\(\int \frac {(e \tan (c+d x))^{13/2}}{(a+a \sec (c+d x))^2} \, dx\) [127]

Optimal result

Integrand size = 25, antiderivative size = 313 \[ \int \frac {(e \tan (c+d x))^{13/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {e^{13/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}-\frac {e^{13/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}+\frac {e^{13/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}+\sqrt {e} \tan (c+d x)}\right )}{\sqrt {2} a^2 d}-\frac {12 e^6 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{5 a^2 d \sqrt {\sin (2 c+2 d x)}}+\frac {2 e^5 (e \tan (c+d x))^{3/2}}{3 a^2 d}+\frac {12 e^5 \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 a^2 d}-\frac {4 e^5 \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{7/2}}{7 a^2 d} \] Output:

1/2*e^(13/2)*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*2^(1/2)/a^2/d- 
1/2*e^(13/2)*arctan(1+2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*2^(1/2)/a^2/d+ 
1/2*e^(13/2)*arctanh(2^(1/2)*(e*tan(d*x+c))^(1/2)/(e^(1/2)+e^(1/2)*tan(d*x 
+c)))*2^(1/2)/a^2/d+12/5*e^6*cos(d*x+c)*EllipticE(cos(c+1/4*Pi+d*x),2^(1/2 
))*(e*tan(d*x+c))^(1/2)/a^2/d/sin(2*d*x+2*c)^(1/2)+2/3*e^5*(e*tan(d*x+c))^ 
(3/2)/a^2/d+12/5*e^5*cos(d*x+c)*(e*tan(d*x+c))^(3/2)/a^2/d-4/5*e^5*sec(d*x 
+c)*(e*tan(d*x+c))^(3/2)/a^2/d+2/7*e^3*(e*tan(d*x+c))^(7/2)/a^2/d
 

Mathematica [F]

\[ \int \frac {(e \tan (c+d x))^{13/2}}{(a+a \sec (c+d x))^2} \, dx=\int \frac {(e \tan (c+d x))^{13/2}}{(a+a \sec (c+d x))^2} \, dx \] Input:

Integrate[(e*Tan[c + d*x])^(13/2)/(a + a*Sec[c + d*x])^2,x]
 

Output:

Integrate[(e*Tan[c + d*x])^(13/2)/(a + a*Sec[c + d*x])^2, x]
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {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.

\(\displaystyle \int \frac {(e \tan (c+d x))^{13/2}}{(a \sec (c+d x)+a)^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (-e \cot \left (c+d x+\frac {\pi }{2}\right )\right )^{13/2}}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2}dx\)

\(\Big \downarrow \) 4376

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4374

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^4 \left (\frac {a^2 e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a^2 e^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d}-\frac {a^2 e^{5/2} \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}+\frac {a^2 e^{5/2} \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}-\frac {12 a^2 e^2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d \sqrt {\sin (2 c+2 d x)}}+\frac {2 a^2 (e \tan (c+d x))^{7/2}}{7 d e}+\frac {2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}+\frac {12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}-\frac {4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}\right )}{a^4}\)

Input:

Int[(e*Tan[c + d*x])^(13/2)/(a + a*Sec[c + d*x])^2,x]
 

Output:

(e^4*((a^2*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sq 
rt[2]*d) - (a^2*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]] 
)/(Sqrt[2]*d) - (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]* 
Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]* 
Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (12*a^2*e^2* 
Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*d*Sqrt[ 
Sin[2*c + 2*d*x]]) + (2*a^2*e*(e*Tan[c + d*x])^(3/2))/(3*d) + (12*a^2*e*Co 
s[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*d) - (4*a^2*e*Sec[c + d*x]*(e*Tan[c 
+ d*x])^(3/2))/(5*d) + (2*a^2*(e*Tan[c + d*x])^(7/2))/(7*d*e)))/a^4
 

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]
 

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.77 (sec) , antiderivative size = 734, normalized size of antiderivative = 2.35

method	result	size
default	\(-\frac {\sqrt {2}\, e^{6} \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}}}\, \sqrt {e \tan \left (d x +c \right )}\, \left (105 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 ) \left (\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )+105 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 ) \left (-\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+252 \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}\, \left (\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )+105 \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 (-\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+105 \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 (-\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+504 \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 (-\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+584 \cot \left (d x +c \right )-672 \csc \left (d x +c \right )-20 \sec \left (d x +c \right ) \csc \left (d x +c \right )+168 \sec \left (d x +c \right )^{2} \csc \left (d x +c \right )-60 \sec \left (d x +c \right )^{3} \csc \left (d x +c \right )\right )}{420 a^{2} d \sqrt {-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}\)	\(734\)

Input:

int((e*tan(d*x+c))^(13/2)/(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/420/a^2/d*2^(1/2)*e^6*(-2*sin(d*x+c)*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2) 
*(e*tan(d*x+c))^(1/2)/(-sin(d*x+c)*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*(105 
*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)*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1 
/2-1/2*I,1/2*2^(1/2))*(csc(d*x+c)+cot(d*x+c))+105*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)*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-c 
ot(d*x+c)-csc(d*x+c))+252*(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)) 
*(-cot(d*x+c)+csc(d*x+c)+1)^(1/2)*(csc(d*x+c)+cot(d*x+c))+105*(-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))*(-cot(d*x+c)-csc(d*x+c))+105*(-cot(d*x+c)+csc(d*x+c)+1)^(1/2)*(2*co 
t(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))*(-cot(d*x+c)-csc(d* 
x+c))+504*(-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)*EllipticE((-cot(d*x+c)+csc(d*x+c)+1)^( 
1/2),1/2*2^(1/2))*(-cot(d*x+c)-csc(d*x+c))+584*cot(d*x+c)-672*csc(d*x+c)-2 
0*sec(d*x+c)*csc(d*x+c)+168*sec(d*x+c)^2*csc(d*x+c)-60*sec(d*x+c)^3*csc(d* 
x+c))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(e \tan (c+d x))^{13/2}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate((e*tan(d*x+c))^(13/2)/(a+a*sec(d*x+c))^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e \tan (c+d x))^{13/2}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate((e*tan(d*x+c))**(13/2)/(a+a*sec(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {(e \tan (c+d x))^{13/2}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate((e*tan(d*x+c))^(13/2)/(a+a*sec(d*x+c))^2,x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {(e \tan (c+d x))^{13/2}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\left (e \tan \left (d x + c\right )\right )^{\frac {13}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate((e*tan(d*x+c))^(13/2)/(a+a*sec(d*x+c))^2,x, algorithm="giac")
 

Output:

integrate((e*tan(d*x + c))^(13/2)/(a*sec(d*x + c) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \tan (c+d x))^{13/2}}{(a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^{13/2}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \] Input:

int((e*tan(c + d*x))^(13/2)/(a + a/cos(c + d*x))^2,x)
 

Output:

int((cos(c + d*x)^2*(e*tan(c + d*x))^(13/2))/(a^2*(cos(c + d*x) + 1)^2), x 
)
 

Reduce [F]

\[ \int \frac {(e \tan (c+d x))^{13/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{6}}{\sec \left (d x +c \right )^{2}+2 \sec \left (d x +c \right )+1}d x \right ) e^{6}}{a^{2}} \] Input:

int((e*tan(d*x+c))^(13/2)/(a+a*sec(d*x+c))^2,x)
 

Output:

(sqrt(e)*int((sqrt(tan(c + d*x))*tan(c + d*x)**6)/(sec(c + d*x)**2 + 2*sec 
(c + d*x) + 1),x)*e**6)/a**2