\(\int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx\) [586]

Optimal result

Integrand size = 23, antiderivative size = 121 \[ \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx=-\frac {6 a d^4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}+\frac {6 a d^3 \sqrt {d \sec (e+f x)} \sin (e+f x)}{5 f}+\frac {2 a d (d \sec (e+f x))^{5/2} \sin (e+f x)}{5 f} \] Output:

-6/5*a*d^4*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))/f/cos(f*x+e)^(1/2)/(d*sec 
(f*x+e))^(1/2)+2/7*b*(d*sec(f*x+e))^(7/2)/f+6/5*a*d^3*(d*sec(f*x+e))^(1/2) 
*sin(f*x+e)/f+2/5*a*d*(d*sec(f*x+e))^(5/2)*sin(f*x+e)/f
 

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.57 \[ \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx=\frac {(d \sec (e+f x))^{7/2} \left (40 b-168 a \cos ^{\frac {7}{2}}(e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+70 a \sin (2 (e+f x))+21 a \sin (4 (e+f x))\right )}{140 f} \] Input:

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

Output:

((d*Sec[e + f*x])^(7/2)*(40*b - 168*a*Cos[e + f*x]^(7/2)*EllipticE[(e + f* 
x)/2, 2] + 70*a*Sin[2*(e + f*x)] + 21*a*Sin[4*(e + f*x)]))/(140*f)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 3967, 3042, 4255, 3042, 4255, 3042, 4258, 3042, 3119}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x))dx\)

\(\Big \downarrow \) 3967

\(\displaystyle a \int (d \sec (e+f x))^{7/2}dx+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle a \int \left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}dx+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\)

\(\Big \downarrow \) 4255

\(\displaystyle a \left (\frac {3}{5} d^2 \int (d \sec (e+f x))^{3/2}dx+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle a \left (\frac {3}{5} d^2 \int \left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\)

\(\Big \downarrow \) 4255

\(\displaystyle a \left (\frac {3}{5} d^2 \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-d^2 \int \frac {1}{\sqrt {d \sec (e+f x)}}dx\right )+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle a \left (\frac {3}{5} d^2 \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-d^2 \int \frac {1}{\sqrt {d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\)

\(\Big \downarrow \) 4258

\(\displaystyle a \left (\frac {3}{5} d^2 \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-\frac {d^2 \int \sqrt {\cos (e+f x)}dx}{\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\right )+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle a \left (\frac {3}{5} d^2 \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-\frac {d^2 \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\right )+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\)

\(\Big \downarrow \) 3119

\(\displaystyle a \left (\frac {3}{5} d^2 \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-\frac {2 d^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\right )+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\)

Input:

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

Output:

(2*b*(d*Sec[e + f*x])^(7/2))/(7*f) + a*((2*d*(d*Sec[e + f*x])^(5/2)*Sin[e 
+ f*x])/(5*f) + (3*d^2*((-2*d^2*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + 
 f*x]]*Sqrt[d*Sec[e + f*x]]) + (2*d*Sqrt[d*Sec[e + f*x]]*Sin[e + f*x])/f)) 
/5)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* 
(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3967

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( 
x_)]), x_Symbol] :> Simp[b*((d*Sec[e + f*x])^m/(f*m)), x] + Simp[a   Int[(d 
*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] 
|| NeQ[a^2 + b^2, 0])
 

rule 4255

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) 
  Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] 
&& IntegerQ[2*n]
 

rule 4258

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] 
)^n*Sin[c + d*x]^n   Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && 
 EqQ[n^2, 1/4]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 45.29 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.86

method	result	size
parts	\(\frac {2 a \sqrt {d \sec \left (f x +e \right )}\, d^{3} \left (3 \sin \left (f x +e \right )+\tan \left (f x +e \right )+\sec \left (f x +e \right ) \tan \left (f x +e \right )+i \left (3 \cos \left (f x +e \right )^{2}+6 \cos \left (f x +e \right )+3\right ) \operatorname {EllipticF}\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}+i \left (-3 \cos \left (f x +e \right )^{2}-6 \cos \left (f x +e \right )-3\right ) \operatorname {EllipticE}\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\right )}{5 f \left (1+\cos \left (f x +e \right )\right )}+\frac {2 b \left (d \sec \left (f x +e \right )\right )^{\frac {7}{2}}}{7 f}\)	\(225\)
default	\(\frac {2 \sqrt {d \sec \left (f x +e \right )}\, \left (21 i \left (-\cos \left (f x +e \right )^{2}-2 \cos \left (f x +e \right )-1\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, a \operatorname {EllipticE}\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )+21 i \left (\cos \left (f x +e \right )^{2}+2 \cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, a \operatorname {EllipticF}\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )+7 a \left (3 \sin \left (f x +e \right )+\tan \left (f x +e \right )+\sec \left (f x +e \right ) \tan \left (f x +e \right )\right )+5 b \left (\sec \left (f x +e \right )^{3}+\sec \left (f x +e \right )^{2}\right )\right ) d^{3}}{35 f \left (1+\cos \left (f x +e \right )\right )}\)	\(231\)

Input:

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

Output:

2/5*a/f*(d*sec(f*x+e))^(1/2)*d^3/(1+cos(f*x+e))*(3*sin(f*x+e)+tan(f*x+e)+s 
ec(f*x+e)*tan(f*x+e)+I*(3*cos(f*x+e)^2+6*cos(f*x+e)+3)*EllipticF(I*(csc(f* 
x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^( 
1/2)+I*(-3*cos(f*x+e)^2-6*cos(f*x+e)-3)*EllipticE(I*(csc(f*x+e)-cot(f*x+e) 
),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2))+2/7*b*(d* 
sec(f*x+e))^(7/2)/f
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.20 \[ \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx=\frac {-21 i \, \sqrt {2} a d^{\frac {7}{2}} \cos \left (f x + e\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 i \, \sqrt {2} a d^{\frac {7}{2}} \cos \left (f x + e\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (5 \, b d^{3} + 7 \, {\left (3 \, a d^{3} \cos \left (f x + e\right )^{3} + a d^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{35 \, f \cos \left (f x + e\right )^{3}} \] Input:

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

Output:

1/35*(-21*I*sqrt(2)*a*d^(7/2)*cos(f*x + e)^3*weierstrassZeta(-4, 0, weiers 
trassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 21*I*sqrt(2)*a*d^(7 
/2)*cos(f*x + e)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f 
*x + e) - I*sin(f*x + e))) + 2*(5*b*d^3 + 7*(3*a*d^3*cos(f*x + e)^3 + a*d^ 
3*cos(f*x + e))*sin(f*x + e))*sqrt(d/cos(f*x + e)))/(f*cos(f*x + e)^3)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((d*sec(f*x+e))^(7/2)*(a+b*tan(f*x+e)),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx=\frac {\sqrt {d}\, d^{3} \left (2 \sqrt {\sec \left (f x +e \right )}\, \sec \left (f x +e \right )^{3} b +7 \left (\int \sqrt {\sec \left (f x +e \right )}\, \sec \left (f x +e \right )^{3}d x \right ) a f \right )}{7 f} \] Input:

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

Output:

(sqrt(d)*d**3*(2*sqrt(sec(e + f*x))*sec(e + f*x)**3*b + 7*int(sqrt(sec(e + 
 f*x))*sec(e + f*x)**3,x)*a*f))/(7*f)