Integrand size = 25, antiderivative size = 267 \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}+\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}+\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}} \]
-1/2*(a^2-2*a*b-b^2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/d/e^(3 /2)*2^(1/2)+1/2*(a^2-2*a*b-b^2)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1 /2))/d/e^(3/2)*2^(1/2)+1/4*(a^2+2*a*b-b^2)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)-2 ^(1/2)*(e*cot(d*x+c))^(1/2))/d/e^(3/2)*2^(1/2)-1/4*(a^2+2*a*b-b^2)*ln(e^(1 /2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/d/e^(3/2)*2^(1/2)+2*a ^2/d/e/(e*cot(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.82 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.70 \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=-\frac {\cot ^{\frac {3}{2}}(c+d x) \left (-\frac {2 b^2}{\sqrt {\cot (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\cot ^2(c+d x)\right )}{\sqrt {\cot (c+d x)}}-\frac {a b \left (2 \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{\sqrt {2}}\right )}{d (e \cot (c+d x))^{3/2}} \]
-((Cot[c + d*x]^(3/2)*((-2*b^2)/Sqrt[Cot[c + d*x]] - (2*(a^2 - b^2)*Hyperg eometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2])/Sqrt[Cot[c + d*x]] - (a*b*(2*A rcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt [2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/Sqrt[2]))/(d*(e*Cot[c + d*x])^(3/ 2)))
Time = 0.59 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.91, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 4025, 3042, 4017, 25, 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.
| \(\displaystyle \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4025 |
| \(\displaystyle \frac {\int \frac {2 a b e-\left (a^2-b^2\right ) e \cot (c+d x)}{\sqrt {e \cot (c+d x)}}dx}{e^2}+\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 a b e+\left (a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) e}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{e^2}+\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {2 \int -\frac {e \left (2 a b e-\left (a^2-b^2\right ) e \cot (c+d x)\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \int \frac {e \left (2 a b e-\left (a^2-b^2\right ) e \cot (c+d x)\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \int \frac {2 a b e-\left (a^2-b^2\right ) e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d e}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d e}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )\right )}{d e}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d e}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d e}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d e}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d e}\) |
(2*a^2)/(d*e*Sqrt[e*Cot[c + d*x]]) - (2*(-1/2*((a^2 - 2*a*b - b^2)*(-(ArcT an[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]))) + ((a^2 + 2*a*b - b^2)*(-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*e)
3.1.59.3.1 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[((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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Time = 0.04 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(-\frac {2 \left (\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a^{2}}{\sqrt {e \cot \left (d x +c \right )}}\right )}{d e}\) | \(301\) |
| default | \(-\frac {2 \left (\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a^{2}}{\sqrt {e \cot \left (d x +c \right )}}\right )}{d e}\) | \(301\) |
| parts | \(-\frac {2 a^{2} e \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {1}{e^{2} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}-\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d e \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{2 d \,e^{2}}\) | \(440\) |
-2/d/e*(1/4*a/e*b*(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))+1/8*(-a^2 +b^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*cot(d*x+c))^(1/2)+1))-a^2/(e*cot(d*x+c))^(1 /2))
Leaf count of result is larger than twice the leaf count of optimal. 1298 vs. \(2 (218) = 436\).
Time = 0.31 (sec) , antiderivative size = 1298, normalized size of antiderivative = 4.86 \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
1/2*(4*a^2*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*sin(2*d*x + 2*c ) + (d*e^2*cos(2*d*x + 2*c) + d*e^2)*sqrt((d^2*e^3*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^6)) + 4*a^3*b - 4*a*b^3)/(d^2*e^3 ))*log((a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + ((a^2 - b^2)*d^3*e^5*sqrt(-(a^8 - 12*a^6*b ^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^6)) - 2*(a^5*b - 6*a^3*b^3 + a* b^5)*d*e^2)*sqrt((d^2*e^3*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^ 6 + b^8)/(d^4*e^6)) + 4*a^3*b - 4*a*b^3)/(d^2*e^3))) - (d*e^2*cos(2*d*x + 2*c) + d*e^2)*sqrt((d^2*e^3*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2* b^6 + b^8)/(d^4*e^6)) + 4*a^3*b - 4*a*b^3)/(d^2*e^3))*log((a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - ((a^2 - b^2)*d^3*e^5*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^ 2*b^6 + b^8)/(d^4*e^6)) - 2*(a^5*b - 6*a^3*b^3 + a*b^5)*d*e^2)*sqrt((d^2*e ^3*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^6)) + 4 *a^3*b - 4*a*b^3)/(d^2*e^3))) - (d*e^2*cos(2*d*x + 2*c) + d*e^2)*sqrt(-(d^ 2*e^3*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^6)) - 4*a^3*b + 4*a*b^3)/(d^2*e^3))*log((a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2* b^6 + b^8)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + ((a^2 - b^2)* d^3*e^5*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^6) ) + 2*(a^5*b - 6*a^3*b^3 + a*b^5)*d*e^2)*sqrt(-(d^2*e^3*sqrt(-(a^8 - 12...
\[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=\int \frac {\left (a + b \cot {\left (c + d x \right )}\right )^{2}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=\int { \frac {{\left (b \cot \left (d x + c\right ) + a\right )}^{2}}{\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Time = 12.92 (sec) , antiderivative size = 1196, normalized size of antiderivative = 4.48 \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=\frac {2\,a^2}{d\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}+2\,\mathrm {atanh}\left (\frac {32\,a^4\,d^3\,e^5\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e^3}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e^3}-\frac {a\,b^3}{d^2\,e^3}+\frac {a^3\,b}{d^2\,e^3}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e^3}}}{-16\,a^6\,d^2\,e^4+a^5\,b\,d^2\,e^4\,32{}\mathrm {i}+112\,a^4\,b^2\,d^2\,e^4-a^3\,b^3\,d^2\,e^4\,192{}\mathrm {i}-112\,a^2\,b^4\,d^2\,e^4+a\,b^5\,d^2\,e^4\,32{}\mathrm {i}+16\,b^6\,d^2\,e^4}+\frac {32\,b^4\,d^3\,e^5\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e^3}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e^3}-\frac {a\,b^3}{d^2\,e^3}+\frac {a^3\,b}{d^2\,e^3}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e^3}}}{-16\,a^6\,d^2\,e^4+a^5\,b\,d^2\,e^4\,32{}\mathrm {i}+112\,a^4\,b^2\,d^2\,e^4-a^3\,b^3\,d^2\,e^4\,192{}\mathrm {i}-112\,a^2\,b^4\,d^2\,e^4+a\,b^5\,d^2\,e^4\,32{}\mathrm {i}+16\,b^6\,d^2\,e^4}-\frac {192\,a^2\,b^2\,d^3\,e^5\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e^3}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e^3}-\frac {a\,b^3}{d^2\,e^3}+\frac {a^3\,b}{d^2\,e^3}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e^3}}}{-16\,a^6\,d^2\,e^4+a^5\,b\,d^2\,e^4\,32{}\mathrm {i}+112\,a^4\,b^2\,d^2\,e^4-a^3\,b^3\,d^2\,e^4\,192{}\mathrm {i}-112\,a^2\,b^4\,d^2\,e^4+a\,b^5\,d^2\,e^4\,32{}\mathrm {i}+16\,b^6\,d^2\,e^4}\right )\,\sqrt {\frac {\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}-2\,\mathrm {atanh}\left (\frac {32\,a^4\,d^3\,e^5\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^3\,b}{d^2\,e^3}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e^3}-\frac {a\,b^3}{d^2\,e^3}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e^3}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e^3}}}{16\,a^6\,d^2\,e^4+a^5\,b\,d^2\,e^4\,32{}\mathrm {i}-112\,a^4\,b^2\,d^2\,e^4-a^3\,b^3\,d^2\,e^4\,192{}\mathrm {i}+112\,a^2\,b^4\,d^2\,e^4+a\,b^5\,d^2\,e^4\,32{}\mathrm {i}-16\,b^6\,d^2\,e^4}+\frac {32\,b^4\,d^3\,e^5\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^3\,b}{d^2\,e^3}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e^3}-\frac {a\,b^3}{d^2\,e^3}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e^3}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e^3}}}{16\,a^6\,d^2\,e^4+a^5\,b\,d^2\,e^4\,32{}\mathrm {i}-112\,a^4\,b^2\,d^2\,e^4-a^3\,b^3\,d^2\,e^4\,192{}\mathrm {i}+112\,a^2\,b^4\,d^2\,e^4+a\,b^5\,d^2\,e^4\,32{}\mathrm {i}-16\,b^6\,d^2\,e^4}-\frac {192\,a^2\,b^2\,d^3\,e^5\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^3\,b}{d^2\,e^3}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e^3}-\frac {a\,b^3}{d^2\,e^3}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e^3}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e^3}}}{16\,a^6\,d^2\,e^4+a^5\,b\,d^2\,e^4\,32{}\mathrm {i}-112\,a^4\,b^2\,d^2\,e^4-a^3\,b^3\,d^2\,e^4\,192{}\mathrm {i}+112\,a^2\,b^4\,d^2\,e^4+a\,b^5\,d^2\,e^4\,32{}\mathrm {i}-16\,b^6\,d^2\,e^4}\right )\,\sqrt {-\frac {\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}} \]
2*atanh((32*a^4*d^3*e^5*(e*cot(c + d*x))^(1/2)*((a^4*1i)/(4*d^2*e^3) + (b^ 4*1i)/(4*d^2*e^3) - (a*b^3)/(d^2*e^3) + (a^3*b)/(d^2*e^3) - (a^2*b^2*3i)/( 2*d^2*e^3))^(1/2))/(16*b^6*d^2*e^4 - 16*a^6*d^2*e^4 + a*b^5*d^2*e^4*32i + a^5*b*d^2*e^4*32i - 112*a^2*b^4*d^2*e^4 - a^3*b^3*d^2*e^4*192i + 112*a^4*b ^2*d^2*e^4) + (32*b^4*d^3*e^5*(e*cot(c + d*x))^(1/2)*((a^4*1i)/(4*d^2*e^3) + (b^4*1i)/(4*d^2*e^3) - (a*b^3)/(d^2*e^3) + (a^3*b)/(d^2*e^3) - (a^2*b^2 *3i)/(2*d^2*e^3))^(1/2))/(16*b^6*d^2*e^4 - 16*a^6*d^2*e^4 + a*b^5*d^2*e^4* 32i + a^5*b*d^2*e^4*32i - 112*a^2*b^4*d^2*e^4 - a^3*b^3*d^2*e^4*192i + 112 *a^4*b^2*d^2*e^4) - (192*a^2*b^2*d^3*e^5*(e*cot(c + d*x))^(1/2)*((a^4*1i)/ (4*d^2*e^3) + (b^4*1i)/(4*d^2*e^3) - (a*b^3)/(d^2*e^3) + (a^3*b)/(d^2*e^3) - (a^2*b^2*3i)/(2*d^2*e^3))^(1/2))/(16*b^6*d^2*e^4 - 16*a^6*d^2*e^4 + a*b ^5*d^2*e^4*32i + a^5*b*d^2*e^4*32i - 112*a^2*b^4*d^2*e^4 - a^3*b^3*d^2*e^4 *192i + 112*a^4*b^2*d^2*e^4))*(((a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b ^2)*1i)/(4*d^2*e^3))^(1/2) - 2*atanh((32*a^4*d^3*e^5*(e*cot(c + d*x))^(1/2 )*((a^3*b)/(d^2*e^3) - (b^4*1i)/(4*d^2*e^3) - (a*b^3)/(d^2*e^3) - (a^4*1i) /(4*d^2*e^3) + (a^2*b^2*3i)/(2*d^2*e^3))^(1/2))/(16*a^6*d^2*e^4 - 16*b^6*d ^2*e^4 + a*b^5*d^2*e^4*32i + a^5*b*d^2*e^4*32i + 112*a^2*b^4*d^2*e^4 - a^3 *b^3*d^2*e^4*192i - 112*a^4*b^2*d^2*e^4) + (32*b^4*d^3*e^5*(e*cot(c + d*x) )^(1/2)*((a^3*b)/(d^2*e^3) - (b^4*1i)/(4*d^2*e^3) - (a*b^3)/(d^2*e^3) - (a ^4*1i)/(4*d^2*e^3) + (a^2*b^2*3i)/(2*d^2*e^3))^(1/2))/(16*a^6*d^2*e^4 -...