\(\int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx\) [251]

Optimal result

Integrand size = 25, antiderivative size = 296 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=-\frac {4 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {2 \cot (c+d x) \csc (c+d x) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sqrt {\sin (2 c+2 d x)}}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \] Output:

-4/3*cot(d*x+c)^3/a^2/d/(e*cot(d*x+c))^(3/2)+4/3*cot(d*x+c)^2*csc(d*x+c)/a 
^2/d/(e*cot(d*x+c))^(3/2)+2/3*cot(d*x+c)*csc(d*x+c)*InverseJacobiAM(c-1/4* 
Pi+d*x,2^(1/2))*sin(2*d*x+2*c)^(1/2)/a^2/d/(e*cot(d*x+c))^(3/2)-1/2*arctan 
(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/a^2/d/(e*cot(d*x+c))^(3/2)/tan(d*x+c 
)^(3/2)-1/2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/a^2/d/(e*cot(d*x+c) 
)^(3/2)/tan(d*x+c)^(3/2)-1/2*arctanh(2^(1/2)*tan(d*x+c)^(1/2)/(1+tan(d*x+c 
)))*2^(1/2)/a^2/d/(e*cot(d*x+c))^(3/2)/tan(d*x+c)^(3/2)
 

Mathematica [F]

\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx \] Input:

Integrate[1/((e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2),x]
 

Output:

Integrate[1/((e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2), x]
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.91, 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.

\(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 (e \cot (c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 (e \cot (c+d x))^{3/2}}dx\)

\(\Big \downarrow \) 4388

\(\displaystyle \frac {\int \frac {\tan ^{\frac {3}{2}}(c+d x)}{(\sec (c+d x) a+a)^2}dx}{\tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4376

\(\displaystyle \frac {\int \frac {(a-a \sec (c+d x))^2}{\tan ^{\frac {5}{2}}(c+d x)}dx}{a^4 \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\left (-\cot \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{a^4 \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}\)

\(\Big \downarrow \) 4374

\(\displaystyle \frac {\int \left (\frac {\sec ^2(c+d x) a^2}{\tan ^{\frac {5}{2}}(c+d x)}-\frac {2 \sec (c+d x) a^2}{\tan ^{\frac {5}{2}}(c+d x)}+\frac {a^2}{\tan ^{\frac {5}{2}}(c+d x)}\right )dx}{a^4 \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {a^2 \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}-\frac {4 a^2}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {a^2 \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {a^2 \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {4 a^2 \sec (c+d x)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{3 d \sqrt {\tan (c+d x)}}}{a^4 \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}\)

Input:

Int[1/((e*Cot[c + d*x])^(3/2)*(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[Tan[ 
c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (4*a^2)/(3*d*Tan[c + d*x]^(3/2) 
) + (4*a^2*Sec[c + d*x])/(3*d*Tan[c + d*x]^(3/2)) + (2*a^2*EllipticF[c - P 
i/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*Sqrt[Tan[c + d*x]] 
))/(a^4*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))
 

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 = 1.47 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.92

method	result	size
default	\(-\frac {\left (i \left (3 \cos \left (d x +c \right )^{2}+6 \cos \left (d x +c \right )+3\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 )}+\left (3 \cos \left (d x +c \right )^{2}+6 \cos \left (d x +c \right )+3\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 (-3 \cos \left (d x +c \right )^{2}-6 \cos \left (d x +c \right )-3\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 )}+\left (3 \cos \left (d x +c \right )^{2}+6 \cos \left (d x +c \right )+3\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 (-10 \cos \left (d x +c \right )^{2}-20 \cos \left (d x +c \right )-10\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 )-8 \cos \left (d x +c \right ) \sin \left (d x +c \right )\right ) \csc \left (d x +c \right )}{6 a^{2} d \left (1+\cos \left (d x +c \right )\right ) e \sqrt {e \cot \left (d x +c \right )}}\)	\(569\)

Input:

int(1/(e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/6/a^2/d*(I*(3*cos(d*x+c)^2+6*cos(d*x+c)+3)*EllipticPi((-cot(d*x+c)+csc( 
d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(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)+(3*cos(d* 
x+c)^2+6*cos(d*x+c)+3)*(-cot(d*x+c)+csc(d*x+c)+1)^(1/2)*(2*cot(d*x+c)-2*cs 
c(d*x+c)+2)^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*EllipticPi((-cot(d*x+c)+c 
sc(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))+I*(-3*cos(d*x+c)^2-6*cos(d*x+c)- 
3)*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-co 
t(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)+(3*cos(d*x+c)^2+6*cos(d*x+c)+3)*(-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))+( 
-10*cos(d*x+c)^2-20*cos(d*x+c)-10)*(-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((-c 
ot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2*2^(1/2))-8*cos(d*x+c)*sin(d*x+c))/(1+cos 
(d*x+c))/e/(e*cot(d*x+c))^(1/2)*csc(d*x+c)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate(1/(e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c))^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate(1/(e*cot(d*x+c))**(3/2)/(a+a*sec(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate(1/(e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c))^2,x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(1/(e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c))^2,x, algorithm="giac")
 

Output:

integrate(1/((e*cot(d*x + c))^(3/2)*(a*sec(d*x + c) + a)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \] Input:

int(1/((e*cot(c + d*x))^(3/2)*(a + a/cos(c + d*x))^2),x)
 

Output:

int(cos(c + d*x)^2/(a^2*(e*cot(c + d*x))^(3/2)*(cos(c + d*x) + 1)^2), x)
 

Reduce [F]

\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{2} \sec \left (d x +c \right )^{2}+2 \cot \left (d x +c \right )^{2} \sec \left (d x +c \right )+\cot \left (d x +c \right )^{2}}d x \right )}{a^{2} e^{2}} \] Input:

int(1/(e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c))^2,x)
 

Output:

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