Integrand size = 23, antiderivative size = 116 \[ \int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x)) \, dx=-\frac {\sqrt {2} a e^{5/2} \text {arctanh}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}+\frac {2 a e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {2 a e (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a (e \cot (c+d x))^{5/2}}{5 d} \]
-2/3*a*e*(e*cot(d*x+c))^(3/2)/d-2/5*a*(e*cot(d*x+c))^(5/2)/d-a*e^(5/2)*arc tanh(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+2*a*e^2*(e*cot(d*x+c))^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.59 \[ \int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x)) \, dx=-\frac {2 a e (e \cot (c+d x))^{3/2} \left (3 \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\tan ^2(c+d x)\right )+5 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\tan ^2(c+d x)\right )\right )}{15 d} \]
(-2*a*e*(e*Cot[c + d*x])^(3/2)*(3*Cot[c + d*x]*Hypergeometric2F1[-5/4, 1, -1/4, -Tan[c + d*x]^2] + 5*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2 ]))/(15*d)
Time = 0.61 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 4011, 3042, 4011, 3042, 4011, 3042, 4015, 221}
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.
| \(\displaystyle \int (a \cot (c+d x)+a) (e \cot (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int (e \cot (c+d x))^{3/2} (a e \cot (c+d x)-a e)dx-\frac {2 a (e \cot (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-a e-a \tan \left (c+d x+\frac {\pi }{2}\right ) e\right )dx-\frac {2 a (e \cot (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \sqrt {e \cot (c+d x)} \left (-a e^2-a \cot (c+d x) e^2\right )dx-\frac {2 a (e \cot (c+d x))^{5/2}}{5 d}-\frac {2 a e (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a e^2 \tan \left (c+d x+\frac {\pi }{2}\right )-a e^2\right )dx-\frac {2 a (e \cot (c+d x))^{5/2}}{5 d}-\frac {2 a e (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \frac {a e^3-a e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}dx+\frac {2 a e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {2 a (e \cot (c+d x))^{5/2}}{5 d}-\frac {2 a e (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a e^3+a \tan \left (c+d x+\frac {\pi }{2}\right ) e^3}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {2 a (e \cot (c+d x))^{5/2}}{5 d}-\frac {2 a e (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle -\frac {2 a^2 e^6 \int \frac {1}{2 a^2 e^6-\left (a e^3+a \cot (c+d x) e^3\right )^2 \tan (c+d x)}d\frac {a e^3+a \cot (c+d x) e^3}{\sqrt {e \cot (c+d x)}}}{d}+\frac {2 a e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {2 a e (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a (e \cot (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\sqrt {2} a e^{5/2} \text {arctanh}\left (\frac {a e^3 \cot (c+d x)+a e^3}{\sqrt {2} a e^{5/2} \sqrt {e \cot (c+d x)}}\right )}{d}+\frac {2 a e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {2 a e (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a (e \cot (c+d x))^{5/2}}{5 d}\) |
-((Sqrt[2]*a*e^(5/2)*ArcTanh[(a*e^3 + a*e^3*Cot[c + d*x])/(Sqrt[2]*a*e^(5/ 2)*Sqrt[e*Cot[c + d*x]])])/d) + (2*a*e^2*Sqrt[e*Cot[c + d*x]])/d - (2*a*e* (e*Cot[c + d*x])^(3/2))/(3*d) - (2*a*(e*Cot[c + d*x])^(5/2))/(5*d)
3.1.2.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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*(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. \(318\) vs. \(2(95)=190\).
Time = 0.16 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.75
| method | result | size |
| derivativedivides | \(-\frac {a \left (\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {e \cot \left (d x +c \right )}\, e^{2}+2 e^{3} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d}\) | \(319\) |
| default | \(-\frac {a \left (\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {e \cot \left (d x +c \right )}\, e^{2}+2 e^{3} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d}\) | \(319\) |
| parts | \(-\frac {2 a e \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}+\frac {a \left (-\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+2 \sqrt {e \cot \left (d x +c \right )}\, e^{2}-\frac {e^{2} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4}\right )}{d}\) | \(323\) |
-1/d*a*(2/5*(e*cot(d*x+c))^(5/2)+2/3*e*(e*cot(d*x+c))^(3/2)-2*(e*cot(d*x+c ))^(1/2)*e^2+2*e^3*(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*c ot(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*arc tan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1))))
Time = 0.29 (sec) , antiderivative size = 377, normalized size of antiderivative = 3.25 \[ \int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x)) \, dx=\left [\frac {15 \, \sqrt {2} {\left (a e^{2} \cos \left (2 \, d x + 2 \, c\right ) - a e^{2}\right )} \sqrt {e} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} + 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) + 4 \, {\left (18 \, a e^{2} \cos \left (2 \, d x + 2 \, c\right ) + 5 \, a e^{2} \sin \left (2 \, d x + 2 \, c\right ) - 12 \, a e^{2}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{30 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )}}, \frac {15 \, \sqrt {2} {\left (a e^{2} \cos \left (2 \, d x + 2 \, c\right ) - a e^{2}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) + 2 \, {\left (18 \, a e^{2} \cos \left (2 \, d x + 2 \, c\right ) + 5 \, a e^{2} \sin \left (2 \, d x + 2 \, c\right ) - 12 \, a e^{2}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{15 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )}}\right ] \]
[1/30*(15*sqrt(2)*(a*e^2*cos(2*d*x + 2*c) - a*e^2)*sqrt(e)*log(sqrt(2)*sqr t(e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*(cos(2*d*x + 2*c) - s in(2*d*x + 2*c) - 1) + 2*e*sin(2*d*x + 2*c) + e) + 4*(18*a*e^2*cos(2*d*x + 2*c) + 5*a*e^2*sin(2*d*x + 2*c) - 12*a*e^2)*sqrt((e*cos(2*d*x + 2*c) + e) /sin(2*d*x + 2*c)))/(d*cos(2*d*x + 2*c) - d), 1/15*(15*sqrt(2)*(a*e^2*cos( 2*d*x + 2*c) - a*e^2)*sqrt(-e)*arctan(1/2*sqrt(2)*sqrt(-e)*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) /(e*cos(2*d*x + 2*c) + e)) + 2*(18*a*e^2*cos(2*d*x + 2*c) + 5*a*e^2*sin(2* d*x + 2*c) - 12*a*e^2)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d *cos(2*d*x + 2*c) - d)]
\[ \int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x)) \, dx=a \left (\int \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx + \int \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot {\left (c + d x \right )}\, dx\right ) \]
a*(Integral((e*cot(c + d*x))**(5/2), x) + Integral((e*cot(c + d*x))**(5/2) *cot(c + d*x), x))
Exception generated. \[ \int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x)) \, dx=\text {Exception raised: ValueError} \]
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
\[ \int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x)) \, dx=\int { {\left (a \cot \left (d x + c\right ) + a\right )} \left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]
Time = 13.78 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.24 \[ \int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x)) \, dx=\frac {2\,a\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}-\frac {2\,a\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}-\frac {2\,a\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d}+\frac {{\left (-1\right )}^{1/4}\,a\,e^{5/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,1{}\mathrm {i}}{\sqrt {e}}\right )}{d}-\frac {{\left (-1\right )}^{1/4}\,a\,e^{5/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,e^{5/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (1+1{}\mathrm {i}\right )}{d} \]
(2*a*e^2*(e*cot(c + d*x))^(1/2))/d - (2*a*e*(e*cot(c + d*x))^(3/2))/(3*d) - (2*a*(e*cot(c + d*x))^(5/2))/(5*d) + ((-1)^(1/4)*a*e^(5/2)*atan(((-1)^(1 /4)*(e*cot(c + d*x))^(1/2))/e^(1/2))*(1 + 1i))/d + ((-1)^(1/4)*a*e^(5/2)*a tan(((-1)^(1/4)*(e*cot(c + d*x))^(1/2)*1i)/e^(1/2)))/d - ((-1)^(1/4)*a*e^( 5/2)*atanh(((-1)^(1/4)*(e*cot(c + d*x))^(1/2))/e^(1/2)))/d