Integrand size = 23, antiderivative size = 252 \[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=-\frac {(a+b) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{5/2}}+\frac {(a+b) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{5/2}}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}+\frac {2 b}{d e^2 \sqrt {e \cot (c+d x)}}-\frac {(a-b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{5/2}}+\frac {(a-b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{5/2}} \]
2/3*a/d/e/(e*cot(d*x+c))^(3/2)-1/2*(a+b)*arctan(1-2^(1/2)*(e*cot(d*x+c))^( 1/2)/e^(1/2))/d/e^(5/2)*2^(1/2)+1/2*(a+b)*arctan(1+2^(1/2)*(e*cot(d*x+c))^ (1/2)/e^(1/2))/d/e^(5/2)*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))/d/e^(5/2)*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))/d/e^(5/2)*2^(1/2)+2*b/d/e^2/(e* cot(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.75 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.78 \[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=\frac {3 b \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+8 \sqrt {\tan (c+d x)}\right )-8 a \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right )\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)} \]
(3*b*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*Sqrt[2]*ArcTan[ 1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x ]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 8*Sqrt[Tan[c + d*x]]) - 8*a*(-1 + Hypergeometric2F1[3/4, 1, 7/4, - Tan[c + d*x]^2])*Tan[c + d*x]^(3/2))/(12*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))
Time = 0.69 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.97, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {3042, 4012, 3042, 4012, 25, 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.
| \(\displaystyle \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a-b \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\int \frac {b e-a e \cot (c+d x)}{(e \cot (c+d x))^{3/2}}dx}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {b e+a \tan \left (c+d x+\frac {\pi }{2}\right ) e}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\int -\frac {a e^2+b \cot (c+d x) e^2}{\sqrt {e \cot (c+d x)}}dx}{e^2}+\frac {2 b}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 b}{d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {a e^2+b \cot (c+d x) e^2}{\sqrt {e \cot (c+d x)}}dx}{e^2}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 b}{d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {a e^2-b e^2 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{e^2}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {2 b}{d \sqrt {e \cot (c+d x)}}-\frac {2 \int -\frac {e^2 (a e+b \cot (c+d x) e)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d e^2}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int \frac {e^2 (a e+b \cot (c+d x) e)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 b}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \int \frac {a e+b \cot (c+d x) e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d}+\frac {2 b}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a+b) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d}+\frac {2 b}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a+b) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )\right )}{d}+\frac {2 b}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a+b) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+\frac {2 b}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+\frac {2 b}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} (a-b) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+\frac {2 b}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} (a-b) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+\frac {2 b}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} (a-b) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+\frac {2 b}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}+\frac {2 b}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
(2*a)/(3*d*e*(e*Cot[c + d*x])^(3/2)) + ((2*b)/(d*Sqrt[e*Cot[c + d*x]]) + ( 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]*S qrt[e])))/2 + ((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]*Sqr t[e]*Sqrt[e*Cot[c + d*x]]]/(2*Sqrt[2]*Sqrt[e])))/2))/d)/e^2
3.1.55.3.1 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[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (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 + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
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 = 311, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{e^{2}}+\frac {2 a}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2 b}{e^{2} \sqrt {e \cot \left (d x +c \right )}}}{d}\) | \(311\) |
| default | \(\frac {-\frac {2 \left (-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{e^{2}}+\frac {2 a}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2 b}{e^{2} \sqrt {e \cot \left (d x +c \right )}}}{d}\) | \(311\) |
| parts | \(-\frac {2 a e \left (-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{4}}-\frac {1}{3 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d}+\frac {b \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e^{2} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {2}{e^{2} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\) | \(314\) |
1/d*(-2/e^2*(-1/8*a/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))-1/8*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*arcta n(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))+2/3*a/e/(e*cot(d*x+c))^(3 /2)+2*b/e^2/(e*cot(d*x+c))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 905 vs. \(2 (199) = 398\).
Time = 0.27 (sec) , antiderivative size = 905, normalized size of antiderivative = 3.59 \[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=-\frac {3 \, {\left (d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + d e^{3}\right )} \sqrt {-\frac {d^{2} e^{5} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{10}}} + 2 \, a b}{d^{2} e^{5}}} \log \left (-{\left (a^{4} - b^{4}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + {\left (b d^{3} e^{8} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{10}}} + {\left (a^{3} - a b^{2}\right )} d e^{3}\right )} \sqrt {-\frac {d^{2} e^{5} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{10}}} + 2 \, a b}{d^{2} e^{5}}}\right ) - 3 \, {\left (d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + d e^{3}\right )} \sqrt {-\frac {d^{2} e^{5} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{10}}} + 2 \, a b}{d^{2} e^{5}}} \log \left (-{\left (a^{4} - b^{4}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} - {\left (b d^{3} e^{8} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{10}}} + {\left (a^{3} - a b^{2}\right )} d e^{3}\right )} \sqrt {-\frac {d^{2} e^{5} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{10}}} + 2 \, a b}{d^{2} e^{5}}}\right ) - 3 \, {\left (d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + d e^{3}\right )} \sqrt {\frac {d^{2} e^{5} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{10}}} - 2 \, a b}{d^{2} e^{5}}} \log \left (-{\left (a^{4} - b^{4}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + {\left (b d^{3} e^{8} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{10}}} - {\left (a^{3} - a b^{2}\right )} d e^{3}\right )} \sqrt {\frac {d^{2} e^{5} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{10}}} - 2 \, a b}{d^{2} e^{5}}}\right ) + 3 \, {\left (d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + d e^{3}\right )} \sqrt {\frac {d^{2} e^{5} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{10}}} - 2 \, a b}{d^{2} e^{5}}} \log \left (-{\left (a^{4} - b^{4}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} - {\left (b d^{3} e^{8} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{10}}} - {\left (a^{3} - a b^{2}\right )} d e^{3}\right )} \sqrt {\frac {d^{2} e^{5} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{10}}} - 2 \, a b}{d^{2} e^{5}}}\right ) + 4 \, {\left (a \cos \left (2 \, d x + 2 \, c\right ) - 3 \, b \sin \left (2 \, d x + 2 \, c\right ) - a\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{6 \, {\left (d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + d e^{3}\right )}} \]
-1/6*(3*(d*e^3*cos(2*d*x + 2*c) + d*e^3)*sqrt(-(d^2*e^5*sqrt(-(a^4 - 2*a^2 *b^2 + b^4)/(d^4*e^10)) + 2*a*b)/(d^2*e^5))*log(-(a^4 - b^4)*sqrt((e*cos(2 *d*x + 2*c) + e)/sin(2*d*x + 2*c)) + (b*d^3*e^8*sqrt(-(a^4 - 2*a^2*b^2 + b ^4)/(d^4*e^10)) + (a^3 - a*b^2)*d*e^3)*sqrt(-(d^2*e^5*sqrt(-(a^4 - 2*a^2*b ^2 + b^4)/(d^4*e^10)) + 2*a*b)/(d^2*e^5))) - 3*(d*e^3*cos(2*d*x + 2*c) + d *e^3)*sqrt(-(d^2*e^5*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/(d^4*e^10)) + 2*a*b)/(d ^2*e^5))*log(-(a^4 - b^4)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - (b*d^3*e^8*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/(d^4*e^10)) + (a^3 - a*b^2)*d*e ^3)*sqrt(-(d^2*e^5*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/(d^4*e^10)) + 2*a*b)/(d^2 *e^5))) - 3*(d*e^3*cos(2*d*x + 2*c) + d*e^3)*sqrt((d^2*e^5*sqrt(-(a^4 - 2* a^2*b^2 + b^4)/(d^4*e^10)) - 2*a*b)/(d^2*e^5))*log(-(a^4 - b^4)*sqrt((e*co s(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + (b*d^3*e^8*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/(d^4*e^10)) - (a^3 - a*b^2)*d*e^3)*sqrt((d^2*e^5*sqrt(-(a^4 - 2*a^2 *b^2 + b^4)/(d^4*e^10)) - 2*a*b)/(d^2*e^5))) + 3*(d*e^3*cos(2*d*x + 2*c) + d*e^3)*sqrt((d^2*e^5*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/(d^4*e^10)) - 2*a*b)/( d^2*e^5))*log(-(a^4 - b^4)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - (b*d^3*e^8*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/(d^4*e^10)) - (a^3 - a*b^2)*d* e^3)*sqrt((d^2*e^5*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/(d^4*e^10)) - 2*a*b)/(d^2 *e^5))) + 4*(a*cos(2*d*x + 2*c) - 3*b*sin(2*d*x + 2*c) - a)*sqrt((e*cos(2* d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*e^3*cos(2*d*x + 2*c) + d*e^3)
\[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=\int \frac {a + b \cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=\int { \frac {b \cot \left (d x + c\right ) + a}{\left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
Time = 13.33 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.63 \[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=\frac {2\,a}{3\,d\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}+\frac {2\,b}{d\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}+\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d\,e^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d\,e^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,1{}\mathrm {i}}{d\,e^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,1{}\mathrm {i}}{d\,e^{5/2}} \]
(2*a)/(3*d*e*(e*cot(c + d*x))^(3/2)) + (2*b)/(d*e^2*(e*cot(c + d*x))^(1/2) ) - ((-1)^(1/4)*a*atan(((-1)^(1/4)*(e*cot(c + d*x))^(1/2))/e^(1/2))*1i)/(d *e^(5/2)) - ((-1)^(1/4)*a*atanh(((-1)^(1/4)*(e*cot(c + d*x))^(1/2))/e^(1/2 ))*1i)/(d*e^(5/2)) + ((-1)^(1/4)*b*atan(((-1)^(1/4)*(e*cot(c + d*x))^(1/2) )/e^(1/2)))/(d*e^(5/2)) - ((-1)^(1/4)*b*atanh(((-1)^(1/4)*(e*cot(c + d*x)) ^(1/2))/e^(1/2)))/(d*e^(5/2))