Integrand size = 25, antiderivative size = 357 \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=-\frac {4 a^2 (e \cot (c+d x))^{5/2} \tan (c+d x)}{3 d}-\frac {4 a^2 (e \cot (c+d x))^{5/2} \sec (c+d x) \tan (c+d x)}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)} \tan ^2(c+d x)}{3 d}+\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}{\sqrt {2} d}-\frac {a^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}{\sqrt {2} d}+\frac {a^2 (e \cot (c+d x))^{5/2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {5}{2}}(c+d x)}{2 \sqrt {2} d}-\frac {a^2 (e \cot (c+d x))^{5/2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {5}{2}}(c+d x)}{2 \sqrt {2} d} \]
-4/3*a^2*(e*cot(d*x+c))^(5/2)*tan(d*x+c)/d-4/3*a^2*(e*cot(d*x+c))^(5/2)*se c(d*x+c)*tan(d*x+c)/d+2/3*a^2*(e*cot(d*x+c))^(5/2)*(sin(c+1/4*Pi+d*x)^2)^( 1/2)/sin(c+1/4*Pi+d*x)*EllipticF(cos(c+1/4*Pi+d*x),2^(1/2))*sec(d*x+c)*sin (2*d*x+2*c)^(1/2)*tan(d*x+c)^2/d-1/2*a^2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2 ))*(e*cot(d*x+c))^(5/2)*tan(d*x+c)^(5/2)/d*2^(1/2)-1/2*a^2*arctan(1+2^(1/2 )*tan(d*x+c)^(1/2))*(e*cot(d*x+c))^(5/2)*tan(d*x+c)^(5/2)/d*2^(1/2)+1/4*a^ 2*(e*cot(d*x+c))^(5/2)*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))*tan(d*x+c )^(5/2)/d*2^(1/2)-1/4*a^2*(e*cot(d*x+c))^(5/2)*ln(1+2^(1/2)*tan(d*x+c)^(1/ 2)+tan(d*x+c))*tan(d*x+c)^(5/2)/d*2^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 12.83 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.26 \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=-\frac {2 a^2 e \cos ^4\left (\frac {1}{2} (c+d x)\right ) (e \cot (c+d x))^{3/2} \left (2+2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-\tan ^2(c+d x)\right )-\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right ) \sec ^4\left (\frac {1}{2} \cot ^{-1}(\cot (c+d x))\right )}{3 d} \]
(-2*a^2*e*Cos[(c + d*x)/2]^4*(e*Cot[c + d*x])^(3/2)*(2 + 2*Hypergeometric2 F1[-3/4, 1/2, 1/4, -Tan[c + d*x]^2] - Hypergeometric2F1[3/4, 1, 7/4, -Cot[ c + d*x]^2])*Sec[ArcCot[Cot[c + d*x]]/2]^4)/(3*d)
Time = 0.65 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.75, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 4388, 3042, 4374, 2009}
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.
| \(\displaystyle \int (a \sec (c+d x)+a)^2 (e \cot (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sec (c+d x)+a)^2 (e \cot (c+d x))^{5/2}dx\) |
| \(\Big \downarrow \) 4388 |
| \(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \int \frac {(\sec (c+d x) a+a)^2}{\tan ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}{\left (-\cot \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \int \left (\frac {\sec ^2(c+d x) a^2}{\tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \sec (c+d x) a^2}{\tan ^{\frac {5}{2}}(c+d x)}+\frac {a^2}{\tan ^{\frac {5}{2}}(c+d x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \left (\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {a^2 \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}-\frac {4 a^2}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {a^2 \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {a^2 \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {4 a^2 \sec (c+d x)}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a^2 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{3 d \sqrt {\tan (c+d x)}}\right )\) |
(e*Cot[c + d*x])^(5/2)*((a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt [2]*d) - (a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + (a^2*L og[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (a^2*Lo g[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (4*a^2)/ (3*d*Tan[c + d*x]^(3/2)) - (4*a^2*Sec[c + d*x])/(3*d*Tan[c + d*x]^(3/2)) - (2*a^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/ (3*d*Sqrt[Tan[c + d*x]]))*Tan[c + d*x]^(5/2)
3.3.38.3.1 Defintions of rubi rules used
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[ c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*((a_) + (b_.)*sec[(c_.) + (d_.)*(x _)])^(n_.), x_Symbol] :> Simp[(e*Cot[c + d*x])^m*Tan[c + d*x]^m Int[(a + b*Sec[c + d*x])^n/Tan[c + d*x]^m, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && !IntegerQ[m]
Result contains complex when optimal does not.
Time = 17.61 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.39
| method | result | size |
| default | \(-\frac {a^{2} e^{2} \sqrt {2}\, \left (\cos \left (d x +c \right )+1\right ) \left (-3 i \sin \left (d x +c \right ) \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+3 i \sin \left (d x +c \right ) \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+3 \sin \left (d x +c \right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+3 \sin \left (d x +c \right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (d x +c \right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+4 \sqrt {2}\, \cos \left (d x +c \right )\right ) \sqrt {e \cot \left (d x +c \right )}\, \sec \left (d x +c \right ) \csc \left (d x +c \right )}{6 d}\) | \(496\) |
| parts | \(-\frac {2 a^{2} 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 {2 a^{2} e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}-\frac {2 a^{2} \sqrt {2}\, e^{2} \sqrt {e \cot \left (d x +c \right )}\, \left (-\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \cot \left (d x +c \right )^{2}-\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )+\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \csc \left (d x +c \right )^{2}+\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \sec \left (d x +c \right ) \csc \left (d x +c \right )^{2}+\sqrt {2}\, \csc \left (d x +c \right )\right )}{3 d}\) | \(568\) |
-1/6*a^2*e^2/d*2^(1/2)*(cos(d*x+c)+1)*(-3*I*sin(d*x+c)*(csc(d*x+c)-cot(d*x +c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2) *EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+3*I*sin (d*x+c)*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(c ot(d*x+c)-csc(d*x+c))^(1/2)*EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2 -1/2*I,1/2*2^(1/2))+3*sin(d*x+c)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(csc(d*x+c) -cot(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*EllipticPi((csc(d*x+c )-cot(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+3*sin(d*x+c)*(cot(d*x+c)-csc( d*x+c))^(1/2)*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)^(1 /2)*EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))-2*si n(d*x+c)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(co t(d*x+c)-csc(d*x+c)+1)^(1/2)*EllipticF((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2 *2^(1/2))+4*2^(1/2)*cos(d*x+c))*(e*cot(d*x+c))^(1/2)*sec(d*x+c)*csc(d*x+c)
Timed out. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]
Timed out. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]
Exception generated. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Exception raised: ValueError} \]
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
\[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\int { \left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\int {\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2 \,d x \]