Integrand size = 12, antiderivative size = 157 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}+\sqrt {c} \cot (a+b x)}\right )}{\sqrt {2} b c^{3/2}}+\frac {2}{b c \sqrt {c \cot (a+b x)}} \] Output:
-1/2*arctan(1-2^(1/2)*(c*cot(b*x+a))^(1/2)/c^(1/2))*2^(1/2)/b/c^(3/2)+1/2* arctan(1+2^(1/2)*(c*cot(b*x+a))^(1/2)/c^(1/2))*2^(1/2)/b/c^(3/2)-1/2*arcta nh(2^(1/2)*(c*cot(b*x+a))^(1/2)/(c^(1/2)+c^(1/2)*cot(b*x+a)))*2^(1/2)/b/c^ (3/2)+2/b/c/(c*cot(b*x+a))^(1/2)
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\frac {2+\arctan \left (\sqrt [4]{-\cot ^2(a+b x)}\right ) \sqrt [4]{-\cot ^2(a+b x)}-\text {arctanh}\left (\sqrt [4]{-\cot ^2(a+b x)}\right ) \sqrt [4]{-\cot ^2(a+b x)}}{b c \sqrt {c \cot (a+b x)}} \] Input:
Integrate[(c*Cot[a + b*x])^(-3/2),x]
Output:
(2 + ArcTan[(-Cot[a + b*x]^2)^(1/4)]*(-Cot[a + b*x]^2)^(1/4) - ArcTanh[(-C ot[a + b*x]^2)^(1/4)]*(-Cot[a + b*x]^2)^(1/4))/(b*c*Sqrt[c*Cot[a + b*x]])
Time = 0.46 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.25, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {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.
| \(\displaystyle \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (-c \tan \left (a+b x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {2}{b c \sqrt {c \cot (a+b x)}}-\frac {\int \sqrt {c \cot (a+b x)}dx}{c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{b c \sqrt {c \cot (a+b x)}}-\frac {\int \sqrt {-c \tan \left (a+b x+\frac {\pi }{2}\right )}dx}{c^2}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {\int \frac {\sqrt {c \cot (a+b x)}}{\cot ^2(a+b x) c^2+c^2}d(c \cot (a+b x))}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 \int \frac {c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {c^2 \cot ^2(a+b x)+c}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}-\frac {1}{2} \int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}+\frac {1}{2} \int \frac {1}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}\right )-\frac {1}{2} \int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-c^2 \cot ^2(a+b x)-1}d\left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{-c^2 \cot ^2(a+b x)-1}d\left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}\right )-\frac {1}{2} \int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )-\frac {1}{2} \int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {c}-2 \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {c}+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {c}-2 \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {c}+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {c}-2 \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {\sqrt {c}+\sqrt {2} \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} c^{3/2} \cot (a+b x)+c^2 \cot ^2(a+b x)+c\right )}{2 \sqrt {2} \sqrt {c}}-\frac {\log \left (\sqrt {2} c^{3/2} \cot (a+b x)+c^2 \cot ^2(a+b x)+c\right )}{2 \sqrt {2} \sqrt {c}}\right )\right )}{b c}+\frac {2}{b c \sqrt {c \cot (a+b x)}}\) |
Input:
Int[(c*Cot[a + b*x])^(-3/2),x]
Output:
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 + 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*c)
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 = 0.14 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(-\frac {2 c \left (-\frac {1}{c^{2} \sqrt {c \cot \left (b x +a \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 c^{2} \left (c^{2}\right )^{\frac {1}{4}}}\right )}{b}\) | \(157\) |
| default | \(-\frac {2 c \left (-\frac {1}{c^{2} \sqrt {c \cot \left (b x +a \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 c^{2} \left (c^{2}\right )^{\frac {1}{4}}}\right )}{b}\) | \(157\) |
Input:
int(1/(c*cot(b*x+a))^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/b*c*(-1/c^2/(c*cot(b*x+a))^(1/2)-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)))
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (124) = 248\).
Time = 0.11 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.40 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\frac {\frac {2 \, \sqrt {2} {\left (c \cos \left (2 \, b x + 2 \, a\right ) + c\right )} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}}{\sqrt {c}} + 1\right )}{\sqrt {c}} + \frac {2 \, \sqrt {2} {\left (c \cos \left (2 \, b x + 2 \, a\right ) + c\right )} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}}{\sqrt {c}} - 1\right )}{\sqrt {c}} - \frac {\sqrt {2} {\left (c \cos \left (2 \, b x + 2 \, a\right ) + c\right )} \log \left (\frac {\frac {\sqrt {2} \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} \sin \left (2 \, b x + 2 \, a\right )}{\sqrt {c}} + \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right ) + 1}{\sin \left (2 \, b x + 2 \, a\right )}\right )}{\sqrt {c}} + \frac {\sqrt {2} {\left (c \cos \left (2 \, b x + 2 \, a\right ) + c\right )} \log \left (-\frac {\frac {\sqrt {2} \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} \sin \left (2 \, b x + 2 \, a\right )}{\sqrt {c}} - \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right ) - 1}{\sin \left (2 \, b x + 2 \, a\right )}\right )}{\sqrt {c}} + 8 \, \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} \sin \left (2 \, b x + 2 \, a\right )}{4 \, {\left (b c^{2} \cos \left (2 \, b x + 2 \, a\right ) + b c^{2}\right )}} \] Input:
integrate(1/(c*cot(b*x+a))^(3/2),x, algorithm="fricas")
Output:
1/4*(2*sqrt(2)*(c*cos(2*b*x + 2*a) + c)*arctan(sqrt(2)*sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))/sqrt(c) + 1)/sqrt(c) + 2*sqrt(2)*(c*cos(2*b*x + 2*a) + c)*arctan(sqrt(2)*sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a) )/sqrt(c) - 1)/sqrt(c) - sqrt(2)*(c*cos(2*b*x + 2*a) + c)*log((sqrt(2)*sqr t((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))*sin(2*b*x + 2*a)/sqrt(c) + co s(2*b*x + 2*a) + sin(2*b*x + 2*a) + 1)/sin(2*b*x + 2*a))/sqrt(c) + sqrt(2) *(c*cos(2*b*x + 2*a) + c)*log(-(sqrt(2)*sqrt((c*cos(2*b*x + 2*a) + c)/sin( 2*b*x + 2*a))*sin(2*b*x + 2*a)/sqrt(c) - cos(2*b*x + 2*a) - sin(2*b*x + 2* a) - 1)/sin(2*b*x + 2*a))/sqrt(c) + 8*sqrt((c*cos(2*b*x + 2*a) + c)/sin(2* b*x + 2*a))*sin(2*b*x + 2*a))/(b*c^2*cos(2*b*x + 2*a) + b*c^2)
\[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\int \frac {1}{\left (c \cot {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(c*cot(b*x+a))**(3/2),x)
Output:
Integral((c*cot(a + b*x))**(-3/2), x)
Time = 0.12 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\frac {c {\left (\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} + 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} - 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right )}{\sqrt {c}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right )}{\sqrt {c}}}{c^{2}} + \frac {8}{c^{2} \sqrt {\frac {c}{\tan \left (b x + a\right )}}}\right )}}{4 \, b} \] Input:
integrate(1/(c*cot(b*x+a))^(3/2),x, algorithm="maxima")
Output:
1/4*c*((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)*sqr t(c/tan(b*x + a)) + c + c/tan(b*x + a))/sqrt(c) + sqrt(2)*log(-sqrt(2)*sqr t(c)*sqrt(c/tan(b*x + a)) + c + c/tan(b*x + a))/sqrt(c))/c^2 + 8/(c^2*sqrt (c/tan(b*x + a))))/b
\[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\int { \frac {1}{\left (c \cot \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(c*cot(b*x+a))^(3/2),x, algorithm="giac")
Output:
integrate((c*cot(b*x + a))^(-3/2), x)
Time = 9.91 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.48 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\frac {2}{b\,c\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )}{b\,c^{3/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )}{b\,c^{3/2}} \] Input:
int(1/(c*cot(a + b*x))^(3/2),x)
Output:
2/(b*c*(c*cot(a + b*x))^(1/2)) + ((-1)^(1/4)*atan(((-1)^(1/4)*(c*cot(a + b *x))^(1/2))/c^(1/2)))/(b*c^(3/2)) - ((-1)^(1/4)*atanh(((-1)^(1/4)*(c*cot(a + b*x))^(1/2))/c^(1/2)))/(b*c^(3/2))
\[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {\cot \left (b x +a \right )}}{\cot \left (b x +a \right )^{2}}d x \right )}{c^{2}} \] Input:
int(1/(c*cot(b*x+a))^(3/2),x)
Output:
(sqrt(c)*int(sqrt(cot(a + b*x))/cot(a + b*x)**2,x))/c**2