\(\int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx\) [253]

Optimal result

Integrand size = 25, antiderivative size = 257 \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\frac {2 \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{7/2}}-\frac {2 \cot ^3(c+d x) \csc (c+d x) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sqrt {\sin (2 c+2 d x)}}{a^2 d (e \cot (c+d x))^{7/2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{7/2} \tan ^{\frac {7}{2}}(c+d x)}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{7/2} \tan ^{\frac {7}{2}}(c+d x)}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{7/2} \tan ^{\frac {7}{2}}(c+d x)} \] Output:

2*cot(d*x+c)^3/a^2/d/(e*cot(d*x+c))^(7/2)-2*cot(d*x+c)^3*csc(d*x+c)*Invers 
eJacobiAM(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))^ 
(7/2)+1/2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/a^2/d/(e*cot(d*x+c)) 
^(7/2)/tan(d*x+c)^(7/2)+1/2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/a^2 
/d/(e*cot(d*x+c))^(7/2)/tan(d*x+c)^(7/2)+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))^(7/2)/tan(d*x+c)^(7/2)
 

Mathematica [F]

\[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx \] Input:

Integrate[1/((e*Cot[c + d*x])^(7/2)*(a + a*Sec[c + d*x])^2),x]
 

Output:

Integrate[1/((e*Cot[c + d*x])^(7/2)*(a + a*Sec[c + d*x])^2), x]
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.93, 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.

\(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 (e \cot (c+d x))^{7/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 (e \cot (c+d x))^{7/2}}dx\)

\(\Big \downarrow \) 4388

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4376

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4374

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {a^2 \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 a^2 \sqrt {\tan (c+d x)}}{d}-\frac {a^2 \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {a^2 \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {2 a^2 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}}{a^4 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\)

Input:

Int[1/((e*Cot[c + d*x])^(7/2)*(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[T 
an[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)/(a^4*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2))
 

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] (verified)

Result contains complex when optimal does not.

Time = 1.84 (sec) , antiderivative size = 551, normalized size of antiderivative = 2.14

method	result	size
default	\(\frac {\sqrt {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 )}\, \left (1+\cos \left (d x +c \right )\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 )}\, \left (-1-\cos \left (d x +c \right )\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 {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \left (1+\cos \left (d x +c \right )\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 {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \left (1+\cos \left (d x +c \right )\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 )-6 \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 (1+\cos \left (d x +c \right )\right ) \operatorname {EllipticF}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+4 \sin \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}}}\, \csc \left (d x +c \right )}{4 a^{2} d \,e^{3} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}\)	\(551\)

Input:

int(1/(e*cot(d*x+c))^(7/2)/(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/4/a^2/d*2^(1/2)*(I*(-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)*(1+cos(d*x+c))*EllipticPi(( 
-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))+I*(-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)*(-1-cos(d*x+c))*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)*(-csc(d*x+c)+cot(d*x+c) 
)^(1/2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(1+cos(d*x+c))*EllipticPi((-co 
t(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)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1 
/2)*(1+cos(d*x+c))*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2+1/2*I,1 
/2*2^(1/2))-6*(-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)*(1+cos(d*x+c))*EllipticF((-cot(d*x 
+c)+csc(d*x+c)+1)^(1/2),1/2*2^(1/2))+4*sin(d*x+c))*(-2*sin(d*x+c)*cos(d*x+ 
c)/(1+cos(d*x+c))^2)^(1/2)/e^3/(e*cot(d*x+c))^(1/2)/(-sin(d*x+c)*cos(d*x+c 
)/(1+cos(d*x+c))^2)^(1/2)*csc(d*x+c)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate(1/(e*cot(d*x+c))^(7/2)/(a+a*sec(d*x+c))^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate(1/(e*cot(d*x+c))**(7/2)/(a+a*sec(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate(1/(e*cot(d*x+c))^(7/2)/(a+a*sec(d*x+c))^2,x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(1/(e*cot(d*x+c))^(7/2)/(a+a*sec(d*x+c))^2,x, algorithm="giac")
 

Output:

integrate(1/((e*cot(d*x + c))^(7/2)*(a*sec(d*x + c) + a)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{7/2}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \] Input:

int(1/((e*cot(c + d*x))^(7/2)*(a + a/cos(c + d*x))^2),x)
 

Output:

int(cos(c + d*x)^2/(a^2*(e*cot(c + d*x))^(7/2)*(cos(c + d*x) + 1)^2), x)
 

Reduce [F]

\[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}+2 \cot \left (d x +c \right )^{4} \sec \left (d x +c \right )+\cot \left (d x +c \right )^{4}}d x \right )}{a^{2} e^{4}} \] Input:

int(1/(e*cot(d*x+c))^(7/2)/(a+a*sec(d*x+c))^2,x)
 

Output:

(sqrt(e)*int(sqrt(cot(c + d*x))/(cot(c + d*x)**4*sec(c + d*x)**2 + 2*cot(c 
 + d*x)**4*sec(c + d*x) + cot(c + d*x)**4),x))/(a**2*e**4)