Integrand size = 25, antiderivative size = 244 \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^2 \, dx=-\frac {\sqrt {2} a^2 \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {\sqrt {2} a^2 \sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}-\frac {4 a^2 \sqrt {e \cot (c+d x)}}{d}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}-\frac {a^2 \sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d}+\frac {a^2 \sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d} \]
-2/3*a^2*(e*cot(d*x+c))^(3/2)/d/e-1/2*a^2*ln(e^(1/2)+cot(d*x+c)*e^(1/2)-2^ (1/2)*(e*cot(d*x+c))^(1/2))*e^(1/2)/d*2^(1/2)+1/2*a^2*ln(e^(1/2)+cot(d*x+c )*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))*e^(1/2)/d*2^(1/2)-a^2*arctan(1-2^( 1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))*2^(1/2)*e^(1/2)/d+a^2*arctan(1+2^(1/2)* (e*cot(d*x+c))^(1/2)/e^(1/2))*2^(1/2)*e^(1/2)/d-4*a^2*(e*cot(d*x+c))^(1/2) /d
Time = 0.51 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.72 \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^2 \, dx=-\frac {a^2 \sqrt {e \cot (c+d x)} \left (6 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-6 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+24 \sqrt {\cot (c+d x)}+4 \cot ^{\frac {3}{2}}(c+d x)+3 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-3 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{6 d \sqrt {\cot (c+d x)}} \]
-1/6*(a^2*Sqrt[e*Cot[c + d*x]]*(6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 24*Sqrt[Cot[c + d*x]] + 4*Cot[c + d*x]^(3/2) + 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d* x]] + Cot[c + d*x]] - 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(d*Sqrt[Cot[c + d*x]])
Time = 0.64 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.96, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3042, 4026, 27, 2030, 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.
| \(\displaystyle \int (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2 \sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4026 |
| \(\displaystyle \int 2 a^2 \cot (c+d x) \sqrt {e \cot (c+d x)}dx-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 a^2 \int \cot (c+d x) \sqrt {e \cot (c+d x)}dx-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 2030 |
| \(\displaystyle \frac {2 a^2 \int (e \cot (c+d x))^{3/2}dx}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 a^2 \int \left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {2 a^2 \left (e^2 \left (-\int \frac {1}{\sqrt {e \cot (c+d x)}}dx\right )-\frac {2 e \sqrt {e \cot (c+d x)}}{d}\right )}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 a^2 \left (e^2 \left (-\int \frac {1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {2 e \sqrt {e \cot (c+d x)}}{d}\right )}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {2 a^2 \left (\frac {e^3 \int \frac {1}{\sqrt {e \cot (c+d x)} \left (\cot ^2(c+d x) e^2+e^2\right )}d(e \cot (c+d x))}{d}-\frac {2 e \sqrt {e \cot (c+d x)}}{d}\right )}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 e^3 \int \frac {1}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{d}-\frac {2 e \sqrt {e \cot (c+d x)}}{d}\right )}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 e^3 \left (\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}+\frac {\int \frac {e^2 \cot ^2(c+d x)+e}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}\right )}{d}-\frac {2 e \sqrt {e \cot (c+d x)}}{d}\right )}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 e^3 \left (\frac {\frac {1}{2} \int \frac {1}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 e}+\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}\right )}{d}-\frac {2 e \sqrt {e \cot (c+d x)}}{d}\right )}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 e^3 \left (\frac {\frac {\int \frac {1}{-e^2 \cot ^2(c+d x)-1}d\left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e^2 \cot ^2(c+d x)-1}d\left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}\right )}{d}-\frac {2 e \sqrt {e \cot (c+d x)}}{d}\right )}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 e^3 \left (\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {2 e \sqrt {e \cot (c+d x)}}{d}\right )}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 e^3 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {2 e \sqrt {e \cot (c+d x)}}{d}\right )}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 e^3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {2 e \sqrt {e \cot (c+d x)}}{d}\right )}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 e^3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {2 e \sqrt {e \cot (c+d x)}}{d}\right )}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 e^3 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {2 e \sqrt {e \cot (c+d x)}}{d}\right )}{e}-\frac {2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}\) |
(-2*a^2*(e*Cot[c + d*x])^(3/2))/(3*d*e) + (2*a^2*((-2*e*Sqrt[e*Cot[c + d*x ]])/d + (2*e^3*((-(ArcTan[1 - Sqrt[2]*Sqrt[e]*Cot[c + d*x]]/(Sqrt[2]*Sqrt[ e])) + ArcTan[1 + Sqrt[2]*Sqrt[e]*Cot[c + d*x]]/(Sqrt[2]*Sqrt[e]))/(2*e) + (-1/2*Log[e - Sqrt[2]*e^(3/2)*Cot[c + d*x] + e^2*Cot[c + d*x]^2]/(Sqrt[2] *Sqrt[e]) + Log[e + Sqrt[2]*e^(3/2)*Cot[c + d*x] + e^2*Cot[c + d*x]^2]/(2* Sqrt[2]*Sqrt[e]))/(2*e)))/d))/e
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[((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[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*( m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e + f* x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !LeQ [m, -1] && !(EqQ[m, 2] && EqQ[a, 0])
Time = 0.05 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(-\frac {2 a^{2} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 e \sqrt {e \cot \left (d x +c \right )}-\frac {e \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4}\right )}{d e}\) | \(170\) |
| default | \(-\frac {2 a^{2} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 e \sqrt {e \cot \left (d x +c \right )}-\frac {e \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4}\right )}{d e}\) | \(170\) |
| parts | \(-\frac {a^{2} e \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {2 a^{2} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d e}+\frac {2 a^{2} \left (-2 \sqrt {e \cot \left (d x +c \right )}+\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4}\right )}{d}\) | \(450\) |
-2/d*a^2/e*(1/3*(e*cot(d*x+c))^(3/2)+2*e*(e*cot(d*x+c))^(1/2)-1/4*e*(e^2)^ (1/4)*2^(1/2)*(ln((e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+( e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^( 1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1 /2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.52 \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^2 \, dx=\frac {3 \, \left (-\frac {a^{8} e^{2}}{d^{4}}\right )^{\frac {1}{4}} d \log \left (a^{2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + \left (-\frac {a^{8} e^{2}}{d^{4}}\right )^{\frac {1}{4}} d\right ) \sin \left (2 \, d x + 2 \, c\right ) + 3 i \, \left (-\frac {a^{8} e^{2}}{d^{4}}\right )^{\frac {1}{4}} d \log \left (a^{2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + i \, \left (-\frac {a^{8} e^{2}}{d^{4}}\right )^{\frac {1}{4}} d\right ) \sin \left (2 \, d x + 2 \, c\right ) - 3 i \, \left (-\frac {a^{8} e^{2}}{d^{4}}\right )^{\frac {1}{4}} d \log \left (a^{2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} - i \, \left (-\frac {a^{8} e^{2}}{d^{4}}\right )^{\frac {1}{4}} d\right ) \sin \left (2 \, d x + 2 \, c\right ) - 3 \, \left (-\frac {a^{8} e^{2}}{d^{4}}\right )^{\frac {1}{4}} d \log \left (a^{2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} - \left (-\frac {a^{8} e^{2}}{d^{4}}\right )^{\frac {1}{4}} d\right ) \sin \left (2 \, d x + 2 \, c\right ) - 2 \, {\left (a^{2} \cos \left (2 \, d x + 2 \, c\right ) + 6 \, a^{2} \sin \left (2 \, d x + 2 \, c\right ) + a^{2}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{3 \, d \sin \left (2 \, d x + 2 \, c\right )} \]
1/3*(3*(-a^8*e^2/d^4)^(1/4)*d*log(a^2*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2* d*x + 2*c)) + (-a^8*e^2/d^4)^(1/4)*d)*sin(2*d*x + 2*c) + 3*I*(-a^8*e^2/d^4 )^(1/4)*d*log(a^2*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + I*(-a^ 8*e^2/d^4)^(1/4)*d)*sin(2*d*x + 2*c) - 3*I*(-a^8*e^2/d^4)^(1/4)*d*log(a^2* sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - I*(-a^8*e^2/d^4)^(1/4)*d )*sin(2*d*x + 2*c) - 3*(-a^8*e^2/d^4)^(1/4)*d*log(a^2*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - (-a^8*e^2/d^4)^(1/4)*d)*sin(2*d*x + 2*c) - 2 *(a^2*cos(2*d*x + 2*c) + 6*a^2*sin(2*d*x + 2*c) + a^2)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*sin(2*d*x + 2*c))
\[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^2 \, dx=a^{2} \left (\int \sqrt {e \cot {\left (c + d x \right )}}\, dx + \int 2 \sqrt {e \cot {\left (c + d x \right )}} \cot {\left (c + d x \right )}\, dx + \int \sqrt {e \cot {\left (c + d x \right )}} \cot ^{2}{\left (c + d x \right )}\, dx\right ) \]
a**2*(Integral(sqrt(e*cot(c + d*x)), x) + Integral(2*sqrt(e*cot(c + d*x))* cot(c + d*x), x) + Integral(sqrt(e*cot(c + d*x))*cot(c + d*x)**2, x))
Exception generated. \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^2 \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^2 \, dx=\int { {\left (a \cot \left (d x + c\right ) + a\right )}^{2} \sqrt {e \cot \left (d x + c\right )} \,d x } \]
Time = 13.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.43 \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^2 \, dx=-\frac {4\,a^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}-\frac {2\,a^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d\,e}-\frac {{\left (-1\right )}^{1/4}\,a^2\,\sqrt {e}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,2{}\mathrm {i}}{d}-\frac {2\,{\left (-1\right )}^{1/4}\,a^2\,\sqrt {e}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,1{}\mathrm {i}}{\sqrt {e}}\right )}{d} \]