Integrand size = 12, antiderivative size = 234 \[ \int \frac {1}{(c \cot (a+b x))^{7/2}} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{7/2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{7/2}}+\frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {2}{b c^3 \sqrt {c \cot (a+b x)}}-\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{7/2}}+\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{7/2}} \]
2/5/b/c/(c*cot(b*x+a))^(5/2)+1/2*arctan(1-2^(1/2)*(c*cot(b*x+a))^(1/2)/c^( 1/2))/b/c^(7/2)*2^(1/2)-1/2*arctan(1+2^(1/2)*(c*cot(b*x+a))^(1/2)/c^(1/2)) /b/c^(7/2)*2^(1/2)-1/4*ln(c^(1/2)+cot(b*x+a)*c^(1/2)-2^(1/2)*(c*cot(b*x+a) )^(1/2))/b/c^(7/2)*2^(1/2)+1/4*ln(c^(1/2)+cot(b*x+a)*c^(1/2)+2^(1/2)*(c*co t(b*x+a))^(1/2))/b/c^(7/2)*2^(1/2)-2/b/c^3/(c*cot(b*x+a))^(1/2)
Time = 0.36 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.41 \[ \int \frac {1}{(c \cot (a+b x))^{7/2}} \, dx=\frac {-5 \arctan \left (\sqrt [4]{-\cot ^2(a+b x)}\right ) \sqrt [4]{-\cot ^2(a+b x)}+5 \text {arctanh}\left (\sqrt [4]{-\cot ^2(a+b x)}\right ) \sqrt [4]{-\cot ^2(a+b x)}+2 \left (-5+\tan ^2(a+b x)\right )}{5 b c^3 \sqrt {c \cot (a+b x)}} \]
(-5*ArcTan[(-Cot[a + b*x]^2)^(1/4)]*(-Cot[a + b*x]^2)^(1/4) + 5*ArcTanh[(- Cot[a + b*x]^2)^(1/4)]*(-Cot[a + b*x]^2)^(1/4) + 2*(-5 + Tan[a + b*x]^2))/ (5*b*c^3*Sqrt[c*Cot[a + b*x]])
Time = 0.56 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.96, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {3042, 3955, 3042, 3955, 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.
| \(\displaystyle \int \frac {1}{(c \cot (a+b x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (-c \tan \left (a+b x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\int \frac {1}{(c \cot (a+b x))^{3/2}}dx}{c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\int \frac {1}{\left (-c \tan \left (a+b x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\frac {2}{b c \sqrt {c \cot (a+b x)}}-\frac {\int \sqrt {c \cot (a+b x)}dx}{c^2}}{c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\frac {2}{b c \sqrt {c \cot (a+b x)}}-\frac {\int \sqrt {-c \tan \left (a+b x+\frac {\pi }{2}\right )}dx}{c^2}}{c^2}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\frac {\int \frac {\sqrt {c \cot (a+b x)}}{\cot ^2(a+b x) c^2+c^2}d(c \cot (a+b x))}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}}{c^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\frac {2 \int \frac {c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}}{c^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \int \frac {c^2 \cot ^2(a+b x)+c}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}-\frac {1}{2} \int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}}{c^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}+\frac {1}{2} \int \frac {1}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}\right )-\frac {1}{2} \int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}}{c^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-c^2 \cot ^2(a+b x)-1}d\left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{-c^2 \cot ^2(a+b x)-1}d\left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}\right )-\frac {1}{2} \int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}}{c^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )-\frac {1}{2} \int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}}{c^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {c}-2 \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {c}+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {c}-2 \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {c}+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {c}-2 \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {\sqrt {c}+\sqrt {2} \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}}{c^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2}{5 b c (c \cot (a+b x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} c^{3/2} \cot (a+b x)+c^2 \cot ^2(a+b x)+c\right )}{2 \sqrt {2} \sqrt {c}}-\frac {\log \left (\sqrt {2} c^{3/2} \cot (a+b x)+c^2 \cot ^2(a+b x)+c\right )}{2 \sqrt {2} \sqrt {c}}\right )\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}}{c^2}\) |
2/(5*b*c*(c*Cot[a + b*x])^(5/2)) - (2/(b*c*Sqrt[c*Cot[a + b*x]]) + (2*((-( ArcTan[1 - Sqrt[2]*Sqrt[c]*Cot[a + b*x]]/(Sqrt[2]*Sqrt[c])) + ArcTan[1 + S qrt[2]*Sqrt[c]*Cot[a + b*x]]/(Sqrt[2]*Sqrt[c]))/2 + (Log[c - Sqrt[2]*c^(3/ 2)*Cot[a + b*x] + c^2*Cot[a + b*x]^2]/(2*Sqrt[2]*Sqrt[c]) - Log[c + Sqrt[2 ]*c^(3/2)*Cot[a + b*x] + c^2*Cot[a + b*x]^2]/(2*Sqrt[2]*Sqrt[c]))/2))/(b*c ))/c^2
3.1.16.3.1 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*Tan[c + d*x] )^(n + 1)/(b*d*(n + 1)), x] - Simp[1/b^2 Int[(b*Tan[c + d*x])^(n + 2), x] , x] /; FreeQ[{b, c, d}, x] && LtQ[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]
Time = 0.03 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.73
| method | result | size |
| derivativedivides | \(-\frac {2 c \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 c^{4} \left (c^{2}\right )^{\frac {1}{4}}}-\frac {1}{5 c^{2} \left (c \cot \left (b x +a \right )\right )^{\frac {5}{2}}}+\frac {1}{c^{4} \sqrt {c \cot \left (b x +a \right )}}\right )}{b}\) | \(171\) |
| default | \(-\frac {2 c \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 c^{4} \left (c^{2}\right )^{\frac {1}{4}}}-\frac {1}{5 c^{2} \left (c \cot \left (b x +a \right )\right )^{\frac {5}{2}}}+\frac {1}{c^{4} \sqrt {c \cot \left (b x +a \right )}}\right )}{b}\) | \(171\) |
-2/b*c*(1/8/c^4/(c^2)^(1/4)*2^(1/2)*(ln((c*cot(b*x+a)-(c^2)^(1/4)*(c*cot(b *x+a))^(1/2)*2^(1/2)+(c^2)^(1/2))/(c*cot(b*x+a)+(c^2)^(1/4)*(c*cot(b*x+a)) ^(1/2)*2^(1/2)+(c^2)^(1/2)))+2*arctan(2^(1/2)/(c^2)^(1/4)*(c*cot(b*x+a))^( 1/2)+1)-2*arctan(-2^(1/2)/(c^2)^(1/4)*(c*cot(b*x+a))^(1/2)+1))-1/5/c^2/(c* cot(b*x+a))^(5/2)+1/c^4/(c*cot(b*x+a))^(1/2))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.05 \[ \int \frac {1}{(c \cot (a+b x))^{7/2}} \, dx=-\frac {8 \, \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} {\left (3 \, \cos \left (2 \, b x + 2 \, a\right ) + 2\right )} \sin \left (2 \, b x + 2 \, a\right ) + 5 \, {\left (b c^{4} \cos \left (2 \, b x + 2 \, a\right )^{2} + 2 \, b c^{4} \cos \left (2 \, b x + 2 \, a\right ) + b c^{4}\right )} \left (-\frac {1}{b^{4} c^{14}}\right )^{\frac {1}{4}} \log \left (b^{3} c^{11} \left (-\frac {1}{b^{4} c^{14}}\right )^{\frac {3}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) + 5 \, {\left (-i \, b c^{4} \cos \left (2 \, b x + 2 \, a\right )^{2} - 2 i \, b c^{4} \cos \left (2 \, b x + 2 \, a\right ) - i \, b c^{4}\right )} \left (-\frac {1}{b^{4} c^{14}}\right )^{\frac {1}{4}} \log \left (i \, b^{3} c^{11} \left (-\frac {1}{b^{4} c^{14}}\right )^{\frac {3}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) + 5 \, {\left (i \, b c^{4} \cos \left (2 \, b x + 2 \, a\right )^{2} + 2 i \, b c^{4} \cos \left (2 \, b x + 2 \, a\right ) + i \, b c^{4}\right )} \left (-\frac {1}{b^{4} c^{14}}\right )^{\frac {1}{4}} \log \left (-i \, b^{3} c^{11} \left (-\frac {1}{b^{4} c^{14}}\right )^{\frac {3}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) - 5 \, {\left (b c^{4} \cos \left (2 \, b x + 2 \, a\right )^{2} + 2 \, b c^{4} \cos \left (2 \, b x + 2 \, a\right ) + b c^{4}\right )} \left (-\frac {1}{b^{4} c^{14}}\right )^{\frac {1}{4}} \log \left (-b^{3} c^{11} \left (-\frac {1}{b^{4} c^{14}}\right )^{\frac {3}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right )}{10 \, {\left (b c^{4} \cos \left (2 \, b x + 2 \, a\right )^{2} + 2 \, b c^{4} \cos \left (2 \, b x + 2 \, a\right ) + b c^{4}\right )}} \]
-1/10*(8*sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))*(3*cos(2*b*x + 2* a) + 2)*sin(2*b*x + 2*a) + 5*(b*c^4*cos(2*b*x + 2*a)^2 + 2*b*c^4*cos(2*b*x + 2*a) + b*c^4)*(-1/(b^4*c^14))^(1/4)*log(b^3*c^11*(-1/(b^4*c^14))^(3/4) + sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))) + 5*(-I*b*c^4*cos(2*b*x + 2*a)^2 - 2*I*b*c^4*cos(2*b*x + 2*a) - I*b*c^4)*(-1/(b^4*c^14))^(1/4)*lo g(I*b^3*c^11*(-1/(b^4*c^14))^(3/4) + sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b *x + 2*a))) + 5*(I*b*c^4*cos(2*b*x + 2*a)^2 + 2*I*b*c^4*cos(2*b*x + 2*a) + I*b*c^4)*(-1/(b^4*c^14))^(1/4)*log(-I*b^3*c^11*(-1/(b^4*c^14))^(3/4) + sq rt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))) - 5*(b*c^4*cos(2*b*x + 2*a) ^2 + 2*b*c^4*cos(2*b*x + 2*a) + b*c^4)*(-1/(b^4*c^14))^(1/4)*log(-b^3*c^11 *(-1/(b^4*c^14))^(3/4) + sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a)))) /(b*c^4*cos(2*b*x + 2*a)^2 + 2*b*c^4*cos(2*b*x + 2*a) + b*c^4)
\[ \int \frac {1}{(c \cot (a+b x))^{7/2}} \, dx=\int \frac {1}{\left (c \cot {\left (a + b x \right )}\right )^{\frac {7}{2}}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(c \cot (a+b x))^{7/2}} \, dx=-\frac {c {\left (\frac {5 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} + 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} - 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right )}{\sqrt {c}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right )}{\sqrt {c}}\right )}}{c^{4}} - \frac {8 \, {\left (c^{2} - \frac {5 \, c^{2}}{\tan \left (b x + a\right )^{2}}\right )}}{c^{4} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {5}{2}}}\right )}}{20 \, b} \]
-1/20*c*(5*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(c) + 2*sqrt(c/tan(b *x + a)))/sqrt(c))/sqrt(c) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(c ) - 2*sqrt(c/tan(b*x + a)))/sqrt(c))/sqrt(c) - sqrt(2)*log(sqrt(2)*sqrt(c) *sqrt(c/tan(b*x + a)) + c + c/tan(b*x + a))/sqrt(c) + sqrt(2)*log(-sqrt(2) *sqrt(c)*sqrt(c/tan(b*x + a)) + c + c/tan(b*x + a))/sqrt(c))/c^4 - 8*(c^2 - 5*c^2/tan(b*x + a)^2)/(c^4*(c/tan(b*x + a))^(5/2)))/b
\[ \int \frac {1}{(c \cot (a+b x))^{7/2}} \, dx=\int { \frac {1}{\left (c \cot \left (b x + a\right )\right )^{\frac {7}{2}}} \,d x } \]
Time = 12.47 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(c \cot (a+b x))^{7/2}} \, dx=\frac {\frac {2}{5\,c}-\frac {2\,{\mathrm {cot}\left (a+b\,x\right )}^2}{c}}{b\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )}{b\,c^{7/2}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )}{b\,c^{7/2}} \]