\(\int \frac {(a+a \sec (c+d x))^2}{(e \tan (c+d x))^{7/2}} \, dx\) [117]

Optimal result

Integrand size = 25, antiderivative size = 312 \[ \int \frac {(a+a \sec (c+d x))^2}{(e \tan (c+d x))^{7/2}} \, dx=-\frac {a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{7/2}}+\frac {a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{7/2}}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}+\sqrt {e} \tan (c+d x)}\right )}{\sqrt {2} d e^{7/2}}-\frac {4 a^2}{5 d e (e \tan (c+d x))^{5/2}}-\frac {4 a^2 \sec (c+d x)}{5 d e (e \tan (c+d x))^{5/2}}+\frac {2 a^2}{d e^3 \sqrt {e \tan (c+d x)}}+\frac {12 a^2 \cos (c+d x)}{5 d e^3 \sqrt {e \tan (c+d x)}}+\frac {12 a^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d e^4 \sqrt {\sin (2 c+2 d x)}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 12.51 (sec) , antiderivative size = 2820, normalized size of antiderivative = 9.04 \[ \int \frac {(a+a \sec (c+d x))^2}{(e \tan (c+d x))^{7/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

(Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*((7*Cot[c/2])/(10*d) - (Cot[c 
/2]*Csc[c/2 + (d*x)/2]^2)/(20*d) - (3*(4*Cos[c/2] - Cos[(3*c)/2] + Cos[(5* 
c)/2])*Cos[d*x]*Sec[2*c]*Sin[c/2])/(10*d*(-1 + 2*Cos[c])) - (7*Csc[c/2]*Cs 
c[c/2 + (d*x)/2]*Sin[(d*x)/2])/(10*d) + (Csc[c/2]*Csc[c/2 + (d*x)/2]^3*Sin 
[(d*x)/2])/(20*d) - (3*(2 - 5*Cos[c] + 6*Cos[2*c] + Cos[3*c])*Sec[2*c]*Sin 
[d*x])/(20*d*(-1 + 2*Cos[c])))*Sin[c + d*x]^2*Tan[c + d*x]^2)/(e*Tan[c + d 
*x])^(7/2) + ((E^((2*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + 
 E^((4*I)*(c + d*x))]] - 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I 
)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d 
*x)))]])*Cos[c + d*x]^2*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x]) 
^2*Tan[c + d*x]^(7/2))/(16*d*E^(I*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))) 
)/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e* 
Tan[c + d*x])^(7/2)) + ((-(E^((4*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcT 
an[Sqrt[-1 + E^((4*I)*(c + d*x))]]) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqr 
t[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^ 
((2*I)*(c + d*x)))]])*Cos[c + d*x]^2*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a* 
Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(16*d*E^((2*I)*c)*Sqrt[((-I)*(-1 + E^( 
(2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(- 
1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) - ((-(E^((6*I)*c)*Sqrt[-1 + E^((4*I) 
*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]]) + 2*Sqrt[-1 + E^((...
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4374

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + a*Sec[c + d*x])^2/(e*Tan[c + d*x])^(7/2),x]
 

Output:

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

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.50

method	result	size
parts	\(\frac {2 a^{2} e \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {e \tan \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \tan \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \tan \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \tan \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{4} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {1}{5 e^{2} \left (e \tan \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {1}{e^{4} \sqrt {e \tan \left (d x +c \right )}}\right )}{d}-\frac {2 a^{2}}{5 d e \left (e \tan \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {a^{2} \sqrt {2}\, \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 (\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 (-6-6 \sec \left (d x +c \right )\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 (3+3 \sec \left (d x +c \right )\right )+6-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )\right )}{5 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}}}\, \sqrt {e \tan \left (d x +c \right )}\, e^{3}}\)	\(468\)
default	\(-\frac {a^{2} \sqrt {2}\, \left (5 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 ) \sin \left (d x +c \right )^{2}-5 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 ) \sin \left (d x +c \right )^{2}-5 \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 ) \sin \left (d x +c \right )^{2}-5 \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 ) \sin \left (d x +c \right )^{2}-24 \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 ) \sin \left (d x +c \right )^{2}+12 \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 ) \sin \left (d x +c \right )^{2}-52 \cos \left (d x +c \right )^{2}+44 \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}}}\, \sec \left (d x +c \right )}{20 d \left (-1+\cos \left (d x +c \right )\right ) \sqrt {e \tan \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}}}\, e^{3}}\)	\(660\)

Input:

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

Output:

2*a^2/d*e*(1/8/e^4/(e^2)^(1/4)*2^(1/2)*(ln((e*tan(d*x+c)-(e^2)^(1/4)*(e*ta 
n(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*tan(d*x+c)+(e^2)^(1/4)*(e*tan(d*x+ 
c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*tan(d*x+c) 
)^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*tan(d*x+c))^(1/2)+1))-1/5/e^2/ 
(e*tan(d*x+c))^(5/2)+1/e^4/(e*tan(d*x+c))^(1/2))-2/5*a^2/d/e/(e*tan(d*x+c) 
)^(5/2)+1/5*a^2/d*2^(1/2)*(-2*sin(d*x+c)*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2 
)/(-sin(d*x+c)*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/(e*tan(d*x+c))^(1/2)/e^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)*(-c 
sc(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))*(-6-6*sec(d*x+c))+(-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))*(3+3*sec(d*x+c))+6-2*csc(d*x+c)*cot(d* 
x+c))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+a \sec (c+d x))^2}{(e \tan (c+d x))^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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