Integrand size = 12, antiderivative size = 154 \[ \int (c \cot (a+b x))^{3/2} \, dx=-\frac {c^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}+\frac {c^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}+\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}+\sqrt {c} \cot (a+b x)}\right )}{\sqrt {2} b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b} \] Output:
-1/2*c^(3/2)*arctan(1-2^(1/2)*(c*cot(b*x+a))^(1/2)/c^(1/2))*2^(1/2)/b+1/2* c^(3/2)*arctan(1+2^(1/2)*(c*cot(b*x+a))^(1/2)/c^(1/2))*2^(1/2)/b+1/2*c^(3/ 2)*arctanh(2^(1/2)*(c*cot(b*x+a))^(1/2)/(c^(1/2)+c^(1/2)*cot(b*x+a)))*2^(1 /2)/b-2*c*(c*cot(b*x+a))^(1/2)/b
Time = 0.14 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.05 \[ \int (c \cot (a+b x))^{3/2} \, dx=-\frac {(c \cot (a+b x))^{3/2} \left (\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (a+b x)}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot (a+b x)}\right )}{\sqrt {2}}+2 \sqrt {\cot (a+b x)}+\frac {\log \left (1-\sqrt {2} \sqrt {\cot (a+b x)}+\cot (a+b x)\right )}{2 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt {\cot (a+b x)}+\cot (a+b x)\right )}{2 \sqrt {2}}\right )}{b \cot ^{\frac {3}{2}}(a+b x)} \] Input:
Integrate[(c*Cot[a + b*x])^(3/2),x]
Output:
-(((c*Cot[a + b*x])^(3/2)*(ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*x]]]/Sqrt[2] - ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*x]]]/Sqrt[2] + 2*Sqrt[Cot[a + b*x]] + Log[1 - Sqrt[2]*Sqrt[Cot[a + b*x]] + Cot[a + b*x]]/(2*Sqrt[2]) - Log[1 + S qrt[2]*Sqrt[Cot[a + b*x]] + Cot[a + b*x]]/(2*Sqrt[2])))/(b*Cot[a + b*x]^(3 /2)))
Time = 0.48 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.31, 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, 755, 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.
| \(\displaystyle \int (c \cot (a+b x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (-c \tan \left (a+b x+\frac {\pi }{2}\right )\right )^{3/2}dx\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle c^2 \left (-\int \frac {1}{\sqrt {c \cot (a+b x)}}dx\right )-\frac {2 c \sqrt {c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c^2 \left (-\int \frac {1}{\sqrt {-c \tan \left (a+b x+\frac {\pi }{2}\right )}}dx\right )-\frac {2 c \sqrt {c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {c^3 \int \frac {1}{\sqrt {c \cot (a+b x)} \left (\cot ^2(a+b x) c^2+c^2\right )}d(c \cot (a+b x))}{b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 c^3 \int \frac {1}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}}{b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}}{2 c}+\frac {\int \frac {c^2 \cot ^2(a+b x)+c}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}}{2 c}\right )}{b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\frac {1}{2} \int \frac {1}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}+\frac {1}{2} \int \frac {1}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 c}+\frac {\int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}}{2 c}\right )}{b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\frac {\int \frac {1}{-c^2 \cot ^2(a+b x)-1}d\left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{-c^2 \cot ^2(a+b x)-1}d\left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}}{2 c}+\frac {\int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}}{2 c}\right )}{b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\int \frac {c-c^2 \cot ^2(a+b x)}{c^4 \cot ^4(a+b x)+c^2}d\sqrt {c \cot (a+b x)}}{2 c}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}}{2 c}\right )}{b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 c^3 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {c}-2 \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {c}+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}}{2 c}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}}{2 c}\right )}{b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {c}-2 \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {c}+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}}{2 c}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}}{2 c}\right )}{b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {c}-2 \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {2} \sqrt {c}}+\frac {\int \frac {\sqrt {c}+\sqrt {2} \sqrt {c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {2} c^{3/2} \cot (a+b x)+c}d\sqrt {c \cot (a+b x)}}{2 \sqrt {c}}}{2 c}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}}{2 c}\right )}{b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {c} \cot (a+b x)+1\right )}{\sqrt {2} \sqrt {c}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {c} \cot (a+b x)\right )}{\sqrt {2} \sqrt {c}}}{2 c}+\frac {\frac {\log \left (\sqrt {2} c^{3/2} \cot (a+b x)+c^2 \cot ^2(a+b x)+c\right )}{2 \sqrt {2} \sqrt {c}}-\frac {\log \left (-\sqrt {2} c^{3/2} \cot (a+b x)+c^2 \cot ^2(a+b x)+c\right )}{2 \sqrt {2} \sqrt {c}}}{2 c}\right )}{b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}\) |
Input:
Int[(c*Cot[a + b*x])^(3/2),x]
Output:
(-2*c*Sqrt[c*Cot[a + b*x]])/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]]/(S qrt[2]*Sqrt[c]))/(2*c) + (-1/2*Log[c - Sqrt[2]*c^(3/2)*Cot[a + b*x] + c^2* Cot[a + b*x]^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*c)))/b
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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 = 0.14 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.97
| method | result | size |
| derivativedivides | \(-\frac {2 c \left (\sqrt {c \cot \left (b x +a \right )}-\frac {\left (c^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{b}\) | \(149\) |
| default | \(-\frac {2 c \left (\sqrt {c \cot \left (b x +a \right )}-\frac {\left (c^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{b}\) | \(149\) |
Input:
int((c*cot(b*x+a))^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/b*c*((c*cot(b*x+a))^(1/2)-1/8*(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. 316 vs. \(2 (122) = 244\).
Time = 0.09 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.05 \[ \int (c \cot (a+b x))^{3/2} \, dx=\frac {2 \, \sqrt {2} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} + c}{c}\right ) + 2 \, \sqrt {2} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} - c}{c}\right ) + \sqrt {2} c^{\frac {3}{2}} \log \left (\frac {\sqrt {2} \sqrt {c} \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} \sin \left (2 \, b x + 2 \, a\right ) + c \cos \left (2 \, b x + 2 \, a\right ) + c \sin \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right ) - \sqrt {2} c^{\frac {3}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} \sin \left (2 \, b x + 2 \, a\right ) - c \cos \left (2 \, b x + 2 \, a\right ) - c \sin \left (2 \, b x + 2 \, a\right ) - c}{\sin \left (2 \, b x + 2 \, a\right )}\right ) - 8 \, c \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}}{4 \, b} \] Input:
integrate((c*cot(b*x+a))^(3/2),x, algorithm="fricas")
Output:
1/4*(2*sqrt(2)*c^(3/2)*arctan((sqrt(2)*sqrt(c)*sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a)) + c)/c) + 2*sqrt(2)*c^(3/2)*arctan((sqrt(2)*sqrt(c)*s qrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a)) - c)/c) + sqrt(2)*c^(3/2)*l og((sqrt(2)*sqrt(c)*sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))*sin(2* b*x + 2*a) + c*cos(2*b*x + 2*a) + c*sin(2*b*x + 2*a) + c)/sin(2*b*x + 2*a) ) - sqrt(2)*c^(3/2)*log(-(sqrt(2)*sqrt(c)*sqrt((c*cos(2*b*x + 2*a) + c)/si n(2*b*x + 2*a))*sin(2*b*x + 2*a) - c*cos(2*b*x + 2*a) - c*sin(2*b*x + 2*a) - c)/sin(2*b*x + 2*a)) - 8*c*sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2* a)))/b
\[ \int (c \cot (a+b x))^{3/2} \, dx=\int \left (c \cot {\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((c*cot(b*x+a))**(3/2),x)
Output:
Integral((c*cot(a + b*x))**(3/2), x)
Time = 0.11 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.16 \[ \int (c \cot (a+b x))^{3/2} \, dx=\frac {{\left (2 \, \sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} + 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right ) + 2 \, \sqrt {2} \sqrt {c} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} - 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right ) + \sqrt {2} \sqrt {c} \log \left (\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right ) - \sqrt {2} \sqrt {c} \log \left (-\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right ) - 8 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )} c}{4 \, b} \] Input:
integrate((c*cot(b*x+a))^(3/2),x, algorithm="maxima")
Output:
1/4*(2*sqrt(2)*sqrt(c)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(c) + 2*sqrt(c/tan( b*x + a)))/sqrt(c)) + 2*sqrt(2)*sqrt(c)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt( c) - 2*sqrt(c/tan(b*x + a)))/sqrt(c)) + sqrt(2)*sqrt(c)*log(sqrt(2)*sqrt(c )*sqrt(c/tan(b*x + a)) + c + c/tan(b*x + a)) - sqrt(2)*sqrt(c)*log(-sqrt(2 )*sqrt(c)*sqrt(c/tan(b*x + a)) + c + c/tan(b*x + a)) - 8*sqrt(c/tan(b*x + a)))*c/b
\[ \int (c \cot (a+b x))^{3/2} \, dx=\int { \left (c \cot \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((c*cot(b*x+a))^(3/2),x, algorithm="giac")
Output:
integrate((c*cot(b*x + a))^(3/2), x)
Time = 8.68 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.49 \[ \int (c \cot (a+b x))^{3/2} \, dx=-\frac {2\,c\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{b}-\frac {{\left (-1\right )}^{1/4}\,c^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{b}-\frac {{\left (-1\right )}^{1/4}\,c^{3/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{b} \] Input:
int((c*cot(a + b*x))^(3/2),x)
Output:
- (2*c*(c*cot(a + b*x))^(1/2))/b - ((-1)^(1/4)*c^(3/2)*atan(((-1)^(1/4)*(c *cot(a + b*x))^(1/2))/c^(1/2))*1i)/b - ((-1)^(1/4)*c^(3/2)*atanh(((-1)^(1/ 4)*(c*cot(a + b*x))^(1/2))/c^(1/2))*1i)/b
\[ \int (c \cot (a+b x))^{3/2} \, dx=\frac {\sqrt {c}\, c \left (-2 \sqrt {\cot \left (b x +a \right )}-\left (\int \frac {\sqrt {\cot \left (b x +a \right )}}{\cot \left (b x +a \right )}d x \right ) b \right )}{b} \] Input:
int((c*cot(b*x+a))^(3/2),x)
Output:
(sqrt(c)*c*( - 2*sqrt(cot(a + b*x)) - int(sqrt(cot(a + b*x))/cot(a + b*x), x)*b))/b