\(\int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx\) [250]

Optimal result

Integrand size = 25, antiderivative size = 350 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\frac {2 \cot (c+d x)}{a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {4 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{5 a^2 d \sqrt {e \cot (c+d x)} \sqrt {\sin (2 c+2 d x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right )}{\sqrt {2} a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \] Output:

2*cot(d*x+c)/a^2/d/(e*cot(d*x+c))^(1/2)-12/5*cos(d*x+c)*cot(d*x+c)/a^2/d/( 
e*cot(d*x+c))^(1/2)-4/5*cot(d*x+c)^3/a^2/d/(e*cot(d*x+c))^(1/2)+4/5*cot(d* 
x+c)^2*csc(d*x+c)/a^2/d/(e*cot(d*x+c))^(1/2)+12/5*cos(d*x+c)*EllipticE(cos 
(c+1/4*Pi+d*x),2^(1/2))/a^2/d/(e*cot(d*x+c))^(1/2)/sin(2*d*x+2*c)^(1/2)+1/ 
2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/a^2/d/(e*cot(d*x+c))^(1/2)/t 
an(d*x+c)^(1/2)+1/2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/a^2/d/(e*co 
t(d*x+c))^(1/2)/tan(d*x+c)^(1/2)-1/2*arctanh(2^(1/2)*tan(d*x+c)^(1/2)/(1+t 
an(d*x+c)))*2^(1/2)/a^2/d/(e*cot(d*x+c))^(1/2)/tan(d*x+c)^(1/2)
 

Mathematica [F]

\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx \] Input:

Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])^2),x]
 

Output:

Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])^2), x]
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 4388, 3042, 4376, 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.

\(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 \sqrt {e \cot (c+d x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}dx\)

\(\Big \downarrow \) 4388

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4376

\(\displaystyle \frac {\int \frac {(a-a \sec (c+d x))^2}{\tan ^{\frac {7}{2}}(c+d x)}dx}{a^4 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\left (-\cot \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{a^4 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\)

\(\Big \downarrow \) 4374

\(\displaystyle \frac {\int \left (\frac {\sec ^2(c+d x) a^2}{\tan ^{\frac {7}{2}}(c+d x)}-\frac {2 \sec (c+d x) a^2}{\tan ^{\frac {7}{2}}(c+d x)}+\frac {a^2}{\tan ^{\frac {7}{2}}(c+d x)}\right )dx}{a^4 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {a^2 \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}-\frac {4 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 a^2}{d \sqrt {\tan (c+d x)}}+\frac {a^2 \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {a^2 \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {12 a^2 \cos (c+d x)}{5 d \sqrt {\tan (c+d x)}}+\frac {4 a^2 \sec (c+d x)}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {12 a^2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{5 d \sqrt {\sin (2 c+2 d x)}}}{a^4 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\)

Input:

Int[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])^2),x]
 

Output:

(-((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*Log[1 + Sqrt[2]*Sqrt[T 
an[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (4*a^2)/(5*d*Tan[c + d*x]^(5 
/2)) + (4*a^2*Sec[c + d*x])/(5*d*Tan[c + d*x]^(5/2)) + (2*a^2)/(d*Sqrt[Tan 
[c + d*x]]) - (12*a^2*Cos[c + d*x])/(5*d*Sqrt[Tan[c + d*x]]) - (12*a^2*Cos 
[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[Tan[c + d*x]])/(5*d*Sqrt[Sin[2 
*c + 2*d*x]]))/(a^4*Sqrt[e*Cot[c + d*x]]*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 4376

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( 
a_))^(n_), x_Symbol] :> Simp[a^(2*n)/e^(2*n)   Int[(e*Cot[c + d*x])^(m + 2* 
n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a 
^2 - b^2, 0] && ILtQ[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 [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.02 (sec) , antiderivative size = 669, normalized size of antiderivative = 1.91

method	result	size
default	\(\frac {\left (i \left (5 \cos \left (d x +c \right )^{2}+10 \cos \left (d x +c \right )+5\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 )+i \left (-5 \cos \left (d x +c \right )^{2}-10 \cos \left (d x +c \right )-5\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 )+\left (-5 \cos \left (d x +c \right )^{2}-10 \cos \left (d x +c \right )-5\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 )+\left (-5 \cos \left (d x +c \right )^{2}-10 \cos \left (d x +c \right )-5\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 )+\left (24 \cos \left (d x +c \right )^{2}+48 \cos \left (d x +c \right )+24\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 {EllipticE}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+\left (-12 \cos \left (d x +c \right )^{2}-24 \cos \left (d x +c \right )-12\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 )+\cos \left (d x +c \right ) \left (4 \cos \left (d x +c \right )-4\right )\right ) \csc \left (d x +c \right )}{10 a^{2} d \left (1+\cos \left (d x +c \right )\right ) \sqrt {e \cot \left (d x +c \right )}}\)	\(669\)

Input:

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

Output:

1/10/a^2/d*(I*(5*cos(d*x+c)^2+10*cos(d*x+c)+5)*(-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)*E 
llipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))+I*(-5*co 
s(d*x+c)^2-10*cos(d*x+c)-5)*(-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)*EllipticPi((-cot(d*x 
+c)+csc(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+(-5*cos(d*x+c)^2-10*cos(d*x 
+c)-5)*(-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)*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/ 
2),1/2-1/2*I,1/2*2^(1/2))+(-5*cos(d*x+c)^2-10*cos(d*x+c)-5)*(-cot(d*x+c)+c 
sc(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)*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1 
/2))+(24*cos(d*x+c)^2+48*cos(d*x+c)+24)*(-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)*Elliptic 
E((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2*2^(1/2))+(-12*cos(d*x+c)^2-24*cos(d 
*x+c)-12)*(-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)*EllipticF((-cot(d*x+c)+csc(d*x+c)+1)^( 
1/2),1/2*2^(1/2))+cos(d*x+c)*(4*cos(d*x+c)-4))/(1+cos(d*x+c))/(e*cot(d*x+c 
))^(1/2)*csc(d*x+c)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {1}{\sqrt {e \cot {\left (c + d x \right )}} \sec ^{2}{\left (c + d x \right )} + 2 \sqrt {e \cot {\left (c + d x \right )}} \sec {\left (c + d x \right )} + \sqrt {e \cot {\left (c + d x \right )}}}\, dx}{a^{2}} \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\sqrt {e \cot \left (d x + c\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\sqrt {e \cot \left (d x + c\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right ) \sec \left (d x +c \right )^{2}+2 \cot \left (d x +c \right ) \sec \left (d x +c \right )+\cot \left (d x +c \right )}d x \right )}{a^{2} e} \] Input:

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

Output:

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