\(\int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx\) [338]

Optimal result

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

[Out]

-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*arcta
nh(cos(x)^(1/2))*sin(x)/cos(x)^(1/2)/(sin(x)*tan(x))^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {4482, 4485, 2677, 2681, 2645, 335, 218, 212, 209} \[ \int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx=\frac {\sin (x) \arctan \left (\sqrt {\cos (x)}\right )}{4 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}+\frac {\sin (x) \text {arctanh}\left (\sqrt {\cos (x)}\right )}{4 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}-\frac {\csc (x)}{2 \sqrt {\sin (x) \tan (x)}} \]

[In]

Int[(-Cos[x] + Sec[x])^(-3/2),x]

[Out]

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

Rule 209

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

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

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2645

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(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 2677

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] - Dist[(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] && Integer
sQ[2*m, 2*n] &&  !(EqQ[n, -3/2] && EqQ[m, 1])

Rule 2681

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[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 4482

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

Rule 4485

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = Ac
tivateTrig[w]}, Dist[(vv^m*ww^n)^FracPart[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])

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(\sin (x) \tan (x))^{3/2}} \, dx \\ & = \frac {\left (\sqrt {\sin (x)} \sqrt {\tan (x)}\right ) \int \frac {1}{\sin ^{\frac {3}{2}}(x) \tan ^{\frac {3}{2}}(x)} \, dx}{\sqrt {\sin (x) \tan (x)}} \\ & = -\frac {\csc (x)}{2 \sqrt {\sin (x) \tan (x)}}-\frac {\left (\sqrt {\sin (x)} \sqrt {\tan (x)}\right ) \int \frac {\sqrt {\tan (x)}}{\sin ^{\frac {3}{2}}(x)} \, dx}{4 \sqrt {\sin (x) \tan (x)}} \\ & = -\frac {\csc (x)}{2 \sqrt {\sin (x) \tan (x)}}-\frac {\sin (x) \int \frac {\csc (x)}{\sqrt {\cos (x)}} \, dx}{4 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ & = -\frac {\csc (x)}{2 \sqrt {\sin (x) \tan (x)}}+\frac {\sin (x) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-x^2\right )} \, dx,x,\cos (x)\right )}{4 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ & = -\frac {\csc (x)}{2 \sqrt {\sin (x) \tan (x)}}+\frac {\sin (x) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\sqrt {\cos (x)}\right )}{2 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ & = -\frac {\csc (x)}{2 \sqrt {\sin (x) \tan (x)}}+\frac {\sin (x) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\cos (x)}\right )}{4 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}+\frac {\sin (x) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\cos (x)}\right )}{4 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ & = -\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)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx=\frac {\cot (x) \left (\arctan \left (\sqrt [4]{\cos ^2(x)}\right )+\text {arctanh}\left (\sqrt [4]{\cos ^2(x)}\right )-2 \sqrt [4]{\cos ^2(x)} \csc ^2(x)\right ) \sqrt {\sin (x) \tan (x)}}{4 \sqrt [4]{\cos ^2(x)}} \]

[In]

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

[Out]

(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))

Maple [B] (verified)

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

Time = 1.32 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.31

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

[In]

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

[Out]

1/8*csc(x)*(cos(x)*arctan(1/2/(-cos(x)/(cos(x)+1)^2)^(1/2))-cos(x)*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))-arctan(1/2/(-cos(x)/(cos(x)+1)^2)^(1/2))+ln((2*cos(x)*(-co
s(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))/
(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 = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx=-\frac {{\left (\cos \left (x\right )^{2} - 1\right )} \arctan \left (\frac {2 \, \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) - {\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac {{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) + 2 \, \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) - 4 \, \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{8 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]

[In]

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

[Out]

-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(-(
cos(x)^2 - 1)/cos(x))*cos(x))/((cos(x)^2 - 1)*sin(x))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{3/2}} \, dx=-\frac {\tan \left (\frac {1}{2} \, x\right )^{2}}{16 \, {\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}} + \frac {1}{8} \, \sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} + \frac {\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1}{16 \, \tan \left (\frac {1}{2} \, x\right )^{2}} + \frac {1}{8} \, \arcsin \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right ) + \frac {1}{8} \, \log \left (-\frac {\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1}{\tan \left (\frac {1}{2} \, x\right )^{2}}\right ) \]

[In]

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

[Out]

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

[In]

int(1/(1/cos(x) - cos(x))^(3/2),x)

[Out]

int(1/(1/cos(x) - cos(x))^(3/2), x)