Integrand size = 25, antiderivative size = 294 \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=-\frac {4 a^2 (e \cot (c+d x))^{5/2} \tan (c+d x)}{3 d}-\frac {4 a^2 (e \cot (c+d x))^{5/2} \sec (c+d x) \tan (c+d x)}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)} \tan ^2(c+d x)}{3 d}+\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}{\sqrt {2} d}-\frac {a^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}{\sqrt {2} d}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right ) (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}{\sqrt {2} d} \] Output:
-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)*InverseJacobiAM(c-1/4*P i+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*arct an(-1+2^(1/2)*tan(d*x+c)^(1/2))*(e*cot(d*x+c))^(5/2)*tan(d*x+c)^(5/2)*2^(1 /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)*2^(1/2)/d-1/2*a^2*arctanh(2^(1/2)*tan(d*x+c)^(1/2)/(1+tan(d*x +c)))*(e*cot(d*x+c))^(5/2)*tan(d*x+c)^(5/2)*2^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 12.53 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.32 \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=-\frac {2 a^2 e \cos ^4\left (\frac {1}{2} (c+d x)\right ) (e \cot (c+d x))^{3/2} \left (2+2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-\tan ^2(c+d x)\right )-\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right ) \sec ^4\left (\frac {1}{2} \cot ^{-1}(\cot (c+d x))\right )}{3 d} \] Input:
Integrate[(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2,x]
Output:
(-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 = 0.68 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.90, 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 |
| \(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \left (\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {a^2 \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}-\frac {4 a^2}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {a^2 \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {a^2 \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {4 a^2 \sec (c+d x)}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a^2 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{3 d \sqrt {\tan (c+d x)}}\right )\) |
Input:
Int[(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2,x]
Output:
(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)
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]
Time = 7.73 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00
| method | result | size |
| parts | \(-\frac {2 a^{2} e \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}-\frac {2 a^{2} e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}-\frac {2 a^{2} e^{2} \sqrt {e \cot \left (d x +c \right )}\, \left (\sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \left (1+\sec \left (d x +c \right )\right ) \operatorname {EllipticF}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+2 \csc \left (d x +c \right )\right )}{3 d}\) | \(295\) |
| default | \(-\frac {a^{2} \left (1+\cos \left (d x +c \right )\right ) e^{2} \sqrt {e \cot \left (d x +c \right )}\, \left (3 i \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sin \left (d x +c \right )-3 i \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sin \left (d x +c \right )+3 \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sin \left (d x +c \right )+3 \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sin \left (d x +c \right )-2 \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \sin \left (d x +c \right )+8 \cos \left (d x +c \right )\right ) \sec \left (d x +c \right ) \csc \left (d x +c \right )}{6 d}\) | \(500\) |
Input:
int((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-2*a^2/d*e*(1/3*(e*cot(d*x+c))^(3/2)-1/8*e^2/(e^2)^(1/4)*2^(1/2)*(ln((e*co t(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)))-2/3*a^2/d*e*(e*cot(d*x+c))^(3/2)-2/3*a^2/d*e^2*(e*cot(d *x+c))^(1/2)*((-csc(d*x+c)+cot(d*x+c))^(1/2)*(-cot(d*x+c)+csc(d*x+c)+1)^(1 /2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(1+sec(d*x+c))*EllipticF((-cot(d*x +c)+csc(d*x+c)+1)^(1/2),1/2*2^(1/2))+2*csc(d*x+c))
Timed out. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c))^2,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate((e*cot(d*x+c))**(5/2)*(a+a*sec(d*x+c))**2,x)
Output:
Timed out
Exception generated. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c))^2,x, algorithm="maxima")
Output:
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
\[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\int { \left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:
integrate((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c))^2,x, algorithm="giac")
Output:
integrate((e*cot(d*x + c))^(5/2)*(a*sec(d*x + c) + a)^2, x)
Timed out. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\int {\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2 \,d x \] Input:
int((e*cot(c + d*x))^(5/2)*(a + a/cos(c + d*x))^2,x)
Output:
int((e*cot(c + d*x))^(5/2)*(a + a/cos(c + d*x))^2, x)
\[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\sqrt {e}\, a^{2} e^{2} \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2} \sec \left (d x +c \right )^{2}d x +2 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2} \sec \left (d x +c \right )d x \right )+\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}d x \right ) \] Input:
int((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c))^2,x)
Output:
sqrt(e)*a**2*e**2*(int(sqrt(cot(c + d*x))*cot(c + d*x)**2*sec(c + d*x)**2, x) + 2*int(sqrt(cot(c + d*x))*cot(c + d*x)**2*sec(c + d*x),x) + int(sqrt(c ot(c + d*x))*cot(c + d*x)**2,x))