Integrand size = 23, antiderivative size = 226 \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x)) \, dx=\frac {(a-b) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {(a-b) \sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {2 b \sqrt {e \cot (c+d x)}}{d}-\frac {(a+b) \sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d}+\frac {(a+b) \sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d} \]
1/2*(a-b)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))*e^(1/2)/d*2^(1/2) -1/2*(a-b)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))*e^(1/2)/d*2^(1/2 )-1/4*(a+b)*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)+1/4*(a+b)*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)-2*b*(e*cot(d*x+c))^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.32 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.69 \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x)) \, dx=-\frac {\sqrt {e \cot (c+d x)} \left (8 b \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\tan ^2(c+d x)\right )+\sqrt {2} a \left (2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right ) \sqrt {\tan (c+d x)}\right )}{4 d} \]
-1/4*(Sqrt[e*Cot[c + d*x]]*(8*b*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d *x]^2] + Sqrt[2]*a*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])*Sqrt[Tan[c + d*x]]))/d
Time = 0.50 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.96, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 4011, 3042, 4017, 27, 1482, 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.
| \(\displaystyle \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \frac {a e \cot (c+d x)-b e}{\sqrt {e \cot (c+d x)}}dx-\frac {2 b \sqrt {e \cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {-b e-a \tan \left (c+d x+\frac {\pi }{2}\right ) e}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b \sqrt {e \cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {2 \int \frac {e (b e-a e \cot (c+d x))}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d}-\frac {2 b \sqrt {e \cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 e \int \frac {b e-a e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d}-\frac {2 b \sqrt {e \cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 e \left (\frac {1}{2} (a+b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} (a-b) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d}-\frac {2 b \sqrt {e \cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 e \left (\frac {1}{2} (a+b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} (a-b) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )\right )}{d}-\frac {2 b \sqrt {e \cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 e \left (\frac {1}{2} (a+b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} (a-b) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {2 b \sqrt {e \cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 e \left (\frac {1}{2} (a+b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {2 b \sqrt {e \cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 e \left (\frac {1}{2} (a+b) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {2 b \sqrt {e \cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 e \left (\frac {1}{2} (a+b) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {2 b \sqrt {e \cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 e \left (\frac {1}{2} (a+b) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {2 b \sqrt {e \cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 e \left (\frac {1}{2} (a+b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {2 b \sqrt {e \cot (c+d x)}}{d}\) |
(-2*b*Sqrt[e*Cot[c + d*x]])/d + (2*e*(-1/2*((a - b)*(-(ArcTan[1 - (Sqrt[2] *Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*S qrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e]))) + ((a + b)*(-1/2*Log[e + e*Cot[c + d*x] - Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(Sqrt[2]*Sqrt[e]) + Log[e + e*Cot[c + d*x] + Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(2*Sqrt[2 ]*Sqrt[e])))/2))/d
3.1.52.3.1 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[((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[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Time = 0.03 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.27
| method | result | size |
| parts | \(-\frac {a e \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {b \left (-2 \sqrt {e \cot \left (d x +c \right )}+\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4}\right )}{d}\) | \(287\) |
| derivativedivides | \(\frac {-2 \sqrt {e \cot \left (d x +c \right )}\, b -2 e \left (-\frac {b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {a \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}\) | \(290\) |
| default | \(\frac {-2 \sqrt {e \cot \left (d x +c \right )}\, b -2 e \left (-\frac {b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {a \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}\) | \(290\) |
-1/4*a/d*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))+b/d*(-2*(e*cot(d *x+c))^(1/2)+1/4*(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)))
Leaf count of result is larger than twice the leaf count of optimal. 730 vs. \(2 (177) = 354\).
Time = 0.27 (sec) , antiderivative size = 730, normalized size of antiderivative = 3.23 \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x)) \, dx=\frac {d \sqrt {\frac {2 \, a b e + d^{2} \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{2}}{d^{4}}}}{d^{2}}} \log \left (-{\left (a^{4} - b^{4}\right )} e \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + {\left (a d^{3} \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{2}}{d^{4}}} - {\left (a^{2} b - b^{3}\right )} d e\right )} \sqrt {\frac {2 \, a b e + d^{2} \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{2}}{d^{4}}}}{d^{2}}}\right ) - d \sqrt {\frac {2 \, a b e + d^{2} \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{2}}{d^{4}}}}{d^{2}}} \log \left (-{\left (a^{4} - b^{4}\right )} e \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} - {\left (a d^{3} \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{2}}{d^{4}}} - {\left (a^{2} b - b^{3}\right )} d e\right )} \sqrt {\frac {2 \, a b e + d^{2} \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{2}}{d^{4}}}}{d^{2}}}\right ) - d \sqrt {\frac {2 \, a b e - d^{2} \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{2}}{d^{4}}}}{d^{2}}} \log \left (-{\left (a^{4} - b^{4}\right )} e \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + {\left (a d^{3} \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{2}}{d^{4}}} + {\left (a^{2} b - b^{3}\right )} d e\right )} \sqrt {\frac {2 \, a b e - d^{2} \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{2}}{d^{4}}}}{d^{2}}}\right ) + d \sqrt {\frac {2 \, a b e - d^{2} \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{2}}{d^{4}}}}{d^{2}}} \log \left (-{\left (a^{4} - b^{4}\right )} e \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} - {\left (a d^{3} \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{2}}{d^{4}}} + {\left (a^{2} b - b^{3}\right )} d e\right )} \sqrt {\frac {2 \, a b e - d^{2} \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{2}}{d^{4}}}}{d^{2}}}\right ) - 4 \, b \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{2 \, d} \]
1/2*(d*sqrt((2*a*b*e + d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^2/d^4))/d^2)*lo g(-(a^4 - b^4)*e*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + (a*d^3* sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^2/d^4) - (a^2*b - b^3)*d*e)*sqrt((2*a*b*e + d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^2/d^4))/d^2)) - d*sqrt((2*a*b*e + d^ 2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^2/d^4))/d^2)*log(-(a^4 - b^4)*e*sqrt((e* cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - (a*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^2/d^4) - (a^2*b - b^3)*d*e)*sqrt((2*a*b*e + d^2*sqrt(-(a^4 - 2*a^2* b^2 + b^4)*e^2/d^4))/d^2)) - d*sqrt((2*a*b*e - d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^2/d^4))/d^2)*log(-(a^4 - b^4)*e*sqrt((e*cos(2*d*x + 2*c) + e)/sin (2*d*x + 2*c)) + (a*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^2/d^4) + (a^2*b - b^3)*d*e)*sqrt((2*a*b*e - d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^2/d^4))/d^2) ) + d*sqrt((2*a*b*e - d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^2/d^4))/d^2)*log (-(a^4 - b^4)*e*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - (a*d^3*s qrt(-(a^4 - 2*a^2*b^2 + b^4)*e^2/d^4) + (a^2*b - b^3)*d*e)*sqrt((2*a*b*e - d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^2/d^4))/d^2)) - 4*b*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/d
\[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x)) \, dx=\int \sqrt {e \cot {\left (c + d x \right )}} \left (a + b \cot {\left (c + d x \right )}\right )\, dx \]
Exception generated. \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x)) \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x)) \, dx=\int { {\left (b \cot \left (d x + c\right ) + a\right )} \sqrt {e \cot \left (d x + c\right )} \,d x } \]
Time = 13.49 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.57 \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x)) \, dx=-\frac {2\,b\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}-\frac {{\left (-1\right )}^{1/4}\,a\,\sqrt {e}\,\left (\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )-\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\right )}{d}-\frac {{\left (-1\right )}^{1/4}\,b\,\sqrt {e}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,1{}\mathrm {i}}{d}-\frac {{\left (-1\right )}^{1/4}\,b\,\sqrt {e}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,1{}\mathrm {i}}{d} \]
- (2*b*(e*cot(c + d*x))^(1/2))/d - ((-1)^(1/4)*b*e^(1/2)*atan(((-1)^(1/4)* (e*cot(c + d*x))^(1/2))/e^(1/2))*1i)/d - ((-1)^(1/4)*b*e^(1/2)*atanh(((-1) ^(1/4)*(e*cot(c + d*x))^(1/2))/e^(1/2))*1i)/d - ((-1)^(1/4)*a*e^(1/2)*(ata n(((-1)^(1/4)*(e*cot(c + d*x))^(1/2))/e^(1/2)) - atanh(((-1)^(1/4)*(e*cot( c + d*x))^(1/2))/e^(1/2))))/d