 3*(1/sin(x)^2 - 1)^2 + 3/sin(x)^2 - 2) - 3/8*((1/sin(x)^ 2 - 1)^(3/2) - sqrt(1/sin(x)^2 - 1))/((1/sin(x)^2 - 1)^2 + 2/sin(x)^2 - 1) + 35/16*arctan(sqrt(1/sin(x)^2 - 1))
Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.28 \[ \int \cot (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=-\frac {1}{48} \, {\left ({\left (2 \, {\left (4 \, \sin \left (x\right )^{2} - 19\right )} \sin \left (x\right )^{2} + 87\right )} \sqrt {-\sin \left (x\right )^{2} + 1} \sin \left (x\right ) - 105 \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor - x\right )} \left (-1\right )^{\left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor } + \frac {24 \, {\left (\sqrt {-\sin \left (x\right )^{2} + 1} - 1\right )}}{\sin \left (x\right )} - \frac {24 \, \sin \left (x\right )}{\sqrt {-\sin \left (x\right )^{2} + 1} - 1}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \]
-1/48*((2*(4*sin(x)^2 - 19)*sin(x)^2 + 87)*sqrt(-sin(x)^2 + 1)*sin(x) - 10 5*(pi*floor(x/pi + 1/2) - x)*(-1)^floor(x/pi + 1/2) + 24*(sqrt(-sin(x)^2 + 1) - 1)/sin(x) - 24*sin(x)/(sqrt(-sin(x)^2 + 1) - 1))*sgn(sin(x))
Timed out. \[ \int \cot (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=\int -\mathrm {cot}\left (x\right )\,\sqrt {\frac {1}{{\sin \left (x\right )}^2}-1}\,{\left ({\sin \left (x\right )}^2-1\right )}^3 \,d x \]