Integrand size = 23, antiderivative size = 94 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=-\frac {\sqrt {2} a e^{3/2} \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 a (e \cot (c+d x))^{3/2}}{3 d} \]
-2/3*a*(e*cot(d*x+c))^(3/2)/d-a*e^(3/2)*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-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 = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.71 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=-\frac {2 a e \sqrt {e \cot (c+d x)} \left (\cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\tan ^2(c+d x)\right )+3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\tan ^2(c+d x)\right )\right )}{3 d} \]
(-2*a*e*Sqrt[e*Cot[c + d*x]]*(Cot[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2] + 3*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2]))/( 3*d)
Time = 0.45 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 4011, 3042, 4011, 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.
| \(\displaystyle \int (a \cot (c+d x)+a) (e \cot (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \sqrt {e \cot (c+d x)} (a e \cot (c+d x)-a e)dx-\frac {2 a (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (-a e-a \tan \left (c+d x+\frac {\pi }{2}\right ) e\right )dx-\frac {2 a (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \frac {-a e^2-a \cot (c+d x) e^2}{\sqrt {e \cot (c+d x)}}dx-\frac {2 a (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a e^2 \tan \left (c+d x+\frac {\pi }{2}\right )-a e^2}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle -\frac {2 a^2 e^4 \int \frac {1}{-2 a^2 e^4-\left (a e^2-a e^2 \cot (c+d x)\right )^2 \tan (c+d x)}d\left (-\frac {a e^2-a e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 a (e \cot (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\sqrt {2} a e^{3/2} \arctan \left (\frac {a e^2-a e^2 \cot (c+d x)}{\sqrt {2} a e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 a (e \cot (c+d x))^{3/2}}{3 d}\) |
-((Sqrt[2]*a*e^(3/2)*ArcTan[(a*e^2 - a*e^2*Cot[c + d*x])/(Sqrt[2]*a*e^(3/2 )*Sqrt[e*Cot[c + d*x]])])/d) - (2*a*e*Sqrt[e*Cot[c + d*x]])/d - (2*a*(e*Co t[c + d*x])^(3/2))/(3*d)
3.1.3.3.1 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[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. \(302\) vs. \(2(77)=154\).
Time = 0.04 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.22
| method | result | size |
| parts | \(-\frac {2 a e \left (\sqrt {e \cot \left (d x +c \right )}-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{d}+\frac {a \left (-\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}\) | \(303\) |
| derivativedivides | \(-\frac {a \left (\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 e \sqrt {e \cot \left (d x +c \right )}-2 e^{2} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d}\) | \(304\) |
| default | \(-\frac {a \left (\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 e \sqrt {e \cot \left (d x +c \right )}-2 e^{2} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d}\) | \(304\) |
-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)))+a/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. 166 vs. \(2 (78) = 156\).
Time = 0.28 (sec) , antiderivative size = 334, normalized size of antiderivative = 3.55 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=\left [\frac {3 \, \sqrt {2} a \sqrt {-e} e \log \left (\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) \sin \left (2 \, d x + 2 \, c\right ) - 4 \, {\left (a e \cos \left (2 \, d x + 2 \, c\right ) + 3 \, a e \sin \left (2 \, d x + 2 \, c\right ) + a e\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{6 \, d \sin \left (2 \, d x + 2 \, c\right )}, -\frac {3 \, \sqrt {2} a e^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) \sin \left (2 \, d x + 2 \, c\right ) + 2 \, {\left (a e \cos \left (2 \, d x + 2 \, c\right ) + 3 \, a e \sin \left (2 \, d x + 2 \, c\right ) + a e\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{3 \, d \sin \left (2 \, d x + 2 \, c\right )}\right ] \]
[1/6*(3*sqrt(2)*a*sqrt(-e)*e*log(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) - 2*e*si n(2*d*x + 2*c) + e)*sin(2*d*x + 2*c) - 4*(a*e*cos(2*d*x + 2*c) + 3*a*e*sin (2*d*x + 2*c) + a*e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*s in(2*d*x + 2*c)), -1/3*(3*sqrt(2)*a*e^(3/2)*arctan(-1/2*sqrt(2)*sqrt(e)*sq rt((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))*sin(2*d*x + 2*c) + 2*(a*e*cos(2*d* x + 2*c) + 3*a*e*sin(2*d*x + 2*c) + a*e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin (2*d*x + 2*c)))/(d*sin(2*d*x + 2*c))]
\[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=a \left (\int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot {\left (c + d x \right )}\, dx\right ) \]
a*(Integral((e*cot(c + d*x))**(3/2), x) + Integral((e*cot(c + d*x))**(3/2) *cot(c + d*x), x))
Exception generated. \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=\text {Exception raised: ValueError} \]
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
\[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=\int { {\left (a \cot \left (d x + c\right ) + a\right )} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}} \,d x } \]
Time = 13.51 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.04 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=-\frac {2\,a\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}-\frac {2\,a\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,e^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (1-\mathrm {i}\right )}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,e^{3/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (-1-\mathrm {i}\right )}{d} \]