\(\int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx\) [288]

Optimal result

Integrand size = 25, antiderivative size = 240 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=-\frac {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\frac {2 a^2 e \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a^2 e \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}-\frac {5 a^2 e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d}+\frac {3 a^2 e \sqrt {e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d} \] Output:

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

Mathematica [C] (warning: unable to verify)

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

Time = 14.29 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.81 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=\frac {2 a^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) (e \csc (c+d x))^{3/2} \left (3 \arctan \left (\sqrt {\csc (c+d x)}\right ) \sqrt {\cos ^2(c+d x)}+3 \text {arctanh}\left (\sqrt {\csc (c+d x)}\right ) \sqrt {\cos ^2(c+d x)}-6 \sqrt {\csc (c+d x)}-6 \sqrt {\cos ^2(c+d x)} \sqrt {\csc (c+d x)}+5 \sqrt {-\cot ^2(c+d x)} \sqrt {\csc (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\csc ^2(c+d x)\right )\right ) \sec (c+d x) \sec ^4\left (\frac {1}{2} \csc ^{-1}(\csc (c+d x))\right )}{3 d \csc ^{\frac {3}{2}}(c+d x)} \] Input:

Integrate[(e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2,x]
 

Output:

(2*a^2*Cos[(c + d*x)/2]^4*(e*Csc[c + d*x])^(3/2)*(3*ArcTan[Sqrt[Csc[c + d* 
x]]]*Sqrt[Cos[c + d*x]^2] + 3*ArcTanh[Sqrt[Csc[c + d*x]]]*Sqrt[Cos[c + d*x 
]^2] - 6*Sqrt[Csc[c + d*x]] - 6*Sqrt[Cos[c + d*x]^2]*Sqrt[Csc[c + d*x]] + 
5*Sqrt[-Cot[c + d*x]^2]*Sqrt[Csc[c + d*x]]*Hypergeometric2F1[3/4, 3/2, 7/4 
, Csc[c + d*x]^2])*Sec[c + d*x]*Sec[ArcCsc[Csc[c + d*x]]/2]^4)/(3*d*Csc[c 
+ d*x]^(3/2))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.74, 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.

\(\displaystyle \int (a \sec (c+d x)+a)^2 (e \csc (c+d x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2 \left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2}dx\)

\(\Big \downarrow \) 4366

\(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {(\sec (c+d x) a+a)^2}{\sin ^{\frac {3}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^{3/2}}dx\)

\(\Big \downarrow \) 4360

\(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {(-\cos (c+d x) a-a)^2 \sec ^2(c+d x)}{\sin ^{\frac {3}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\left (a \sin \left (c+d x-\frac {\pi }{2}\right )-a\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^{3/2} \sin \left (c+d x-\frac {\pi }{2}\right )^2}dx\)

\(\Big \downarrow \) 3352

\(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \left (\frac {\sec ^2(c+d x) a^2}{\sin ^{\frac {3}{2}}(c+d x)}+\frac {2 \sec (c+d x) a^2}{\sin ^{\frac {3}{2}}(c+d x)}+\frac {a^2}{\sin ^{\frac {3}{2}}(c+d x)}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[(e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2,x]
 

Output:

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

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 = 1.99 (sec) , antiderivative size = 615, normalized size of antiderivative = 2.56

method	result	size
parts	\(-\frac {a^{2} \sqrt {2}\, \left (\sqrt {2}-\left (2 \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\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 )\right ) \sqrt {1+i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \sqrt {1-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \left (1+\cos \left (d x +c \right )\right )\right ) e \sqrt {e \csc \left (d x +c \right )}}{d}-\frac {a^{2} \sqrt {2}\, \left (\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 (-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 {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 (3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )\right )+\left (3 \cos \left (d x +c \right )-1\right ) \sqrt {2}\right ) e \sqrt {e \csc \left (d x +c \right )}\, \sec \left (d x +c \right )}{2 d}-\frac {2 a^{2} \left (2 \cos \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}+\sin \left (d x +c \right ) \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 )+\sin \left (d x +c \right ) \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 )+2 \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\right ) e \sqrt {e \csc \left (d x +c \right )}}{d \left (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}}}}\)	\(615\)
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 {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 ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \left (2 \cos \left (d x +c \right )^{2}+2 \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 {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \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 (-2 \cos \left (d x +c \right )^{2}-2 \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 {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 ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \left (-2 \cos \left (d x +c \right )^{2}-2 \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 )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \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 (-2 \cos \left (d x +c \right )^{2}-2 \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 )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \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 (-10 \cos \left (d x +c \right )^{2}-10 \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 )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \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 (5 \cos \left (d x +c \right )^{2}+5 \cos \left (d x +c \right )\right )+\left (9 \cos \left (d x +c \right )-1\right ) \sqrt {2}\right ) \sqrt {e \csc \left (d x +c \right )}\, e \sec \left (d x +c \right )}{2 d}\)	\(732\)

Input:

int((e*csc(d*x+c))^(3/2)*(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

-a^2/d*2^(1/2)*(2^(1/2)-(2*EllipticE((1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2),1 
/2*2^(1/2))-EllipticF((1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2),1/2*2^(1/2)))*(1 
+I*(-csc(d*x+c)+cot(d*x+c)))^(1/2)*(1-I*(-csc(d*x+c)+cot(d*x+c)))^(1/2)*(- 
I*(-csc(d*x+c)+cot(d*x+c)))^(1/2)*(1+cos(d*x+c)))*e*(e*csc(d*x+c))^(1/2)-1 
/2*a^2/d*2^(1/2)*((1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2)*(1-I*cot(d*x+c)+I*cs 
c(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))*(-6*cos(d*x+c)^2-6*cos(d*x+c))+(1+I*c 
ot(d*x+c)-I*csc(d*x+c))^(1/2)*(1-I*cot(d*x+c)+I*csc(d*x+c))^(1/2)*(-I*(-cs 
c(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))*(3*cos(d*x+c)^2+3*cos(d*x+c))+(3*cos(d*x+c)-1)*2^(1/2))*e*(e* 
csc(d*x+c))^(1/2)*sec(d*x+c)-2*a^2/d*(2*cos(d*x+c)*(sin(d*x+c)/(1+cos(d*x+ 
c))^2)^(1/2)+sin(d*x+c)*arctan(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)*arctanh(sin(d*x+c)*(sin(d*x+c)/(1+cos(d*x+ 
c))^2)^(1/2)/(-1+cos(d*x+c)))+2*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2))*e*(e* 
csc(d*x+c))^(1/2)/(1+cos(d*x+c))/(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.04 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx =\text {Too large to display} \] Input:

integrate((e*csc(d*x+c))^(3/2)*(a+a*sec(d*x+c))^2,x, algorithm="fricas")
 

Output:

[-1/4*(2*a^2*sqrt(-e)*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) - a^2*sqr 
t(-e)*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)*s 
qrt(-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)) + 10*a^2*sqrt(2*I*e)*e*cos(d*x + c)*weierstrassZeta(4, 0, weier 
strassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + 10*a^2*sqrt(-2*I*e) 
*e*cos(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + 
c) - I*sin(d*x + c))) + 4*(5*a^2*e*cos(d*x + c)^2 + 4*a^2*e*cos(d*x + c) - 
 a^2*e)*sqrt(e/sin(d*x + c)))/(d*cos(d*x + c)), 1/4*(2*a^2*e^(3/2)*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) + a^2*e^(3/2)*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*c 
os(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)) - 10*a^2*sqrt(2*I*e)*e*cos 
(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I 
*sin(d*x + c))) - 10*a^2*sqrt(-2*I*e)*e*cos(d*x + c)*weierstrassZeta(4, 0, 
 weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c))) - 4*(5*a^2*e...
 

Sympy [F(-1)]

Timed out. \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \] Input:

integrate((e*csc(d*x+c))**(3/2)*(a+a*sec(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \] Input:

integrate((e*csc(d*x+c))^(3/2)*(a+a*sec(d*x+c))^2,x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

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

integrate((e*csc(d*x+c))^(3/2)*(a+a*sec(d*x+c))^2,x, algorithm="giac")
 

Output:

integrate((e*csc(d*x + c))^(3/2)*(a*sec(d*x + c) + a)^2, x)
 

Mupad [F(-1)]

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

int((a + a/cos(c + d*x))^2*(e/sin(c + d*x))^(3/2),x)
 

Output:

int((a + a/cos(c + d*x))^2*(e/sin(c + d*x))^(3/2), x)
 

Reduce [F]

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

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

Output:

sqrt(e)*a**2*e*(int(sqrt(csc(c + d*x))*csc(c + d*x)*sec(c + d*x)**2,x) + 2 
*int(sqrt(csc(c + d*x))*csc(c + d*x)*sec(c + d*x),x) + int(sqrt(csc(c + d* 
x))*csc(c + d*x),x))