Integrand size = 12, antiderivative size = 212 \[ \int (c \cot (a+b x))^{5/2} \, dx=-\frac {c^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}+\frac {c^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}-\frac {2 c (c \cot (a+b x))^{3/2}}{3 b}+\frac {c^{5/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}-\frac {c^{5/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b} \]
-2/3*c*(c*cot(b*x+a))^(3/2)/b-1/2*c^(5/2)*arctan(1-2^(1/2)*(c*cot(b*x+a))^ (1/2)/c^(1/2))/b*2^(1/2)+1/2*c^(5/2)*arctan(1+2^(1/2)*(c*cot(b*x+a))^(1/2) /c^(1/2))/b*2^(1/2)+1/4*c^(5/2)*ln(c^(1/2)+cot(b*x+a)*c^(1/2)-2^(1/2)*(c*c ot(b*x+a))^(1/2))/b*2^(1/2)-1/4*c^(5/2)*ln(c^(1/2)+cot(b*x+a)*c^(1/2)+2^(1 /2)*(c*cot(b*x+a))^(1/2))/b*2^(1/2)
Time = 0.35 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.48 \[ \int (c \cot (a+b x))^{5/2} \, dx=-\frac {c (c \cot (a+b x))^{3/2} \left (-3 \arctan \left (\sqrt [4]{-\cot ^2(a+b x)}\right ) \sqrt [4]{-\cot (a+b x)}+3 \text {arctanh}\left (\sqrt [4]{-\cot ^2(a+b x)}\right ) \sqrt [4]{-\cot (a+b x)}+2 \cot ^{\frac {7}{4}}(a+b x)\right )}{3 b \cot ^{\frac {7}{4}}(a+b x)} \]
-1/3*(c*(c*Cot[a + b*x])^(3/2)*(-3*ArcTan[(-Cot[a + b*x]^2)^(1/4)]*(-Cot[a + b*x])^(1/4) + 3*ArcTanh[(-Cot[a + b*x]^2)^(1/4)]*(-Cot[a + b*x])^(1/4) + 2*Cot[a + b*x]^(7/4)))/(b*Cot[a + b*x]^(7/4))
Time = 0.45 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.93, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {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.
\(\displaystyle \int (c \cot (a+b x))^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (-c \tan \left (a+b x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle c^2 \left (-\int \sqrt {c \cot (a+b x)}dx\right )-\frac {2 c (c \cot (a+b x))^{3/2}}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle c^2 \left (-\int \sqrt {-c \tan \left (a+b x+\frac {\pi }{2}\right )}dx\right )-\frac {2 c (c \cot (a+b x))^{3/2}}{3 b}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {c^3 \int \frac {\sqrt {c \cot (a+b x)}}{\cot ^2(a+b x) c^2+c^2}d(c \cot (a+b x))}{b}-\frac {2 c (c \cot (a+b x))^{3/2}}{3 b}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2 c^3 \int \frac {c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}}{b}-\frac {2 c (c \cot (a+b x))^{3/2}}{3 b}\) |
\(\Big \downarrow \) 826 |
\(\displaystyle \frac {2 c^3 \left (\frac {1}{2} \int \frac {c^2 \cot ^2(a+b x)+c}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}-\frac {1}{2} \int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}\right )}{b}-\frac {2 c (c \cot (a+b x))^{3/2}}{3 b}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {2 c^3 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}+\frac {1}{2} \int \frac {1}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}\right )-\frac {1}{2} \int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}\right )}{b}-\frac {2 c (c \cot (a+b x))^{3/2}}{3 b}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {2 c^3 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-c^2 \cot ^2(a+b x)-1}d\left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{-c^2 \cot ^2(a+b x)-1}d\left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}\right )-\frac {1}{2} \int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}\right )}{b}-\frac {2 c (c \cot (a+b x))^{3/2}}{3 b}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {2 c^3 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )-\frac {1}{2} \int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}\right )}{b}-\frac {2 c (c \cot (a+b x))^{3/2}}{3 b}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {2 c^3 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {c}-2 \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {c}+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )\right )}{b}-\frac {2 c (c \cot (a+b x))^{3/2}}{3 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 c^3 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {c}-2 \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {c}+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )\right )}{b}-\frac {2 c (c \cot (a+b x))^{3/2}}{3 b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 c^3 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {c}-2 \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {\sqrt {c}+\sqrt {2} \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )\right )}{b}-\frac {2 c (c \cot (a+b x))^{3/2}}{3 b}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {2 c^3 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} c^{3/2} \cot (a+b x)+c^2 \cot ^2(a+b x)+c\right )}{2 \sqrt {2} \sqrt {c}}-\frac {\log \left (\sqrt {2} c^{3/2} \cot (a+b x)+c^2 \cot ^2(a+b x)+c\right )}{2 \sqrt {2} \sqrt {c}}\right )\right )}{b}-\frac {2 c (c \cot (a+b x))^{3/2}}{3 b}\) |
(-2*c*(c*Cot[a + b*x])^(3/2))/(3*b) + (2*c^3*((-(ArcTan[1 - Sqrt[2]*Sqrt[c ]*Cot[a + b*x]]/(Sqrt[2]*Sqrt[c])) + ArcTan[1 + Sqrt[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
3.1.10.3.1 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]
Time = 0.03 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(-\frac {2 c \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {3}{2}}}{3}-\frac {c^{2} \sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (c^{2}\right )^{\frac {1}{4}}}\right )}{b}\) | \(154\) |
default | \(-\frac {2 c \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {3}{2}}}{3}-\frac {c^{2} \sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (c^{2}\right )^{\frac {1}{4}}}\right )}{b}\) | \(154\) |
-2/b*c*(1/3*(c*cot(b*x+a))^(3/2)-1/8*c^2/(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)))
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.61 \[ \int (c \cot (a+b x))^{5/2} \, dx=\frac {3 \, \left (-\frac {c^{10}}{b^{4}}\right )^{\frac {1}{4}} b \log \left (c^{7} \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} + \left (-\frac {c^{10}}{b^{4}}\right )^{\frac {3}{4}} b^{3}\right ) \sin \left (2 \, b x + 2 \, a\right ) - 3 i \, \left (-\frac {c^{10}}{b^{4}}\right )^{\frac {1}{4}} b \log \left (c^{7} \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} + i \, \left (-\frac {c^{10}}{b^{4}}\right )^{\frac {3}{4}} b^{3}\right ) \sin \left (2 \, b x + 2 \, a\right ) + 3 i \, \left (-\frac {c^{10}}{b^{4}}\right )^{\frac {1}{4}} b \log \left (c^{7} \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} - i \, \left (-\frac {c^{10}}{b^{4}}\right )^{\frac {3}{4}} b^{3}\right ) \sin \left (2 \, b x + 2 \, a\right ) - 3 \, \left (-\frac {c^{10}}{b^{4}}\right )^{\frac {1}{4}} b \log \left (c^{7} \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} - \left (-\frac {c^{10}}{b^{4}}\right )^{\frac {3}{4}} b^{3}\right ) \sin \left (2 \, b x + 2 \, a\right ) - 4 \, {\left (c^{2} \cos \left (2 \, b x + 2 \, a\right ) + c^{2}\right )} \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}}{6 \, b \sin \left (2 \, b x + 2 \, a\right )} \]
1/6*(3*(-c^10/b^4)^(1/4)*b*log(c^7*sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a)) + (-c^10/b^4)^(3/4)*b^3)*sin(2*b*x + 2*a) - 3*I*(-c^10/b^4)^(1/4) *b*log(c^7*sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a)) + I*(-c^10/b^4) ^(3/4)*b^3)*sin(2*b*x + 2*a) + 3*I*(-c^10/b^4)^(1/4)*b*log(c^7*sqrt((c*cos (2*b*x + 2*a) + c)/sin(2*b*x + 2*a)) - I*(-c^10/b^4)^(3/4)*b^3)*sin(2*b*x + 2*a) - 3*(-c^10/b^4)^(1/4)*b*log(c^7*sqrt((c*cos(2*b*x + 2*a) + c)/sin(2 *b*x + 2*a)) - (-c^10/b^4)^(3/4)*b^3)*sin(2*b*x + 2*a) - 4*(c^2*cos(2*b*x + 2*a) + c^2)*sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a)))/(b*sin(2*b* x + 2*a))
\[ \int (c \cot (a+b x))^{5/2} \, dx=\int \left (c \cot {\left (a + b x \right )}\right )^{\frac {5}{2}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.87 \[ \int (c \cot (a+b x))^{5/2} \, dx=\frac {{\left (3 \, c^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} + 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} - 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right )}{\sqrt {c}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right )}{\sqrt {c}}\right )} - 8 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {3}{2}}\right )} c}{12 \, b} \]
1/12*(3*c^2*(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)) - 8*(c/tan(b *x + a))^(3/2))*c/b
\[ \int (c \cot (a+b x))^{5/2} \, dx=\int { \left (c \cot \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
Time = 12.51 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.35 \[ \int (c \cot (a+b x))^{5/2} \, dx=\frac {{\left (-1\right )}^{1/4}\,c^{5/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )}{b}-\frac {2\,c\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{3/2}}{3\,b}-\frac {{\left (-1\right )}^{1/4}\,c^{5/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )}{b} \]