\(\int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx\) [231]

Optimal result

Integrand size = 11, antiderivative size = 72 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx=-\frac {\csc (x)}{2 \sqrt {\sin (x) \tan (x)}}+\frac {\arctan \left (\sqrt {\cos (x)}\right ) \sin (x)}{4 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}+\frac {\text {arctanh}\left (\sqrt {\cos (x)}\right ) \sin (x)}{4 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \] Output:

-1/2*csc(x)/(sin(x)*tan(x))^(1/2)+1/4*arctan(cos(x)^(1/2))*sin(x)/cos(x)^( 
1/2)/(sin(x)*tan(x))^(1/2)+1/4*arctanh(cos(x)^(1/2))*sin(x)/cos(x)^(1/2)/( 
sin(x)*tan(x))^(1/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx=\frac {\cot (x) \left (\arctan \left (\sqrt [4]{\cos ^2(x)}\right )+\text {arctanh}\left (\sqrt [4]{\cos ^2(x)}\right )-2 \sqrt [4]{\cos ^2(x)} \csc ^2(x)\right ) \sqrt {\sin (x) \tan (x)}}{4 \sqrt [4]{\cos ^2(x)}} \] Input:

Integrate[(-Cos[x] + Sec[x])^(-3/2),x]
 

Output:

(Cot[x]*(ArcTan[(Cos[x]^2)^(1/4)] + ArcTanh[(Cos[x]^2)^(1/4)] - 2*(Cos[x]^ 
2)^(1/4)*Csc[x]^2)*Sqrt[Sin[x]*Tan[x]])/(4*(Cos[x]^2)^(1/4))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.17, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.273, Rules used = {3042, 4897, 3042, 4900, 3042, 3077, 3042, 3081, 3042, 3045, 266, 756, 216, 219}

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.

\(\displaystyle \int \frac {1}{(\sec (x)-\cos (x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(\sec (x)-\cos (x))^{3/2}}dx\)

\(\Big \downarrow \) 4897

\(\displaystyle \int \frac {1}{(\sin (x) \tan (x))^{3/2}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(\sin (x) \tan (x))^{3/2}}dx\)

\(\Big \downarrow \) 4900

\(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \int \frac {1}{\sin ^{\frac {3}{2}}(x) \tan ^{\frac {3}{2}}(x)}dx}{\sqrt {\sin (x) \tan (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \int \frac {1}{\sin (x)^{3/2} \tan (x)^{3/2}}dx}{\sqrt {\sin (x) \tan (x)}}\)

\(\Big \downarrow \) 3077

\(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {1}{4} \int \frac {\sqrt {\tan (x)}}{\sin ^{\frac {3}{2}}(x)}dx-\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )}{\sqrt {\sin (x) \tan (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {1}{4} \int \frac {\sqrt {\tan (x)}}{\sin (x)^{3/2}}dx-\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )}{\sqrt {\sin (x) \tan (x)}}\)

\(\Big \downarrow \) 3081

\(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {\sqrt {\cos (x)} \sqrt {\tan (x)} \int \frac {\csc (x)}{\sqrt {\cos (x)}}dx}{4 \sqrt {\sin (x)}}-\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )}{\sqrt {\sin (x) \tan (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {\sqrt {\cos (x)} \sqrt {\tan (x)} \int \frac {1}{\sqrt {\cos (x)} \sin (x)}dx}{4 \sqrt {\sin (x)}}-\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )}{\sqrt {\sin (x) \tan (x)}}\)

\(\Big \downarrow \) 3045

\(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (\frac {\sqrt {\cos (x)} \sqrt {\tan (x)} \int \frac {1}{\sqrt {\cos (x)} \left (1-\cos ^2(x)\right )}d\cos (x)}{4 \sqrt {\sin (x)}}-\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )}{\sqrt {\sin (x) \tan (x)}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (\frac {\sqrt {\cos (x)} \sqrt {\tan (x)} \int \frac {1}{1-\cos ^2(x)}d\sqrt {\cos (x)}}{2 \sqrt {\sin (x)}}-\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )}{\sqrt {\sin (x) \tan (x)}}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (\frac {\sqrt {\cos (x)} \sqrt {\tan (x)} \left (\frac {1}{2} \int \frac {1}{1-\cos (x)}d\sqrt {\cos (x)}+\frac {1}{2} \int \frac {1}{\cos (x)+1}d\sqrt {\cos (x)}\right )}{2 \sqrt {\sin (x)}}-\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )}{\sqrt {\sin (x) \tan (x)}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (\frac {\sqrt {\cos (x)} \sqrt {\tan (x)} \left (\frac {1}{2} \int \frac {1}{1-\cos (x)}d\sqrt {\cos (x)}+\frac {1}{2} \arctan \left (\sqrt {\cos (x)}\right )\right )}{2 \sqrt {\sin (x)}}-\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )}{\sqrt {\sin (x) \tan (x)}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (\frac {\sqrt {\cos (x)} \sqrt {\tan (x)} \left (\frac {1}{2} \arctan \left (\sqrt {\cos (x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\cos (x)}\right )\right )}{2 \sqrt {\sin (x)}}-\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )}{\sqrt {\sin (x) \tan (x)}}\)

Input:

Int[(-Cos[x] + Sec[x])^(-3/2),x]
 

Output:

(Sqrt[Sin[x]]*(-1/2*1/(Sin[x]^(3/2)*Sqrt[Tan[x]]) + ((ArcTan[Sqrt[Cos[x]]] 
/2 + ArcTanh[Sqrt[Cos[x]]]/2)*Sqrt[Cos[x]]*Sqrt[Tan[x]])/(2*Sqrt[Sin[x]])) 
*Sqrt[Tan[x]])/Sqrt[Sin[x]*Tan[x]]
 

Defintions of rubi rules used

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 3042

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

rule 3045

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ 
Symbol] :> Simp[-(a*f)^(-1)   Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], 
x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && 
 !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
 

rule 3077

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n + 1)/(b*f*(m 
 + n + 1))), x] - Simp[(n + 1)/(b^2*(m + n + 1))   Int[(a*Sin[e + f*x])^m*( 
b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] 
&& NeQ[m + n + 1, 0] && IntegersQ[2*m, 2*n] &&  !(EqQ[n, -3/2] && EqQ[m, 1] 
)
 

rule 3081

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[Cos[e + f*x]^n*((b*Tan[e + f*x])^n/(a*Sin[e + f*x])^ 
n)   Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b, e, 
 f, m, n}, x] &&  !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(- 
1)]) || IntegersQ[m - 1/2, n - 1/2])
 

rule 4897

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]
 

rule 4900

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTri 
g[u], vv = ActivateTrig[v], ww = ActivateTrig[w]}, Simp[(vv^m*ww^n)^FracPar 
t[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p]))   Int[uu*vv^(m*p)*ww^(n*p), x] 
, x]] /; FreeQ[{m, n, p}, x] &&  !IntegerQ[p] && ( !InertTrigFreeQ[v] ||  ! 
InertTrigFreeQ[w])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(165\) vs. \(2(52)=104\).

Time = 0.54 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.31

method	result	size
default	\(\frac {\csc \left (x \right ) \left (\cos \left (x \right ) \arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )-\cos \left (x \right ) \ln \left (\frac {2 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )+1}\right )-4 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )+\ln \left (\frac {2 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )+1}\right )\right )}{8 \sqrt {\sin \left (x \right ) \tan \left (x \right )}\, \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\)	\(166\)

Input:

int(1/(-cos(x)+sec(x))^(3/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/8*csc(x)*(cos(x)*arctan(1/2/(-cos(x)/(cos(x)+1)^2)^(1/2))-cos(x)*ln((2*c 
os(x)*(-cos(x)/(cos(x)+1)^2)^(1/2)+2*(-cos(x)/(cos(x)+1)^2)^(1/2)-cos(x)+1 
)/(cos(x)+1))-4*(-cos(x)/(cos(x)+1)^2)^(1/2)-arctan(1/2/(-cos(x)/(cos(x)+1 
)^2)^(1/2))+ln((2*cos(x)*(-cos(x)/(cos(x)+1)^2)^(1/2)+2*(-cos(x)/(cos(x)+1 
)^2)^(1/2)-cos(x)+1)/(cos(x)+1)))/(sin(x)*tan(x))^(1/2)/(-cos(x)/(cos(x)+1 
)^2)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (52) = 104\).

Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx=-\frac {{\left (\cos \left (x\right )^{2} - 1\right )} \arctan \left (\frac {2 \, \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) - {\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac {{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) + 2 \, \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) - 4 \, \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{8 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \] Input:

integrate(1/(-cos(x)+sec(x))^(3/2),x, algorithm="fricas")
 

Output:

-1/8*((cos(x)^2 - 1)*arctan(2*sqrt(-(cos(x)^2 - 1)/cos(x))*cos(x)/((cos(x) 
 - 1)*sin(x)))*sin(x) - (cos(x)^2 - 1)*log(((cos(x) + 1)*sin(x) + 2*sqrt(- 
(cos(x)^2 - 1)/cos(x))*cos(x))/((cos(x) - 1)*sin(x)))*sin(x) - 4*sqrt(-(co 
s(x)^2 - 1)/cos(x))*cos(x))/((cos(x)^2 - 1)*sin(x))
 

Sympy [F]

\[ \int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx=\int \frac {1}{\left (- \cos {\left (x \right )} + \sec {\left (x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(-cos(x)+sec(x))**(3/2),x)
 

Output:

Integral((-cos(x) + sec(x))**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx=\int { \frac {1}{{\left (-\cos \left (x\right ) + \sec \left (x\right )\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(-cos(x)+sec(x))^(3/2),x, algorithm="maxima")
 

Output:

integrate((-cos(x) + sec(x))^(-3/2), x)
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx=-\frac {\tan \left (\frac {1}{2} \, x\right )^{2}}{16 \, {\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}} + \frac {1}{8} \, \sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} + \frac {\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1}{16 \, \tan \left (\frac {1}{2} \, x\right )^{2}} + \frac {1}{8} \, \arcsin \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right ) + \frac {1}{8} \, \log \left (-\frac {\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1}{\tan \left (\frac {1}{2} \, x\right )^{2}}\right ) \] Input:

integrate(1/(-cos(x)+sec(x))^(3/2),x, algorithm="giac")
 

Output:

-1/16*tan(1/2*x)^2/(sqrt(-tan(1/2*x)^4 + 1) - 1) + 1/8*sqrt(-tan(1/2*x)^4 
+ 1) + 1/16*(sqrt(-tan(1/2*x)^4 + 1) - 1)/tan(1/2*x)^2 + 1/8*arcsin(tan(1/ 
2*x)^2) + 1/8*log(-(sqrt(-tan(1/2*x)^4 + 1) - 1)/tan(1/2*x)^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx=\int \frac {1}{{\left (\frac {1}{\cos \left (x\right )}-\cos \left (x\right )\right )}^{3/2}} \,d x \] Input:

int(1/(1/cos(x) - cos(x))^(3/2),x)
 

Output:

int(1/(1/cos(x) - cos(x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx=\int \frac {\sqrt {-\cos \left (x \right )+\sec \left (x \right )}}{\cos \left (x \right )^{2}-2 \cos \left (x \right ) \sec \left (x \right )+\sec \left (x \right )^{2}}d x \] Input:

int(1/(-cos(x)+sec(x))^(3/2),x)
 

Output:

int(sqrt( - cos(x) + sec(x))/(cos(x)**2 - 2*cos(x)*sec(x) + sec(x)**2),x)