\(\int (a+b \sec (e+f x))^{3/2} \, dx\) [251]

Optimal result

Integrand size = 14, antiderivative size = 309 \[ \int (a+b \sec (e+f x))^{3/2} \, dx=-\frac {2 (a-b) \sqrt {a+b} \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}+\frac {2 (2 a-b) \sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}-\frac {2 a \sqrt {a+b} \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(684\) vs. \(2(309)=618\).

Time = 6.30 (sec) , antiderivative size = 684, normalized size of antiderivative = 2.21 \[ \int (a+b \sec (e+f x))^{3/2} \, dx=\frac {2 b \cos (e+f x) (a+b \sec (e+f x))^{3/2} \sin (e+f x)}{f (b+a \cos (e+f x))}+\frac {2 (a+b \sec (e+f x))^{3/2} \left (a b \tan \left (\frac {1}{2} (e+f x)\right )+b^2 \tan \left (\frac {1}{2} (e+f x)\right )-2 a b \tan ^3\left (\frac {1}{2} (e+f x)\right )+a b \tan ^5\left (\frac {1}{2} (e+f x)\right )-b^2 \tan ^5\left (\frac {1}{2} (e+f x)\right )-2 a^2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}-2 a^2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+b (a+b) E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+\left (a^2-2 a b-b^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}\right )}{f (b+a \cos (e+f x))^{3/2} \sec ^{\frac {3}{2}}(e+f x) \sqrt {\frac {1}{1-\tan ^2\left (\frac {1}{2} (e+f x)\right )}} \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^{3/2} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}}} \] Input:

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

Output:

(2*b*Cos[e + f*x]*(a + b*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(f*(b + a*Cos[e 
 + f*x])) + (2*(a + b*Sec[e + f*x])^(3/2)*(a*b*Tan[(e + f*x)/2] + b^2*Tan[ 
(e + f*x)/2] - 2*a*b*Tan[(e + f*x)/2]^3 + a*b*Tan[(e + f*x)/2]^5 - b^2*Tan 
[(e + f*x)/2]^5 - 2*a^2*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/( 
a + b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + 
b*Tan[(e + f*x)/2]^2)/(a + b)] - 2*a^2*EllipticPi[-1, ArcSin[Tan[(e + f*x) 
/2]], (a - b)/(a + b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqr 
t[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + b*(a + 
b)*EllipticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(e + 
f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b 
*Tan[(e + f*x)/2]^2)/(a + b)] + (a^2 - 2*a*b - b^2)*EllipticF[ArcSin[Tan[( 
e + f*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + 
f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + 
 b)]))/(f*(b + a*Cos[e + f*x])^(3/2)*Sec[e + f*x]^(3/2)*Sqrt[(1 - Tan[(e + 
 f*x)/2]^2)^(-1)]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)^(3/2) 
*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + 
f*x)/2]^2)])
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4268, 3042, 4409, 3042, 4271, 4319, 4492}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4268

\(\displaystyle \int \frac {a^2+(2 a-b) b \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}}dx+b^2 \int \frac {\sec (e+f x) (\sec (e+f x)+1)}{\sqrt {a+b \sec (e+f x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {a^2+(2 a-b) b \csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+b^2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 4409

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

\(\Big \downarrow \) 3042

\(\displaystyle a^2 \int \frac {1}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+b^2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+b (2 a-b) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 4271

\(\displaystyle b^2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+b (2 a-b) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {2 a \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{f}\)

\(\Big \downarrow \) 4319

\(\displaystyle b^2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+\frac {2 (2 a-b) \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{f}-\frac {2 a \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{f}\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {2 (2 a-b) \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{f}-\frac {2 (a-b) \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{f}-\frac {2 a \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{f}\)

Input:

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

Output:

(-2*(a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f 
*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*S 
qrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f + (2*(2*a - b)*Sqrt[a + b]*Cot[e 
 + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a 
 - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/ 
(a - b))])/f - (2*a*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[ 
Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e 
 + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f
 

Defintions of rubi rules used

rule 3042

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

rule 4268

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(3/2), x_Symbol] :> Int[(a^2 + b 
*(2*a - b)*Csc[c + d*x])/Sqrt[a + b*Csc[c + d*x]], x] + Simp[b^2   Int[Csc[ 
c + d*x]*((1 + Csc[c + d*x])/Sqrt[a + b*Csc[c + d*x]]), x], x] /; FreeQ[{a, 
 b, c, d}, x] && NeQ[a^2 - b^2, 0]
 

rule 4271

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a 
 + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) 
*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ 
c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && 
NeQ[a^2 - b^2, 0]
 

rule 4319

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* 
x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt 
[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, 
 f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4409

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ 
.) + (a_)], x_Symbol] :> Simp[c   Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + 
Simp[d   Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, 
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
 

rule 4492

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c 
sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a 
 + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e 
+ f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + 
 f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, 
 f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(634\) vs. \(2(282)=564\).

Time = 7.03 (sec) , antiderivative size = 635, normalized size of antiderivative = 2.06

method	result	size
default	\(\frac {\left (\cos \left (f x +e \right )+1\right )^{2} \left (\left (\left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right ) a b +\left (-\left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right ) b^{2}-2 \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a^{2}+4 \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a b +2 \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b^{2}-2 \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a b -2 \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b^{2}+4 \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {a -b}{a +b}}\right ) a^{2}\right ) \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right ) \sqrt {a +b \sec \left (f x +e \right )}\, \sec \left (f x +e \right )}{2 f \left (a \cos \left (f x +e \right )+b \right )}\)	\(635\)

Input:

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

Output:

1/2/f/(a*cos(f*x+e)+b)*(cos(f*x+e)+1)^2*(((1-cos(f*x+e))^3*csc(f*x+e)^3-cs 
c(f*x+e)+cot(f*x+e))*a*b+(-(1-cos(f*x+e))^3*csc(f*x+e)^3-csc(f*x+e)+cot(f* 
x+e))*b^2-2*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(c 
os(f*x+e)+1))^(1/2)*EllipticF(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*a 
^2+4*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(cos(f*x+ 
e)+1))^(1/2)*EllipticF(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*a*b+2*(c 
os(f*x+e)/(cos(f*x+e)+1))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(cos(f*x+e)+1))^ 
(1/2)*EllipticF(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*b^2-2*(cos(f*x+ 
e)/(cos(f*x+e)+1))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(cos(f*x+e)+1))^(1/2)*E 
llipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*a*b-2*(cos(f*x+e)/(cos 
(f*x+e)+1))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(cos(f*x+e)+1))^(1/2)*Elliptic 
E(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*b^2+4*(cos(f*x+e)/(cos(f*x+e) 
+1))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(cos(f*x+e)+1))^(1/2)*EllipticPi(cot( 
f*x+e)-csc(f*x+e),-1,((a-b)/(a+b))^(1/2))*a^2)*((1-cos(f*x+e))^2*csc(f*x+e 
)^2-1)*(a+b*sec(f*x+e))^(1/2)*sec(f*x+e)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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