Integrand size = 23, antiderivative size = 247 \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x)) \, dx=-\frac {(a+b) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}+\frac {(a+b) e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}-\frac {(a-b) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d}+\frac {(a-b) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d} \]
-2/3*b*(e*cot(d*x+c))^(3/2)/d-1/2*(a+b)*e^(3/2)*arctan(1-2^(1/2)*(e*cot(d* x+c))^(1/2)/e^(1/2))/d*2^(1/2)+1/2*(a+b)*e^(3/2)*arctan(1+2^(1/2)*(e*cot(d *x+c))^(1/2)/e^(1/2))/d*2^(1/2)-1/4*(a-b)*e^(3/2)*ln(e^(1/2)+cot(d*x+c)*e^ (1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2))/d*2^(1/2)+1/4*(a-b)*e^(3/2)*ln(e^(1/2) +cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/d*2^(1/2)-2*a*e*(e*cot(d *x+c))^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.28 \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x)) \, dx=-\frac {2 e \sqrt {e \cot (c+d x)} \left (b \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\tan ^2(c+d x)\right )+3 a \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\tan ^2(c+d x)\right )\right )}{3 d} \]
(-2*e*Sqrt[e*Cot[c + d*x]]*(b*Cot[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2] + 3*a*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2])) /(3*d)
Time = 0.63 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.97, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 4011, 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.
| \(\displaystyle \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \sqrt {e \cot (c+d x)} (a e \cot (c+d x)-b e)dx-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (-b e-a \tan \left (c+d x+\frac {\pi }{2}\right ) e\right )dx-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \frac {-a e^2-b \cot (c+d x) e^2}{\sqrt {e \cot (c+d x)}}dx-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {b e^2 \tan \left (c+d x+\frac {\pi }{2}\right )-a e^2}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {2 \int \frac {e^2 (a e+b \cot (c+d x) e)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 e^2 \int \frac {a e+b \cot (c+d x) e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 e^2 \left (\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a+b) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 e^2 \left (\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a+b) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 e^2 \left (\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a+b) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 e^2 \left (\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 e^2 \left (\frac {1}{2} (a-b) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 e^2 \left (\frac {1}{2} (a-b) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 e^2 \left (\frac {1}{2} (a-b) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 e^2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b (e \cot (c+d x))^{3/2}}{3 d}\) |
(-2*a*e*Sqrt[e*Cot[c + d*x]])/d - (2*b*(e*Cot[c + d*x])^(3/2))/(3*d) + (2* e^2*(((a + b)*(-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[ 2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2] *Sqrt[e])))/2 + ((a - b)*(-1/2*Log[e + e*Cot[c + d*x] - Sqrt[2]*Sqrt[e]*Sq rt[e*Cot[c + d*x]]]/(Sqrt[2]*Sqrt[e]) + Log[e + e*Cot[c + d*x] + Sqrt[2]*S qrt[e]*Sqrt[e*Cot[c + d*x]]]/(2*Sqrt[2]*Sqrt[e])))/2))/d
3.1.51.3.1 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 = 0.04 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.23
| method | result | size |
| parts | \(-\frac {2 a e \left (\sqrt {e \cot \left (d x +c \right )}-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{d}+\frac {b \left (-\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}\) | \(303\) |
| derivativedivides | \(\frac {-\frac {2 b \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 a e \sqrt {e \cot \left (d x +c \right )}+2 e^{2} \left (\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}\) | \(306\) |
| default | \(\frac {-\frac {2 b \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 a e \sqrt {e \cot \left (d x +c \right )}+2 e^{2} \left (\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}\) | \(306\) |
-2*a/d*e*((e*cot(d*x+c))^(1/2)-1/8*(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*(-2/3*(e*cot(d*x+c))^(3/2)+1/4*e^2/(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*c ot(d*x+c))^(1/2)+1)))
Leaf count of result is larger than twice the leaf count of optimal. 843 vs. \(2 (194) = 388\).
Time = 0.28 (sec) , antiderivative size = 843, normalized size of antiderivative = 3.41 \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x)) \, dx=-\frac {3 \, d \sqrt {-\frac {2 \, a b e^{3} + \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{6}}{d^{4}}} d^{2}}{d^{2}}} \log \left (-{\left (a^{4} - b^{4}\right )} e^{4} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + {\left ({\left (a^{3} - a b^{2}\right )} d e^{3} + \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{6}}{d^{4}}} b d^{3}\right )} \sqrt {-\frac {2 \, a b e^{3} + \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{6}}{d^{4}}} d^{2}}{d^{2}}}\right ) \sin \left (2 \, d x + 2 \, c\right ) - 3 \, d \sqrt {-\frac {2 \, a b e^{3} + \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{6}}{d^{4}}} d^{2}}{d^{2}}} \log \left (-{\left (a^{4} - b^{4}\right )} e^{4} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} - {\left ({\left (a^{3} - a b^{2}\right )} d e^{3} + \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{6}}{d^{4}}} b d^{3}\right )} \sqrt {-\frac {2 \, a b e^{3} + \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{6}}{d^{4}}} d^{2}}{d^{2}}}\right ) \sin \left (2 \, d x + 2 \, c\right ) + 3 \, d \sqrt {-\frac {2 \, a b e^{3} - \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{6}}{d^{4}}} d^{2}}{d^{2}}} \log \left (-{\left (a^{4} - b^{4}\right )} e^{4} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + {\left ({\left (a^{3} - a b^{2}\right )} d e^{3} - \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{6}}{d^{4}}} b d^{3}\right )} \sqrt {-\frac {2 \, a b e^{3} - \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{6}}{d^{4}}} d^{2}}{d^{2}}}\right ) \sin \left (2 \, d x + 2 \, c\right ) - 3 \, d \sqrt {-\frac {2 \, a b e^{3} - \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{6}}{d^{4}}} d^{2}}{d^{2}}} \log \left (-{\left (a^{4} - b^{4}\right )} e^{4} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} - {\left ({\left (a^{3} - a b^{2}\right )} d e^{3} - \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{6}}{d^{4}}} b d^{3}\right )} \sqrt {-\frac {2 \, a b e^{3} - \sqrt {-\frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{6}}{d^{4}}} d^{2}}{d^{2}}}\right ) \sin \left (2 \, d x + 2 \, c\right ) + 4 \, {\left (b e \cos \left (2 \, d x + 2 \, c\right ) + 3 \, a e \sin \left (2 \, d x + 2 \, c\right ) + b e\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{6 \, d \sin \left (2 \, d x + 2 \, c\right )} \]
-1/6*(3*d*sqrt(-(2*a*b*e^3 + sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^6/d^4)*d^2)/d ^2)*log(-(a^4 - b^4)*e^4*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + ((a^3 - a*b^2)*d*e^3 + sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^6/d^4)*b*d^3)*sqrt (-(2*a*b*e^3 + sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^6/d^4)*d^2)/d^2))*sin(2*d*x + 2*c) - 3*d*sqrt(-(2*a*b*e^3 + sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^6/d^4)*d^ 2)/d^2)*log(-(a^4 - b^4)*e^4*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c )) - ((a^3 - a*b^2)*d*e^3 + sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^6/d^4)*b*d^3)* sqrt(-(2*a*b*e^3 + sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^6/d^4)*d^2)/d^2))*sin(2 *d*x + 2*c) + 3*d*sqrt(-(2*a*b*e^3 - sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^6/d^4 )*d^2)/d^2)*log(-(a^4 - b^4)*e^4*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + ((a^3 - a*b^2)*d*e^3 - sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^6/d^4)*b*d ^3)*sqrt(-(2*a*b*e^3 - sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^6/d^4)*d^2)/d^2))*s in(2*d*x + 2*c) - 3*d*sqrt(-(2*a*b*e^3 - sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^6 /d^4)*d^2)/d^2)*log(-(a^4 - b^4)*e^4*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d *x + 2*c)) - ((a^3 - a*b^2)*d*e^3 - sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^6/d^4) *b*d^3)*sqrt(-(2*a*b*e^3 - sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^6/d^4)*d^2)/d^2 ))*sin(2*d*x + 2*c) + 4*(b*e*cos(2*d*x + 2*c) + 3*a*e*sin(2*d*x + 2*c) + b *e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*sin(2*d*x + 2*c))
\[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x)) \, dx=\int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (a + b \cot {\left (c + d x \right )}\right )\, dx \]
Exception generated. \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x)) \, dx=\text {Exception raised: ValueError} \]
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
\[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x)) \, dx=\int { {\left (b \cot \left (d x + c\right ) + a\right )} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}} \,d x } \]
Time = 13.86 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.62 \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x)) \, dx=\frac {{\left (-1\right )}^{1/4}\,b\,e^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d}-\frac {2\,a\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}-\frac {2\,b\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}-\frac {{\left (-1\right )}^{1/4}\,b\,e^{3/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d}-\frac {{\left (-1\right )}^{1/4}\,a\,e^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,1{}\mathrm {i}}{d}-\frac {{\left (-1\right )}^{1/4}\,a\,e^{3/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,1{}\mathrm {i}}{d} \]
((-1)^(1/4)*b*e^(3/2)*atan(((-1)^(1/4)*(e*cot(c + d*x))^(1/2))/e^(1/2)))/d - (2*a*e*(e*cot(c + d*x))^(1/2))/d - ((-1)^(1/4)*a*e^(3/2)*atan(((-1)^(1/ 4)*(e*cot(c + d*x))^(1/2))/e^(1/2))*1i)/d - ((-1)^(1/4)*a*e^(3/2)*atanh((( -1)^(1/4)*(e*cot(c + d*x))^(1/2))/e^(1/2))*1i)/d - (2*b*(e*cot(c + d*x))^( 3/2))/(3*d) - ((-1)^(1/4)*b*e^(3/2)*atanh(((-1)^(1/4)*(e*cot(c + d*x))^(1/ 2))/e^(1/2)))/d