3.3.17 \(\int \frac {(c+d \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx\) [217]

3.3.17.1 Optimal result

Integrand size = 29, antiderivative size = 652 \[ \int \frac {(c+d \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx=-\frac {2 c (c+d) \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\sqrt {\frac {(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))^{3/2} \sqrt {\frac {(a+b) (b c-a d) (-1+\sec (e+f x)) (c+d \sec (e+f x))}{(c+d)^2 (a+b \sec (e+f x))^2}}}{a (a+b) f \sqrt {c+d \sec (e+f x)}}+\frac {2 d (c+d) \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\sqrt {\frac {(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))^{3/2} \sqrt {-\frac {(a+b) (-b c+a d) (-1+\sec (e+f x)) (c+d \sec (e+f x))}{(c+d)^2 (a+b \sec (e+f x))^2}}}{b (a+b) f \sqrt {c+d \sec (e+f x)}}+\frac {2 (b c-a d) \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {\frac {(b c-a d) (-1+\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} \sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a b f \sqrt {\frac {(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}} \]

output

-2*c*(c+d)*cot(f*x+e)*EllipticPi(((a+b)*(c+d*sec(f*x+e))/(c+d)/(a+b*sec(f* 
x+e)))^(1/2),a*(c+d)/(a+b)/c,((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*(a+b*sec(f*x 
+e))^(3/2)*((-a*d+b*c)*(1+sec(f*x+e))/(c-d)/(a+b*sec(f*x+e)))^(1/2)*((a+b) 
*(-a*d+b*c)*(-1+sec(f*x+e))*(c+d*sec(f*x+e))/(c+d)^2/(a+b*sec(f*x+e))^2)^( 
1/2)/a/(a+b)/f/(c+d*sec(f*x+e))^(1/2)+2*d*(c+d)*cot(f*x+e)*EllipticPi(((a+ 
b)*(c+d*sec(f*x+e))/(c+d)/(a+b*sec(f*x+e)))^(1/2),b*(c+d)/(a+b)/d,((a-b)*( 
c+d)/(a+b)/(c-d))^(1/2))*(a+b*sec(f*x+e))^(3/2)*((-a*d+b*c)*(1+sec(f*x+e)) 
/(c-d)/(a+b*sec(f*x+e)))^(1/2)*(-(a+b)*(a*d-b*c)*(-1+sec(f*x+e))*(c+d*sec( 
f*x+e))/(c+d)^2/(a+b*sec(f*x+e))^2)^(1/2)/b/(a+b)/f/(c+d*sec(f*x+e))^(1/2) 
+2*(-a*d+b*c)*cot(f*x+e)*EllipticF(((a+b)*(c+d*sec(f*x+e))/(c+d)/(a+b*sec( 
f*x+e)))^(1/2),((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*((-a*d+b*c)*(-1+sec(f*x+e) 
)/(c+d)/(a+b*sec(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sec(f*x+e))/(c-d)/(a+b*sec( 
f*x+e)))^(1/2)*(a+b*sec(f*x+e))^(1/2)*(c+d*sec(f*x+e))^(1/2)/a/b/f/((a+b)* 
(c+d*sec(f*x+e))/(c+d)/(a+b*sec(f*x+e)))^(1/2)
 

3.3.17.2 Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 37.55 (sec) , antiderivative size = 50041, normalized size of antiderivative = 76.75 \[ \int \frac {(c+d \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx=\text {Result too large to show} \]

input

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

output

Result too large to show
 

3.3.17.3 Rubi [F]

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.

\(\displaystyle \int \frac {(c+d \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4433

\(\displaystyle \int \frac {(c+d \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}}dx\)

input

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

output

$Aborted
 

3.3.17.3.1 Defintions of rubi rules used

rule 3042

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

rule 4433

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*( 
d_.) + (c_))^(n_.), x_Symbol] :> Unintegrable[(a + b*Csc[e + f*x])^m*(c + d 
*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

3.3.17.4 Maple [A] (verified)

Time = 17.07 (sec) , antiderivative size = 457, normalized size of antiderivative = 0.70

method	result	size
default	\(-\frac {2 \sqrt {c +d \sec \left (f x +e \right )}\, \sqrt {a +b \sec \left (f x +e \right )}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) c^{2}-2 \operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) c d +\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) d^{2}-2 \operatorname {EllipticPi}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \frac {a +b}{a -b}, \frac {\sqrt {\frac {c -d}{c +d}}}{\sqrt {\frac {a -b}{a +b}}}\right ) d^{2}-2 \operatorname {EllipticPi}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), -\frac {a +b}{a -b}, \frac {\sqrt {\frac {c -d}{c +d}}}{\sqrt {\frac {a -b}{a +b}}}\right ) c^{2}\right ) \left (\cos \left (f x +e \right )^{2}+\cos \left (f x +e \right )\right )}{f \sqrt {\frac {a -b}{a +b}}\, \left (d +c \cos \left (f x +e \right )\right ) \left (b +a \cos \left (f x +e \right )\right )}\)	\(457\)

input

int((c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
 

output

-2/f/((a-b)/(a+b))^(1/2)*(c+d*sec(f*x+e))^(1/2)*(a+b*sec(f*x+e))^(1/2)*(1/ 
(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(co 
s(f*x+e)+1))^(1/2)*(EllipticF(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)) 
,((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*c^2-2*EllipticF(((a-b)/(a+b))^(1/2)*(-co 
t(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*c*d+EllipticF(((a-b) 
/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*d^ 
2-2*EllipticPi(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),(a+b)/(a-b),(( 
c-d)/(c+d))^(1/2)/((a-b)/(a+b))^(1/2))*d^2-2*EllipticPi(((a-b)/(a+b))^(1/2 
)*(-cot(f*x+e)+csc(f*x+e)),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a+b))^ 
(1/2))*c^2)/(d+c*cos(f*x+e))/(b+a*cos(f*x+e))*(cos(f*x+e)^2+cos(f*x+e))
 

3.3.17.5 Fricas [F(-1)]

Timed out. \[ \int \frac {(c+d \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx=\text {Timed out} \]

input

integrate((c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="fric 
as")
 

output

Timed out
 

3.3.17.6 Sympy [F]

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

input

integrate((c+d*sec(f*x+e))**(3/2)/(a+b*sec(f*x+e))**(1/2),x)
 

output

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

3.3.17.7 Maxima [F]

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

input

integrate((c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="maxi 
ma")
 

output

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

3.3.17.8 Giac [F]

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

input

integrate((c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="giac 
")
 

output

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

3.3.17.9 Mupad [F(-1)]

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

input

int((c + d/cos(e + f*x))^(3/2)/(a + b/cos(e + f*x))^(1/2),x)
 

output

int((c + d/cos(e + f*x))^(3/2)/(a + b/cos(e + f*x))^(1/2), x)