\(\int \sec ^3(e+f x) (a+b \sec ^2(e+f x))^{3/2} \, dx\) [242]

Optimal result

Integrand size = 25, antiderivative size = 375 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{15 b f}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{15 b f \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}}}+\frac {(a+b) (9 a+8 b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{15 f \left (a+b-a \sin ^2(e+f x)\right )}+\frac {2 (3 a+2 b) \sec (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{15 f}+\frac {b \sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{5 f} \] Output:

1/15*(3*a^2+13*a*b+8*b^2)*sin(f*x+e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^( 
1/2)/b/f-1/15*(3*a^2+13*a*b+8*b^2)*(cos(f*x+e)^2)^(1/2)*EllipticE(sin(f*x+ 
e),(a/(a+b))^(1/2))*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)/b/f/((a+b-a* 
sin(f*x+e)^2)/(a+b))^(1/2)+1/15*(a+b)*(9*a+8*b)*(cos(f*x+e)^2)^(1/2)*Ellip 
ticF(sin(f*x+e),(a/(a+b))^(1/2))*((a+b-a*sin(f*x+e)^2)/(a+b))^(1/2)*(sec(f 
*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)/f/(a+b-a*sin(f*x+e)^2)+2/15*(3*a+2*b)* 
sec(f*x+e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)*tan(f*x+e)/f+1/5*b*se 
c(f*x+e)^3*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)*tan(f*x+e)/f
 

Mathematica [F]

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

Integrate[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2),x]
 

Output:

Integrate[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2), x]
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4636, 2057, 2058, 315, 25, 402, 27, 402, 25, 27, 399, 323, 321, 330, 327}

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.

\(\displaystyle \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4636

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 315

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}-\frac {1}{5} \int -\frac {(a+b) (5 a+4 b)-a (5 a+3 b) \sin ^2(e+f x)}{\left (1-\sin ^2(e+f x)\right )^{5/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \int \frac {(a+b) (5 a+4 b)-a (5 a+3 b) \sin ^2(e+f x)}{\left (1-\sin ^2(e+f x)\right )^{5/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {\int \frac {b \left ((a+b) (9 a+8 b)-2 a (3 a+2 b) \sin ^2(e+f x)\right )}{\left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{3 b}+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {(a+b) (9 a+8 b)-2 a (3 a+2 b) \sin ^2(e+f x)}{\left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\int -\frac {a \left ((a+b) (3 a+4 b)-\left (3 a^2+13 b a+8 b^2\right ) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}+\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {\int \frac {a \left ((a+b) (3 a+4 b)-\left (3 a^2+13 b a+8 b^2\right ) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \int \frac {(a+b) (3 a+4 b)-\left (3 a^2+13 b a+8 b^2\right ) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) (9 a+8 b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a}\right )}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) (9 a+8 b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}d\sin (e+f x)}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) (9 a+8 b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a+b) (9 a+8 b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {-a \sin ^2(e+f x)+a+b} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a+b) (9 a+8 b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

Input:

Int[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2),x]
 

Output:

(Sqrt[1 - Sin[e + f*x]^2]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(1 - Sin[e + f*x 
]^2)]*((b*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(5*(1 - Sin[e + f*x 
]^2)^(5/2)) + ((2*(3*a + 2*b)*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2]) 
/(3*(1 - Sin[e + f*x]^2)^(3/2)) + (((3*a^2 + 13*a*b + 8*b^2)*Sin[e + f*x]* 
Sqrt[a + b - a*Sin[e + f*x]^2])/(b*Sqrt[1 - Sin[e + f*x]^2]) - (a*(((3*a^2 
 + 13*a*b + 8*b^2)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - 
 a*Sin[e + f*x]^2])/(a*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(a + b)* 
(9*a + 8*b)*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + 
 f*x]^2)/(a + b)])/(a*Sqrt[a + b - a*Sin[e + f*x]^2])))/b)/3)/5))/(f*Sqrt[ 
a + b - a*Sin[e + f*x]^2])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 323

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 330

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]   Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ[a, 
0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

rule 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

rule 3042

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

rule 4636

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ 
))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f 
 Subst[Int[(a + b/(1 - ff^2*x^2)^(n/2))^p/(1 - ff^2*x^2)^((m + 1)/2), x], x 
, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] 
 && IntegerQ[n/2] &&  !IntegerQ[p]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 33.15 (sec) , antiderivative size = 3442, normalized size of antiderivative = 9.18

method	result	size
default	\(\text {Expression too large to display}\)	\(3442\)

Input:

int(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/15/f/(2*I*a^(1/2)*b^(1/2)-a+b)/((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)/b 
*(a+b*sec(f*x+e)^2)^(3/2)/(cos(f*x+e)^4*(1+cos(f*x+e))*a^2+cos(f*x+e)^2*(2 
*cos(f*x+e)+2)*a*b+(1+cos(f*x+e))*b^2)*((1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f* 
x+e)-I*a^(1/2)*b^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x+e)))^(1/2)*(-1/(a+b)*(I* 
a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a-b)/(1+cos(f*x+e) 
))^(1/2)*a^4*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e) 
-cot(f*x+e)),(-(4*I*a^(3/2)*b^(1/2)-4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/(a+ 
b)^2)^(1/2))*(3*cos(f*x+e)^5+6*cos(f*x+e)^4+3*cos(f*x+e)^3)+(1/(a+b)*(I*a^ 
(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x+e))) 
^(1/2)*(-1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)-cos(f*x+e 
)*a-b)/(1+cos(f*x+e)))^(1/2)*a^3*b*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a 
+b))^(1/2)*(csc(f*x+e)-cot(f*x+e)),(-(4*I*a^(3/2)*b^(1/2)-4*I*a^(1/2)*b^(3 
/2)-a^2+6*a*b-b^2)/(a+b)^2)^(1/2))*(19*cos(f*x+e)^5+38*cos(f*x+e)^4+19*cos 
(f*x+e)^3)+(1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)+cos(f* 
x+e)*a+b)/(1+cos(f*x+e)))^(1/2)*(-1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I* 
a^(1/2)*b^(1/2)-cos(f*x+e)*a-b)/(1+cos(f*x+e)))^(1/2)*a^2*b^2*EllipticE((( 
2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot(f*x+e)),(-(4*I*a^(3/ 
2)*b^(1/2)-4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/(a+b)^2)^(1/2))*(37*cos(f*x+ 
e)^5+74*cos(f*x+e)^4+37*cos(f*x+e)^3)+(1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+ 
e)-I*a^(1/2)*b^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x+e)))^(1/2)*(-1/(a+b)*(I...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.37 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="fricas")
 

Output:

1/30*((2*(3*I*a^3 + 13*I*a^2*b + 8*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)* 
cos(f*x + e)^4 - (3*I*a^3 + 19*I*a^2*b + 34*I*a*b^2 + 16*I*b^3)*sqrt(a)*co 
s(f*x + e)^4)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_e(arc 
sin(sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) + I*sin(f* 
x + e))), (a^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^ 
2) + (2*(-3*I*a^3 - 13*I*a^2*b - 8*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)* 
cos(f*x + e)^4 - (-3*I*a^3 - 19*I*a^2*b - 34*I*a*b^2 - 16*I*b^3)*sqrt(a)*c 
os(f*x + e)^4)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_e(ar 
csin(sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) - I*sin(f 
*x + e))), (a^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a 
^2) - 2*(4*(3*I*a^2*b + 2*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)*cos(f*x + 
 e)^4 + (-3*I*a^3 - 13*I*a^2*b - 18*I*a*b^2 - 8*I*b^3)*sqrt(a)*cos(f*x + e 
)^4)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_f(arcsin(sqrt( 
(2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) + I*sin(f*x + e))), 
 (a^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) - 2*(4 
*(-3*I*a^2*b - 2*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)*cos(f*x + e)^4 + ( 
3*I*a^3 + 13*I*a^2*b + 18*I*a*b^2 + 8*I*b^3)*sqrt(a)*cos(f*x + e)^4)*sqrt( 
(2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_f(arcsin(sqrt((2*a*sqrt( 
(a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) - I*sin(f*x + e))), (a^2 + 8* 
a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) + 2*((3*a^3 +...
                                                                                    
                                                                                    
 

Sympy [F]

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

integrate(sec(f*x+e)**3*(a+b*sec(f*x+e)**2)**(3/2),x)
 

Output:

Integral((a + b*sec(e + f*x)**2)**(3/2)*sec(e + f*x)**3, x)
 

Maxima [F]

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

integrate(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*sec(f*x + e)^2 + a)^(3/2)*sec(f*x + e)^3, x)
 

Giac [F]

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

integrate(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*sec(f*x + e)^2 + a)^(3/2)*sec(f*x + e)^3, x)
 

Mupad [F(-1)]

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

int((a + b/cos(e + f*x)^2)^(3/2)/cos(e + f*x)^3,x)
 

Output:

int((a + b/cos(e + f*x)^2)^(3/2)/cos(e + f*x)^3, x)
 

Reduce [F]

\[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\left (\int \sqrt {\sec \left (f x +e \right )^{2} b +a}\, \sec \left (f x +e \right )^{5}d x \right ) b +\left (\int \sqrt {\sec \left (f x +e \right )^{2} b +a}\, \sec \left (f x +e \right )^{3}d x \right ) a \] Input:

int(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^(3/2),x)
 

Output:

int(sqrt(sec(e + f*x)**2*b + a)*sec(e + f*x)**5,x)*b + int(sqrt(sec(e + f* 
x)**2*b + a)*sec(e + f*x)**3,x)*a