\(\int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2 \, dx\) [240]

Optimal result

Integrand size = 25, antiderivative size = 247 \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2 \, dx=\frac {2 a^2 \sqrt {e \cot (c+d x)} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{d}-\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} d}+\frac {a^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} d}+\frac {a^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} d}+\frac {2 a^2 \sqrt {e \cot (c+d x)} \tan (c+d x)}{d} \] Output:

2*a^2*(e*cot(d*x+c))^(1/2)*InverseJacobiAM(c-1/4*Pi+d*x,2^(1/2))*sec(d*x+c 
)*sin(2*d*x+2*c)^(1/2)/d+1/2*a^2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*(e*co 
t(d*x+c))^(1/2)*tan(d*x+c)^(1/2)*2^(1/2)/d+1/2*a^2*arctan(1+2^(1/2)*tan(d* 
x+c)^(1/2))*(e*cot(d*x+c))^(1/2)*tan(d*x+c)^(1/2)*2^(1/2)/d+1/2*a^2*arctan 
h(2^(1/2)*tan(d*x+c)^(1/2)/(1+tan(d*x+c)))*(e*cot(d*x+c))^(1/2)*tan(d*x+c) 
^(1/2)*2^(1/2)/d+2*a^2*(e*cot(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 = 11.69 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.48 \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2 \, dx=\frac {a^2 e (1+\cos (c+d x))^2 \left (3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\cot ^2(c+d x)\right )+6 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\tan ^2(c+d x)\right )-2 \cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right ) \sec ^4\left (\frac {1}{2} \cot ^{-1}(\cot (c+d x))\right )}{6 d \sqrt {e \cot (c+d x)}} \] Input:

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

Output:

(a^2*e*(1 + Cos[c + d*x])^2*(3*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d* 
x]^2] + 6*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[c + d*x]^2] - 2*Cot[c + d* 
x]^2*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])*Sec[ArcCot[Cot[c + d 
*x]]/2]^4)/(6*d*Sqrt[e*Cot[c + d*x]])
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.96, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4388

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4374

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

\(\Big \downarrow \) 2009

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (-\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {a^2 \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 a^2 \sqrt {\tan (c+d x)}}{d}-\frac {a^2 \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {a^2 \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 a^2 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )\)

Input:

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

Output:

Sqrt[e*Cot[c + d*x]]*(-((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* 
Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (a^2*L 
og[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*a^2* 
EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(d*Sqrt[ 
Tan[c + d*x]]) + (2*a^2*Sqrt[Tan[c + d*x]])/d)*Sqrt[Tan[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 4374

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]
 

rule 4388

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]
 

Maple [A] (verified)

Time = 2.17 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.07

method	result	size
parts	\(-\frac {a^{2} e \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {2 a^{2} e}{d \sqrt {e \cot \left (d x +c \right )}}+\frac {2 a^{2} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (1+\sec \left (d x +c \right )\right )}{d}\)	\(264\)
default	\(-\frac {a^{2} \sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}\, \left (1-\cos \left (d x +c \right )\right ) \sqrt {-\frac {2 \sin \left (d x +c \right ) \cos \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \left (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 )-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 )+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 )+\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 )+\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 )+4 \csc \left (d x +c \right )-4 \cot \left (d x +c \right )\right ) \csc \left (d x +c \right )}{4 d \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}\right )^{\frac {3}{2}}}\)	\(524\)

Input:

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

Output:

-1/4*a^2/d*e/(e^2)^(1/4)*2^(1/2)*(ln((e*cot(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*a^2/d*e/(e*c 
ot(d*x+c))^(1/2)+2*a^2/d*(e*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)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*El 
lipticF((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2*2^(1/2))*(1+sec(d*x+c))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2 \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*cot(d*x+c))^(1/2)*(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
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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