Integrand size = 23, antiderivative size = 99 \[ \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=-\frac {\sqrt {2} a \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{5/2}}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}+\frac {2 a}{d e^2 \sqrt {e \cot (c+d x)}} \]
2/3*a/d/e/(e*cot(d*x+c))^(3/2)-a*arctan(1/2*(e^(1/2)-cot(d*x+c)*e^(1/2))*2 ^(1/2)/(e*cot(d*x+c))^(1/2))*2^(1/2)/d/e^(5/2)+2*a/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 = 0.44 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.05 \[ \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=\frac {a \left (6 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-6 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+3 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-3 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+24 \sqrt {\tan (c+d x)}+8 \tan ^{\frac {3}{2}}(c+d x)-8 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)\right )}{12 d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)} \]
(a*(6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x ]] + Tan[c + d*x]] - 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 24*Sqrt[Tan[c + d*x]] + 8*Tan[c + d*x]^(3/2) - 8*Hypergeometric2 F1[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 = 0.51 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 4012, 3042, 4012, 25, 3042, 4015, 218}
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.
| \(\displaystyle \int \frac {a \cot (c+d x)+a}{(e \cot (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a-a \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\int \frac {a e-a e \cot (c+d x)}{(e \cot (c+d x))^{3/2}}dx}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a e+a \tan \left (c+d x+\frac {\pi }{2}\right ) e}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\int -\frac {a e^2+a \cot (c+d x) e^2}{\sqrt {e \cot (c+d x)}}dx}{e^2}+\frac {2 a}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a}{d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {a e^2+a \cot (c+d x) e^2}{\sqrt {e \cot (c+d x)}}dx}{e^2}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 a}{d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {a e^2-a e^2 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{e^2}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle \frac {\frac {2 a^2 e^2 \int \frac {1}{-2 a^2 e^4-\left (a e^2-a e^2 \cot (c+d x)\right )^2 \tan (c+d x)}d\frac {a e^2-a e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}}{d}+\frac {2 a}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {2 a}{d \sqrt {e \cot (c+d x)}}-\frac {\sqrt {2} a \arctan \left (\frac {a e^2-a e^2 \cot (c+d x)}{\sqrt {2} a e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{d \sqrt {e}}}{e^2}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}\) |
(2*a)/(3*d*e*(e*Cot[c + d*x])^(3/2)) + (-((Sqrt[2]*a*ArcTan[(a*e^2 - a*e^2 *Cot[c + d*x])/(Sqrt[2]*a*e^(3/2)*Sqrt[e*Cot[c + d*x]])])/(d*Sqrt[e])) + ( 2*a)/(d*Sqrt[e*Cot[c + d*x]]))/e^2
3.1.7.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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*(d^2/f) Subst[Int[1/(2*c*d + b*x^2), x], x, (c - d*Tan[e + f*x])/Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(308\) vs. \(2(82)=164\).
Time = 0.05 (sec) , antiderivative size = 309, normalized size of antiderivative = 3.12
| method | result | size |
| derivativedivides | \(-\frac {a \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{e^{2}}-\frac {2}{e^{2} \sqrt {e \cot \left (d x +c \right )}}-\frac {2}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d}\) | \(309\) |
| default | \(-\frac {a \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{e^{2}}-\frac {2}{e^{2} \sqrt {e \cot \left (d x +c \right )}}-\frac {2}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d}\) | \(309\) |
| parts | \(-\frac {2 a e \left (-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{4}}-\frac {1}{3 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d}+\frac {a \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e^{2} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {2}{e^{2} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\) | \(314\) |
-1/d*a*(2/e^2*(-1/8/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/( 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)))-2/e^2/(e*cot(d*x+c))^(1/2)- 2/3/e/(e*cot(d*x+c))^(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (83) = 166\).
Time = 0.28 (sec) , antiderivative size = 358, normalized size of antiderivative = 3.62 \[ \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=\left [\frac {3 \, \sqrt {2} {\left (a e \cos \left (2 \, d x + 2 \, c\right ) + a e\right )} \sqrt {-\frac {1}{e}} \log \left (\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sqrt {-\frac {1}{e}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right ) - 4 \, {\left (a \cos \left (2 \, d x + 2 \, c\right ) - 3 \, a \sin \left (2 \, d x + 2 \, c\right ) - a\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{6 \, {\left (d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + d e^{3}\right )}}, -\frac {\frac {3 \, \sqrt {2} {\left (a e \cos \left (2 \, d x + 2 \, c\right ) + a e\right )} \arctan \left (-\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, \sqrt {e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right )}{\sqrt {e}} + 2 \, {\left (a \cos \left (2 \, d x + 2 \, c\right ) - 3 \, a \sin \left (2 \, d x + 2 \, c\right ) - a\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{3 \, {\left (d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + d e^{3}\right )}}\right ] \]
[1/6*(3*sqrt(2)*(a*e*cos(2*d*x + 2*c) + a*e)*sqrt(-1/e)*log(sqrt(2)*sqrt(( e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*sqrt(-1/e)*(cos(2*d*x + 2*c) + s in(2*d*x + 2*c) - 1) - 2*sin(2*d*x + 2*c) + 1) - 4*(a*cos(2*d*x + 2*c) - 3 *a*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), -1/3*(3*sqrt(2)*(a*e*cos(2*d*x + 2*c) + a*e)*arctan(-1/2*sqrt(2)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*( cos(2*d*x + 2*c) - sin(2*d*x + 2*c) + 1)/(sqrt(e)*(cos(2*d*x + 2*c) + 1))) /sqrt(e) + 2*(a*cos(2*d*x + 2*c) - 3*a*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)]
\[ \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=a \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {\cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx\right ) \]
a*(Integral((e*cot(c + d*x))**(-5/2), x) + Integral(cot(c + d*x)/(e*cot(c + d*x))**(5/2), x))
Exception generated. \[ \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=\int { \frac {a \cot \left (d x + c\right ) + a}{\left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
Time = 13.90 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.04 \[ \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx=\frac {2\,a}{d\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}+\frac {2\,a}{3\,d\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (1-\mathrm {i}\right )}{d\,e^{5/2}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (-1-\mathrm {i}\right )}{d\,e^{5/2}} \]