\(\int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{5/2}} \, dx\) [292]

Optimal result

Integrand size = 25, antiderivative size = 236 \[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{5/2}} \, dx=-\frac {2 a^2 \arctan \left (\sqrt {\sin (c+d x)}\right )}{d e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {9 a^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{5 d e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {4 a^2 \sin (c+d x)}{3 d e^2 \sqrt {e \csc (c+d x)}}-\frac {2 a^2 \cos (c+d x) \sin (c+d x)}{5 d e^2 \sqrt {e \csc (c+d x)}}+\frac {a^2 \tan (c+d x)}{d e^2 \sqrt {e \csc (c+d x)}} \] Output:

-2*a^2*arctan(sin(d*x+c)^(1/2))/d/e^2/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2 
)+2*a^2*arctanh(sin(d*x+c)^(1/2))/d/e^2/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1 
/2)+9/5*a^2*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))/d/e^2/(e*csc(d*x+ 
c))^(1/2)/sin(d*x+c)^(1/2)-4/3*a^2*sin(d*x+c)/d/e^2/(e*csc(d*x+c))^(1/2)-2 
/5*a^2*cos(d*x+c)*sin(d*x+c)/d/e^2/(e*csc(d*x+c))^(1/2)+a^2*tan(d*x+c)/d/e 
^2/(e*csc(d*x+c))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 13.92 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.64 \[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{5/2}} \, dx=\frac {2 a^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} \csc ^{-1}(\csc (c+d x))\right ) \left (-10 \sqrt {\cos ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},\csc ^2(c+d x)\right )+3 \sqrt {-\cot ^2(c+d x)} \left (-10 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{2},\frac {3}{4},\csc ^2(c+d x)\right )+\operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{2},-\frac {1}{4},\csc ^2(c+d x)\right ) \sin ^2(c+d x)\right )\right ) \tan (c+d x)}{15 d e^2 \sqrt {e \csc (c+d x)}} \] Input:

Integrate[(a + a*Sec[c + d*x])^2/(e*Csc[c + d*x])^(5/2),x]
 

Output:

(2*a^2*Cos[(c + d*x)/2]^4*Sec[ArcCsc[Csc[c + d*x]]/2]^4*(-10*Sqrt[Cos[c + 
d*x]^2]*Hypergeometric2F1[-3/4, 1, 1/4, Csc[c + d*x]^2] + 3*Sqrt[-Cot[c + 
d*x]^2]*(-10*Hypergeometric2F1[-1/4, 3/2, 3/4, Csc[c + d*x]^2] + Hypergeom 
etric2F1[-5/4, 3/2, -1/4, Csc[c + d*x]^2]*Sin[c + d*x]^2))*Tan[c + d*x])/( 
15*d*e^2*Sqrt[e*Csc[c + d*x]])
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.68, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 4366, 3042, 4360, 3042, 3352, 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.

\(\displaystyle \int \frac {(a \sec (c+d x)+a)^2}{(e \csc (c+d x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}{\left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{5/2}}dx\)

\(\Big \downarrow \) 4366

\(\displaystyle \frac {\int (\sec (c+d x) a+a)^2 \sin ^{\frac {5}{2}}(c+d x)dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \cos \left (c+d x-\frac {\pi }{2}\right )^{5/2} \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 4360

\(\displaystyle \frac {\int (-\cos (c+d x) a-a)^2 \sec ^2(c+d x) \sin ^{\frac {5}{2}}(c+d x)dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\left (-\cos \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (-\sin \left (c+d x+\frac {\pi }{2}\right ) a-a\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3352

\(\displaystyle \frac {\int \left (a^2 \sin ^{\frac {5}{2}}(c+d x)+a^2 \sec ^2(c+d x) \sin ^{\frac {5}{2}}(c+d x)+2 a^2 \sec (c+d x) \sin ^{\frac {5}{2}}(c+d x)\right )dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {2 a^2 \arctan \left (\sqrt {\sin (c+d x)}\right )}{d}+\frac {2 a^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d}-\frac {4 a^2 \sin ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {9 a^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{5 d}-\frac {2 a^2 \sin ^{\frac {3}{2}}(c+d x) \cos (c+d x)}{5 d}+\frac {a^2 \sin ^{\frac {3}{2}}(c+d x) \sec (c+d x)}{d}}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

Input:

Int[(a + a*Sec[c + d*x])^2/(e*Csc[c + d*x])^(5/2),x]
 

Output:

((-2*a^2*ArcTan[Sqrt[Sin[c + d*x]]])/d + (2*a^2*ArcTanh[Sqrt[Sin[c + d*x]] 
])/d - (9*a^2*EllipticE[(c - Pi/2 + d*x)/2, 2])/(5*d) - (4*a^2*Sin[c + d*x 
]^(3/2))/(3*d) - (2*a^2*Cos[c + d*x]*Sin[c + d*x]^(3/2))/(5*d) + (a^2*Sec[ 
c + d*x]*Sin[c + d*x]^(3/2))/d)/(e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x 
]])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3352

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig 
[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F 
reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 4360

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si 
n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
 

rule 4366

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*( 
x_)])^(p_), x_Symbol] :> Simp[g^IntPart[p]*(g*Sec[e + f*x])^FracPart[p]*Cos 
[e + f*x]^FracPart[p]   Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x], x] / 
; FreeQ[{a, b, e, f, g, m, p}, x] &&  !IntegerQ[p]
 

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.41 (sec) , antiderivative size = 726, normalized size of antiderivative = 3.08

method	result	size
parts	\(\frac {a^{2} \sqrt {2}\, \left (\left (3 \cos \left (d x +c \right )+3\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+\left (-6 \cos \left (d x +c \right )-6\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+\left (\cos \left (d x +c \right )^{3}-4 \cos \left (d x +c \right )+3\right ) \sqrt {2}\right ) \csc \left (d x +c \right )}{5 d \sqrt {e \csc \left (d x +c \right )}\, e^{2}}+\frac {a^{2} \sqrt {2}\, \left (\sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \left (6 \cos \left (d x +c \right )^{2}+6 \cos \left (d x +c \right )\right )+\sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \left (-3 \cos \left (d x +c \right )^{2}-3 \cos \left (d x +c \right )\right )+\left (2 \cos \left (d x +c \right )^{2}-3 \cos \left (d x +c \right )+1\right ) \sqrt {2}\right ) \sec \left (d x +c \right ) \csc \left (d x +c \right )}{2 d \sqrt {e \csc \left (d x +c \right )}\, e^{2}}+\frac {2 a^{2} \left (-\frac {\sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right ) \left (3 \cot \left (d x +c \right )+3 \csc \left (d x +c \right )\right )}{3}-\frac {\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \left (3 \cot \left (d x +c \right )+3 \csc \left (d x +c \right )\right )}{3}-\frac {2 \sin \left (d x +c \right )}{3}\right )}{d \sqrt {e \csc \left (d x +c \right )}\, e^{2}}\)	\(726\)
default	\(\frac {a^{2} \sqrt {2}\, \left (i \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (30 \cos \left (d x +c \right )^{2}+30 \cos \left (d x +c \right )\right )+i \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (-30 \cos \left (d x +c \right )^{2}-30 \cos \left (d x +c \right )\right )+\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (30 \cos \left (d x +c \right )^{2}+30 \cos \left (d x +c \right )\right )+\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (30 \cos \left (d x +c \right )^{2}+30 \cos \left (d x +c \right )\right )+\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \left (54 \cos \left (d x +c \right )^{2}+54 \cos \left (d x +c \right )\right )+\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \left (-27 \cos \left (d x +c \right )^{2}-27 \cos \left (d x +c \right )\right )+\left (6 \cos \left (d x +c \right )^{4}+20 \cos \left (d x +c \right )^{3}+6 \cos \left (d x +c \right )^{2}-47 \cos \left (d x +c \right )+15\right ) \sqrt {2}\right ) \sec \left (d x +c \right ) \csc \left (d x +c \right )}{30 d \sqrt {e \csc \left (d x +c \right )}\, e^{2}}\)	\(770\)

Input:

int((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/5*a^2/d*2^(1/2)*((3*cos(d*x+c)+3)*(1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2)*(1 
-I*cot(d*x+c)+I*csc(d*x+c))^(1/2)*(-I*(-csc(d*x+c)+cot(d*x+c)))^(1/2)*Elli 
pticF((1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2),1/2*2^(1/2))+(-6*cos(d*x+c)-6)*( 
1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2)*(1-I*cot(d*x+c)+I*csc(d*x+c))^(1/2)*(-I 
*(-csc(d*x+c)+cot(d*x+c)))^(1/2)*EllipticE((1+I*cot(d*x+c)-I*csc(d*x+c))^( 
1/2),1/2*2^(1/2))+(cos(d*x+c)^3-4*cos(d*x+c)+3)*2^(1/2))/(e*csc(d*x+c))^(1 
/2)/e^2*csc(d*x+c)+1/2*a^2/d*2^(1/2)*((1-I*cot(d*x+c)+I*csc(d*x+c))^(1/2)* 
(-I*(-csc(d*x+c)+cot(d*x+c)))^(1/2)*EllipticE((1+I*cot(d*x+c)-I*csc(d*x+c) 
)^(1/2),1/2*2^(1/2))*(1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2)*(6*cos(d*x+c)^2+6 
*cos(d*x+c))+(1-I*cot(d*x+c)+I*csc(d*x+c))^(1/2)*(-I*(-csc(d*x+c)+cot(d*x+ 
c)))^(1/2)*EllipticF((1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2),1/2*2^(1/2))*(1+I 
*cot(d*x+c)-I*csc(d*x+c))^(1/2)*(-3*cos(d*x+c)^2-3*cos(d*x+c))+(2*cos(d*x+ 
c)^2-3*cos(d*x+c)+1)*2^(1/2))/(e*csc(d*x+c))^(1/2)/e^2*sec(d*x+c)*csc(d*x+ 
c)+2*a^2/d*(-1/3*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*arctan(sin(d*x+c)*(si 
n(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/(-1+cos(d*x+c)))*(3*cot(d*x+c)+3*csc(d*x+ 
c))-1/3*arctanh(sin(d*x+c)*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/(-1+cos(d*x 
+c)))*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*(3*cot(d*x+c)+3*csc(d*x+c))-2/3* 
sin(d*x+c))/(e*csc(d*x+c))^(1/2)/e^2
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.41 (sec) , antiderivative size = 776, normalized size of antiderivative = 3.29 \[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

[-1/60*(30*a^2*sqrt(-e)*arctan(-1/4*(cos(d*x + c)^2 - 6*sin(d*x + c) - 2)* 
sqrt(-e)*sqrt(e/sin(d*x + c))/(e*sin(d*x + c) + e))*cos(d*x + c) + 15*a^2* 
sqrt(-e)*cos(d*x + c)*log((e*cos(d*x + c)^4 - 72*e*cos(d*x + c)^2 + 8*(cos 
(d*x + c)^4 - 9*cos(d*x + c)^2 + (7*cos(d*x + c)^2 - 8)*sin(d*x + c) + 8)* 
sqrt(-e)*sqrt(e/sin(d*x + c)) + 28*(e*cos(d*x + c)^2 - 2*e)*sin(d*x + c) + 
 72*e)/(cos(d*x + c)^4 - 8*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 - 2)*sin(d*x 
 + c) + 8)) + 54*a^2*sqrt(2*I*e)*cos(d*x + c)*weierstrassZeta(4, 0, weiers 
trassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + 54*a^2*sqrt(-2*I*e)* 
cos(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) 
- I*sin(d*x + c))) - 4*(6*a^2*cos(d*x + c)^4 + 20*a^2*cos(d*x + c)^3 - 21* 
a^2*cos(d*x + c)^2 - 20*a^2*cos(d*x + c) + 15*a^2)*sqrt(e/sin(d*x + c)))/( 
d*e^3*cos(d*x + c)), 1/60*(30*a^2*sqrt(e)*arctan(1/4*(cos(d*x + c)^2 + 6*s 
in(d*x + c) - 2)*sqrt(e)*sqrt(e/sin(d*x + c))/(e*sin(d*x + c) - e))*cos(d* 
x + c) + 15*a^2*sqrt(e)*cos(d*x + c)*log((e*cos(d*x + c)^4 - 72*e*cos(d*x 
+ c)^2 + 8*(cos(d*x + c)^4 - 9*cos(d*x + c)^2 - (7*cos(d*x + c)^2 - 8)*sin 
(d*x + c) + 8)*sqrt(e)*sqrt(e/sin(d*x + c)) - 28*(e*cos(d*x + c)^2 - 2*e)* 
sin(d*x + c) + 72*e)/(cos(d*x + c)^4 - 8*cos(d*x + c)^2 + 4*(cos(d*x + c)^ 
2 - 2)*sin(d*x + c) + 8)) - 54*a^2*sqrt(2*I*e)*cos(d*x + c)*weierstrassZet 
a(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) - 54*a^2 
*sqrt(-2*I*e)*cos(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4,...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((a+a*sec(d*x+c))**2/(e*csc(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{5/2}} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

integrate((a*sec(d*x + c) + a)^2/(e*csc(d*x + c))^(5/2), x)
 

Giac [F]

\[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{5/2}} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

integrate((a*sec(d*x + c) + a)^2/(e*csc(d*x + c))^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{5/2}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2}{{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{5/2}} \,d x \] Input:

int((a + a/cos(c + d*x))^2/(e/sin(c + d*x))^(5/2),x)
 

Output:

int((a + a/cos(c + d*x))^2/(e/sin(c + d*x))^(5/2), x)
 

Reduce [F]

\[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{5/2}} \, dx=\frac {\sqrt {e}\, a^{2} \left (\int \frac {\sqrt {\csc \left (d x +c \right )}}{\csc \left (d x +c \right )^{3}}d x +\int \frac {\sqrt {\csc \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\csc \left (d x +c \right )^{3}}d x +2 \left (\int \frac {\sqrt {\csc \left (d x +c \right )}\, \sec \left (d x +c \right )}{\csc \left (d x +c \right )^{3}}d x \right )\right )}{e^{3}} \] Input:

int((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(5/2),x)
 

Output:

(sqrt(e)*a**2*(int(sqrt(csc(c + d*x))/csc(c + d*x)**3,x) + int((sqrt(csc(c 
 + d*x))*sec(c + d*x)**2)/csc(c + d*x)**3,x) + 2*int((sqrt(csc(c + d*x))*s 
ec(c + d*x))/csc(c + d*x)**3,x)))/e**3