\(\int \frac {1}{\sqrt {x} (a+b \sec (c+d \sqrt {x}))^2} \, dx\) [68]

Optimal result

Integrand size = 22, antiderivative size = 127 \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {2 \sqrt {x}}{a^2}-\frac {4 b \left (2 a^2-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {2 b^2 \tan \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \sec \left (c+d \sqrt {x}\right )\right )} \] Output:

2*x^(1/2)/a^2-4*b*(2*a^2-b^2)*arctanh((a-b)^(1/2)*tan(1/2*c+1/2*d*x^(1/2)) 
/(a+b)^(1/2))/a^2/(a-b)^(3/2)/(a+b)^(3/2)/d+2*b^2*tan(c+d*x^(1/2))/a/(a^2- 
b^2)/d/(a+b*sec(c+d*x^(1/2)))
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {2 \left (-\frac {2 b \left (-2 a^2+b^2\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {a \left (a^2-b^2\right ) \left (c+d \sqrt {x}\right ) \cos \left (c+d \sqrt {x}\right )+b \left (\left (a^2-b^2\right ) \left (c+d \sqrt {x}\right )+a b \sin \left (c+d \sqrt {x}\right )\right )}{b+a \cos \left (c+d \sqrt {x}\right )}\right )}{a^2 (a-b) (a+b) d} \] Input:

Integrate[1/(Sqrt[x]*(a + b*Sec[c + d*Sqrt[x]])^2),x]
 

Output:

(2*((-2*b*(-2*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*Sqrt[x])/2])/Sqrt[a^ 
2 - b^2]])/Sqrt[a^2 - b^2] + (a*(a^2 - b^2)*(c + d*Sqrt[x])*Cos[c + d*Sqrt 
[x]] + b*((a^2 - b^2)*(c + d*Sqrt[x]) + a*b*Sin[c + d*Sqrt[x]]))/(b + a*Co 
s[c + d*Sqrt[x]])))/(a^2*(a - b)*(a + b)*d)
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4692, 3042, 4272, 25, 3042, 4407, 3042, 4318, 3042, 3138, 221}

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.

\(\displaystyle \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx\)

\(\Big \downarrow \) 4692

\(\displaystyle 2 \int \frac {1}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \int \frac {1}{\left (a+b \csc \left (c+d \sqrt {x}+\frac {\pi }{2}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 4272

\(\displaystyle 2 \left (\frac {b^2 \tan \left (c+d \sqrt {x}\right )}{a d \left (a^2-b^2\right ) \left (a+b \sec \left (c+d \sqrt {x}\right )\right )}-\frac {\int -\frac {a^2-b \sec \left (c+d \sqrt {x}\right ) a-b^2}{a+b \sec \left (c+d \sqrt {x}\right )}d\sqrt {x}}{a \left (a^2-b^2\right )}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \left (\frac {\int \frac {a^2-b \sec \left (c+d \sqrt {x}\right ) a-b^2}{a+b \sec \left (c+d \sqrt {x}\right )}d\sqrt {x}}{a \left (a^2-b^2\right )}+\frac {b^2 \tan \left (c+d \sqrt {x}\right )}{a d \left (a^2-b^2\right ) \left (a+b \sec \left (c+d \sqrt {x}\right )\right )}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \left (\frac {\int \frac {a^2-b \csc \left (c+d \sqrt {x}+\frac {\pi }{2}\right ) a-b^2}{a+b \csc \left (c+d \sqrt {x}+\frac {\pi }{2}\right )}d\sqrt {x}}{a \left (a^2-b^2\right )}+\frac {b^2 \tan \left (c+d \sqrt {x}\right )}{a d \left (a^2-b^2\right ) \left (a+b \sec \left (c+d \sqrt {x}\right )\right )}\right )\)

\(\Big \downarrow \) 4407

\(\displaystyle 2 \left (\frac {\frac {\sqrt {x} \left (a^2-b^2\right )}{a}-\frac {b \left (2 a^2-b^2\right ) \int \frac {\sec \left (c+d \sqrt {x}\right )}{a+b \sec \left (c+d \sqrt {x}\right )}d\sqrt {x}}{a}}{a \left (a^2-b^2\right )}+\frac {b^2 \tan \left (c+d \sqrt {x}\right )}{a d \left (a^2-b^2\right ) \left (a+b \sec \left (c+d \sqrt {x}\right )\right )}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \left (\frac {\frac {\sqrt {x} \left (a^2-b^2\right )}{a}-\frac {b \left (2 a^2-b^2\right ) \int \frac {\csc \left (c+d \sqrt {x}+\frac {\pi }{2}\right )}{a+b \csc \left (c+d \sqrt {x}+\frac {\pi }{2}\right )}d\sqrt {x}}{a}}{a \left (a^2-b^2\right )}+\frac {b^2 \tan \left (c+d \sqrt {x}\right )}{a d \left (a^2-b^2\right ) \left (a+b \sec \left (c+d \sqrt {x}\right )\right )}\right )\)

\(\Big \downarrow \) 4318

\(\displaystyle 2 \left (\frac {\frac {\sqrt {x} \left (a^2-b^2\right )}{a}-\frac {\left (2 a^2-b^2\right ) \int \frac {1}{\frac {a \cos \left (c+d \sqrt {x}\right )}{b}+1}d\sqrt {x}}{a}}{a \left (a^2-b^2\right )}+\frac {b^2 \tan \left (c+d \sqrt {x}\right )}{a d \left (a^2-b^2\right ) \left (a+b \sec \left (c+d \sqrt {x}\right )\right )}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \left (\frac {\frac {\sqrt {x} \left (a^2-b^2\right )}{a}-\frac {\left (2 a^2-b^2\right ) \int \frac {1}{\frac {a \sin \left (c+d \sqrt {x}+\frac {\pi }{2}\right )}{b}+1}d\sqrt {x}}{a}}{a \left (a^2-b^2\right )}+\frac {b^2 \tan \left (c+d \sqrt {x}\right )}{a d \left (a^2-b^2\right ) \left (a+b \sec \left (c+d \sqrt {x}\right )\right )}\right )\)

\(\Big \downarrow \) 3138

\(\displaystyle 2 \left (\frac {\frac {\sqrt {x} \left (a^2-b^2\right )}{a}-\frac {2 \left (2 a^2-b^2\right ) \int \frac {1}{\frac {a+b}{b}+\left (1-\frac {a}{b}\right ) x}d\tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{a d}}{a \left (a^2-b^2\right )}+\frac {b^2 \tan \left (c+d \sqrt {x}\right )}{a d \left (a^2-b^2\right ) \left (a+b \sec \left (c+d \sqrt {x}\right )\right )}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle 2 \left (\frac {\frac {\sqrt {x} \left (a^2-b^2\right )}{a}-\frac {2 b \left (2 a^2-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}}{a \left (a^2-b^2\right )}+\frac {b^2 \tan \left (c+d \sqrt {x}\right )}{a d \left (a^2-b^2\right ) \left (a+b \sec \left (c+d \sqrt {x}\right )\right )}\right )\)

Input:

Int[1/(Sqrt[x]*(a + b*Sec[c + d*Sqrt[x]])^2),x]
 

Output:

2*((((a^2 - b^2)*Sqrt[x])/a - (2*b*(2*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[ 
(c + d*Sqrt[x])/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d))/(a*(a^2 - 
 b^2)) + (b^2*Tan[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*Sqrt[x 
]])))
 

Defintions of rubi rules used

rule 25

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

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 3042

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

rule 3138

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ 
e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + b + 
(a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] 
 && NeQ[a^2 - b^2, 0]
 

rule 4272

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[ 
c + d*x]*((a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Sim 
p[1/(a*(n + 1)*(a^2 - b^2))   Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^2 - 
b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x 
], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ 
erQ[2*n]
 

rule 4318

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo 
l] :> Simp[1/b   Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, 
f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4407

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

rule 4692

Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sec[c + d*x])^ 
p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 
 1)/n], 0] && IntegerQ[p]
 

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.28

method	result	size
derivativedivides	\(\frac {\frac {4 b \left (-\frac {a b \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )^{2} a -\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )^{2} b -a -b \right )}-\frac {\left (2 a^{2}-b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{2}}+\frac {4 \arctan \left (\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )}{a^{2}}}{d}\)	\(162\)
default	\(\frac {\frac {4 b \left (-\frac {a b \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )^{2} a -\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )^{2} b -a -b \right )}-\frac {\left (2 a^{2}-b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{2}}+\frac {4 \arctan \left (\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )}{a^{2}}}{d}\)	\(162\)

Input:

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

Output:

2/d*(2*b/a^2*(-a*b/(a^2-b^2)*tan(1/2*c+1/2*d*x^(1/2))/(tan(1/2*c+1/2*d*x^( 
1/2))^2*a-tan(1/2*c+1/2*d*x^(1/2))^2*b-a-b)-(2*a^2-b^2)/(a-b)/(a+b)/((a-b) 
*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*c+1/2*d*x^(1/2))/((a-b)*(a+b))^(1/2))) 
+2/a^2*arctan(tan(1/2*c+1/2*d*x^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (110) = 220\).

Time = 0.11 (sec) , antiderivative size = 574, normalized size of antiderivative = 4.52 \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\left [\frac {2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \cos \left (d \sqrt {x} + c\right ) + 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sqrt {x} + {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {a^{2} - b^{2}} \cos \left (d \sqrt {x} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {a^{2} - b^{2}}\right )} \log \left (\frac {2 \, a b \cos \left (d \sqrt {x} + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d \sqrt {x} + c\right )^{2} + 2 \, a^{2} - b^{2} - 2 \, {\left (\sqrt {a^{2} - b^{2}} b \cos \left (d \sqrt {x} + c\right ) + \sqrt {a^{2} - b^{2}} a\right )} \sin \left (d \sqrt {x} + c\right )}{a^{2} \cos \left (d \sqrt {x} + c\right )^{2} + 2 \, a b \cos \left (d \sqrt {x} + c\right ) + b^{2}}\right ) + 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \sin \left (d \sqrt {x} + c\right )}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cos \left (d \sqrt {x} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d}, \frac {2 \, {\left ({\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \cos \left (d \sqrt {x} + c\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sqrt {x} - {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {-a^{2} + b^{2}} \cos \left (d \sqrt {x} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {-a^{2} + b^{2}}\right )} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} b \cos \left (d \sqrt {x} + c\right ) + \sqrt {-a^{2} + b^{2}} a}{{\left (a^{2} - b^{2}\right )} \sin \left (d \sqrt {x} + c\right )}\right ) + {\left (a^{3} b^{2} - a b^{4}\right )} \sin \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cos \left (d \sqrt {x} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d}\right ] \] Input:

integrate(1/x^(1/2)/(a+b*sec(c+d*x^(1/2)))^2,x, algorithm="fricas")
 

Output:

[(2*(a^5 - 2*a^3*b^2 + a*b^4)*d*sqrt(x)*cos(d*sqrt(x) + c) + 2*(a^4*b - 2* 
a^2*b^3 + b^5)*d*sqrt(x) + ((2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*cos(d*sqrt(x 
) + c) + (2*a^2*b^2 - b^4)*sqrt(a^2 - b^2))*log((2*a*b*cos(d*sqrt(x) + c) 
- (a^2 - 2*b^2)*cos(d*sqrt(x) + c)^2 + 2*a^2 - b^2 - 2*(sqrt(a^2 - b^2)*b* 
cos(d*sqrt(x) + c) + sqrt(a^2 - b^2)*a)*sin(d*sqrt(x) + c))/(a^2*cos(d*sqr 
t(x) + c)^2 + 2*a*b*cos(d*sqrt(x) + c) + b^2)) + 2*(a^3*b^2 - a*b^4)*sin(d 
*sqrt(x) + c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cos(d*sqrt(x) + c) + (a^6*b 
- 2*a^4*b^3 + a^2*b^5)*d), 2*((a^5 - 2*a^3*b^2 + a*b^4)*d*sqrt(x)*cos(d*sq 
rt(x) + c) + (a^4*b - 2*a^2*b^3 + b^5)*d*sqrt(x) - ((2*a^3*b - a*b^3)*sqrt 
(-a^2 + b^2)*cos(d*sqrt(x) + c) + (2*a^2*b^2 - b^4)*sqrt(-a^2 + b^2))*arct 
an(-(sqrt(-a^2 + b^2)*b*cos(d*sqrt(x) + c) + sqrt(-a^2 + b^2)*a)/((a^2 - b 
^2)*sin(d*sqrt(x) + c))) + (a^3*b^2 - a*b^4)*sin(d*sqrt(x) + c))/((a^7 - 2 
*a^5*b^2 + a^3*b^4)*d*cos(d*sqrt(x) + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d 
)]
 

Sympy [F]

\[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{\sqrt {x} \left (a + b \sec {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \] Input:

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

Output:

Integral(1/(sqrt(x)*(a + b*sec(c + d*sqrt(x)))**2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/x^(1/2)/(a+b*sec(c+d*x^(1/2)))^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(4*a^2-4*b^2>0)', see `assume?` f 
or more de
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {4 \, b^{2} \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right )}{{\left (a^{3} d - a b^{2} d\right )} {\left (a \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right )^{2} - a - b\right )}} + \frac {4 \, {\left (2 \, a^{2} b - b^{3}\right )} {\left (\pi \left \lfloor \frac {d \sqrt {x} + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{4} d - a^{2} b^{2} d\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, {\left (d \sqrt {x} + c\right )}}{a^{2} d} \] Input:

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

Output:

-4*b^2*tan(1/2*d*sqrt(x) + 1/2*c)/((a^3*d - a*b^2*d)*(a*tan(1/2*d*sqrt(x) 
+ 1/2*c)^2 - b*tan(1/2*d*sqrt(x) + 1/2*c)^2 - a - b)) + 4*(2*a^2*b - b^3)* 
(pi*floor(1/2*(d*sqrt(x) + c)/pi + 1/2)*sgn(2*a - 2*b) + arctan((a*tan(1/2 
*d*sqrt(x) + 1/2*c) - b*tan(1/2*d*sqrt(x) + 1/2*c))/sqrt(-a^2 + b^2)))/((a 
^4*d - a^2*b^2*d)*sqrt(-a^2 + b^2)) + 2*(d*sqrt(x) + c)/(a^2*d)
 

Mupad [B] (verification not implemented)

Time = 20.19 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.60 \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {\frac {b^2\,4{}\mathrm {i}}{a\,d\,\left (a^2-b^2\right )}+\frac {b^3\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,\sqrt {x}\,1{}\mathrm {i}}\,4{}\mathrm {i}}{a^2\,d\,\left (a^2-b^2\right )}}{a+a\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,\sqrt {x}\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,\sqrt {x}\,1{}\mathrm {i}}}+\frac {2\,\sqrt {x}}{a^2}+\frac {\ln \left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,\sqrt {x}\,1{}\mathrm {i}}\,\left (4\,a^2\,b-2\,b^3\right )-\frac {\left (4\,a^2\,b-2\,b^3\right )\,\left (a^2-b^2\right )\,\left (a+b\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,\sqrt {x}\,1{}\mathrm {i}}\right )\,1{}\mathrm {i}}{{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}\right )\,\left (4\,a^2\,b-2\,b^3\right )}{a^2\,d\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}-\frac {2\,b\,\ln \left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,\sqrt {x}\,1{}\mathrm {i}}\,\left (4\,a^2\,b-2\,b^3\right )+\frac {b\,\left (a^2-b^2\right )\,\left (2\,a^2-b^2\right )\,\left (a+b\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,\sqrt {x}\,1{}\mathrm {i}}\right )\,2{}\mathrm {i}}{{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}\right )\,\left (2\,a^2-b^2\right )}{a^2\,d\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}} \] Input:

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

Output:

((b^2*4i)/(a*d*(a^2 - b^2)) + (b^3*exp(c*1i + d*x^(1/2)*1i)*4i)/(a^2*d*(a^ 
2 - b^2)))/(a + a*exp(c*2i + d*x^(1/2)*2i) + 2*b*exp(c*1i + d*x^(1/2)*1i)) 
 + (2*x^(1/2))/a^2 + (log(exp(c*1i + d*x^(1/2)*1i)*(4*a^2*b - 2*b^3) - ((4 
*a^2*b - 2*b^3)*(a^2 - b^2)*(a + b*exp(c*1i + d*x^(1/2)*1i))*1i)/((a + b)^ 
(3/2)*(a - b)^(3/2)))*(4*a^2*b - 2*b^3))/(a^2*d*(a + b)^(3/2)*(a - b)^(3/2 
)) - (2*b*log(exp(c*1i + d*x^(1/2)*1i)*(4*a^2*b - 2*b^3) + (b*(a^2 - b^2)* 
(2*a^2 - b^2)*(a + b*exp(c*1i + d*x^(1/2)*1i))*2i)/((a + b)^(3/2)*(a - b)^ 
(3/2)))*(2*a^2 - b^2))/(a^2*d*(a + b)^(3/2)*(a - b)^(3/2))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 405, normalized size of antiderivative = 3.19 \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {-8 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right ) a -\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right ) b}{\sqrt {-a^{2}+b^{2}}}\right ) \cos \left (\sqrt {x}\, d +c \right ) a^{3} b +4 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right ) a -\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right ) b}{\sqrt {-a^{2}+b^{2}}}\right ) \cos \left (\sqrt {x}\, d +c \right ) a \,b^{3}-8 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right ) a -\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right ) b}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2} b^{2}+4 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right ) a -\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right ) b}{\sqrt {-a^{2}+b^{2}}}\right ) b^{4}+2 \sqrt {x}\, \cos \left (\sqrt {x}\, d +c \right ) a^{5} d -4 \sqrt {x}\, \cos \left (\sqrt {x}\, d +c \right ) a^{3} b^{2} d +2 \sqrt {x}\, \cos \left (\sqrt {x}\, d +c \right ) a \,b^{4} d +2 \sqrt {x}\, a^{4} b d -4 \sqrt {x}\, a^{2} b^{3} d +2 \sqrt {x}\, b^{5} d +2 \sin \left (\sqrt {x}\, d +c \right ) a^{3} b^{2}-2 \sin \left (\sqrt {x}\, d +c \right ) a \,b^{4}}{a^{2} d \left (\cos \left (\sqrt {x}\, d +c \right ) a^{5}-2 \cos \left (\sqrt {x}\, d +c \right ) a^{3} b^{2}+\cos \left (\sqrt {x}\, d +c \right ) a \,b^{4}+a^{4} b -2 a^{2} b^{3}+b^{5}\right )} \] Input:

int(1/x^(1/2)/(a+b*sec(c+d*x^(1/2)))^2,x)
 

Output:

(2*( - 4*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*a - tan((sqrt(x 
)*d + c)/2)*b)/sqrt( - a**2 + b**2))*cos(sqrt(x)*d + c)*a**3*b + 2*sqrt( - 
 a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*a - tan((sqrt(x)*d + c)/2)*b)/s 
qrt( - a**2 + b**2))*cos(sqrt(x)*d + c)*a*b**3 - 4*sqrt( - a**2 + b**2)*at 
an((tan((sqrt(x)*d + c)/2)*a - tan((sqrt(x)*d + c)/2)*b)/sqrt( - a**2 + b* 
*2))*a**2*b**2 + 2*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*a - t 
an((sqrt(x)*d + c)/2)*b)/sqrt( - a**2 + b**2))*b**4 + sqrt(x)*cos(sqrt(x)* 
d + c)*a**5*d - 2*sqrt(x)*cos(sqrt(x)*d + c)*a**3*b**2*d + sqrt(x)*cos(sqr 
t(x)*d + c)*a*b**4*d + sqrt(x)*a**4*b*d - 2*sqrt(x)*a**2*b**3*d + sqrt(x)* 
b**5*d + sin(sqrt(x)*d + c)*a**3*b**2 - sin(sqrt(x)*d + c)*a*b**4))/(a**2* 
d*(cos(sqrt(x)*d + c)*a**5 - 2*cos(sqrt(x)*d + c)*a**3*b**2 + cos(sqrt(x)* 
d + c)*a*b**4 + a**4*b - 2*a**2*b**3 + b**5))