\(\int \sqrt {e \csc (c+d x)} (a+a \sec (c+d x))^2 \, dx\) [289]

Optimal result

Integrand size = 25, antiderivative size = 154 \[ \int \sqrt {e \csc (c+d x)} (a+a \sec (c+d x))^2 \, dx=\frac {2 a^2 \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {3 a^2 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{d}+\frac {a^2 \sqrt {e \csc (c+d x)} \tan (c+d x)}{d} \] Output:

2*a^2*arctan(sin(d*x+c)^(1/2))*(e*csc(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/d+2*a 
^2*arctanh(sin(d*x+c)^(1/2))*(e*csc(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/d+3*a^2 
*(e*csc(d*x+c))^(1/2)*InverseJacobiAM(1/2*c-1/4*Pi+1/2*d*x,2^(1/2))*sin(d* 
x+c)^(1/2)/d+a^2*(e*csc(d*x+c))^(1/2)*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 = 12.04 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.09 \[ \int \sqrt {e \csc (c+d x)} (a+a \sec (c+d x))^2 \, dx=-\frac {2 a^2 \cos ^5\left (\frac {1}{2} (c+d x)\right ) \sqrt {e \csc (c+d x)} \left (-1+2 \arctan \left (\sqrt {\csc (c+d x)}\right ) \sqrt {\cos ^2(c+d x)} \sqrt {\csc (c+d x)}-2 \text {arctanh}\left (\sqrt {\csc (c+d x)}\right ) \sqrt {\cos ^2(c+d x)} \sqrt {\csc (c+d x)}+3 \sqrt {-\cot ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\csc ^2(c+d x)\right )\right ) \sec (c+d x) \sec ^4\left (\frac {1}{2} \csc ^{-1}(\csc (c+d x))\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{d} \] Input:

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

Output:

(-2*a^2*Cos[(c + d*x)/2]^5*Sqrt[e*Csc[c + d*x]]*(-1 + 2*ArcTan[Sqrt[Csc[c 
+ d*x]]]*Sqrt[Cos[c + d*x]^2]*Sqrt[Csc[c + d*x]] - 2*ArcTanh[Sqrt[Csc[c + 
d*x]]]*Sqrt[Cos[c + d*x]^2]*Sqrt[Csc[c + d*x]] + 3*Sqrt[-Cot[c + d*x]^2]*H 
ypergeometric2F1[1/4, 1/2, 5/4, Csc[c + d*x]^2])*Sec[c + d*x]*Sec[ArcCsc[C 
sc[c + d*x]]/2]^4*Sin[(c + d*x)/2])/d
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.71, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4366

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4360

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3352

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

\(\Big \downarrow \) 2009

\(\displaystyle \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (\frac {2 a^2 \arctan \left (\sqrt {\sin (c+d x)}\right )}{d}+\frac {2 a^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d}+\frac {3 a^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d}+\frac {a^2 \sqrt {\sin (c+d x)} \sec (c+d x)}{d}\right )\)

Input:

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

Output:

Sqrt[e*Csc[c + d*x]]*((2*a^2*ArcTan[Sqrt[Sin[c + d*x]]])/d + (2*a^2*ArcTan 
h[Sqrt[Sin[c + d*x]]])/d + (3*a^2*EllipticF[(c - Pi/2 + d*x)/2, 2])/d + (a 
^2*Sec[c + d*x]*Sqrt[Sin[c + d*x]])/d)*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 = 1.90 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.52

method	result	size
parts	\(\frac {i a^{2} \left (1+\cos \left (d x +c \right )\right ) \sqrt {2}\, \sqrt {e \csc \left (d x +c \right )}\, \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {a^{2} \sqrt {2}\, \sqrt {e \csc \left (d x +c \right )}\, \left (i \left (1+\cos \left (d x +c \right )\right ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}+\sqrt {2}\, \tan \left (d x +c \right )\right )}{2 d}-\frac {2 a^{2} \sqrt {e \csc \left (d x +c \right )}\, \left (\arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right )-\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right )\right ) \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{d \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}\)	\(388\)
default	\(\frac {a^{2} \sqrt {2}\, \sqrt {e \csc \left (d x +c \right )}\, \left (i \left (-1-\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 )}+i \left (2 \cos \left (d x +c \right )+2\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 )+i \left (2 \cos \left (d x +c \right )+2\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 (2 \cos \left (d x +c \right )+2\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 (-2 \cos \left (d x +c \right )-2\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 )+\sqrt {2}\, \tan \left (d x +c \right )\right )}{2 d}\)	\(566\)

Input:

int((e*csc(d*x+c))^(1/2)*(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

I*a^2/d*(1+cos(d*x+c))*2^(1/2)*(e*csc(d*x+c))^(1/2)*(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)*EllipticF((1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2),1/2*2^(1/2))+1/2 
*a^2/d*2^(1/2)*(e*csc(d*x+c))^(1/2)*(I*(1+cos(d*x+c))*(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)*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)+2 
^(1/2)*tan(d*x+c))-2*a^2/d*(e*csc(d*x+c))^(1/2)*(arctan(sin(d*x+c)*(sin(d* 
x+c)/(1+cos(d*x+c))^2)^(1/2)/(-1+cos(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))))/(sin(d*x+c)/(1+cos(d*x+c))^2) 
^(1/2)*(-csc(d*x+c)+cot(d*x+c))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 664, normalized size of antiderivative = 4.31 \[ \int \sqrt {e \csc (c+d x)} (a+a \sec (c+d x))^2 \, dx =\text {Too large to display} \] Input:

integrate((e*csc(d*x+c))^(1/2)*(a+a*sec(d*x+c))^2,x, algorithm="fricas")
 

Output:

[-1/4*(2*a^2*sqrt(-e)*arctan(-1/4*(cos(d*x + c)^2 - 6*sin(d*x + c) - 2)*sq 
rt(-e)*sqrt(e/sin(d*x + c))/(e*sin(d*x + c) + e))*cos(d*x + c) - 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)) + 6*I*a^2*sqrt(2*I*e)*cos(d*x + c)*weierstrassPInverse(4, 0, cos(d* 
x + c) + I*sin(d*x + c)) - 6*I*a^2*sqrt(-2*I*e)*cos(d*x + c)*weierstrassPI 
nverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) - 4*a^2*sqrt(e/sin(d*x + c))*s 
in(d*x + c))/(d*cos(d*x + c)), -1/4*(2*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) - a^2*sqrt(e)*cos(d*x + c)*log((e*cos(d*x + c)^4 - 72*e*co 
s(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)) + 6*I*a^2*sqrt(2*I*e)*cos(d*x + c)*weierst 
rassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) - 6*I*a^2*sqrt(-2*I*e)*c 
os(d*x + c)*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) - 4*a 
^2*sqrt(e/sin(d*x + c))*sin(d*x + c))/(d*cos(d*x + c))]
 

Sympy [F]

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

integrate((e*csc(d*x+c))**(1/2)*(a+a*sec(d*x+c))**2,x)
 

Output:

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

Maxima [F]

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

integrate((e*csc(d*x+c))^(1/2)*(a+a*sec(d*x+c))^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*csc(d*x+c))^(1/2)*(a+a*sec(d*x+c))^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((a + a/cos(c + d*x))^2*(e/sin(c + d*x))^(1/2),x)
 

Output:

int((a + a/cos(c + d*x))^2*(e/sin(c + d*x))^(1/2), x)
 

Reduce [F]

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

int((e*csc(d*x+c))^(1/2)*(a+a*sec(d*x+c))^2,x)
 

Output:

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