\(\int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx\) [339]

Optimal result

Integrand size = 11, antiderivative size = 91 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=\frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}-\frac {3 \arctan \left (\sqrt {\cos (x)}\right ) \sin (x)}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}+\frac {3 \text {arctanh}\left (\sqrt {\cos (x)}\right ) \sin (x)}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \]

[Out]

3/16*cot(x)/(sin(x)*tan(x))^(1/2)-1/4*cot(x)*csc(x)^2/(sin(x)*tan(x))^(1/2)-3/32*arctan(cos(x)^(1/2))*sin(x)/c
os(x)^(1/2)/(sin(x)*tan(x))^(1/2)+3/32*arctanh(cos(x)^(1/2))*sin(x)/cos(x)^(1/2)/(sin(x)*tan(x))^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {4482, 4485, 2677, 2679, 2681, 2645, 335, 304, 209, 212} \[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=-\frac {3 \sin (x) \arctan \left (\sqrt {\cos (x)}\right )}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}+\frac {3 \sin (x) \text {arctanh}\left (\sqrt {\cos (x)}\right )}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}+\frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}} \]

[In]

Int[(-Cos[x] + Sec[x])^(-5/2),x]

[Out]

(3*Cot[x])/(16*Sqrt[Sin[x]*Tan[x]]) - (Cot[x]*Csc[x]^2)/(4*Sqrt[Sin[x]*Tan[x]]) - (3*ArcTan[Sqrt[Cos[x]]]*Sin[
x])/(32*Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]]) + (3*ArcTanh[Sqrt[Cos[x]]]*Sin[x])/(32*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 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[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 2679

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

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))^{5/2}} \, dx \\ & = \frac {\left (\sqrt {\sin (x)} \sqrt {\tan (x)}\right ) \int \frac {1}{\sin ^{\frac {5}{2}}(x) \tan ^{\frac {5}{2}}(x)} \, dx}{\sqrt {\sin (x) \tan (x)}} \\ & = -\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}-\frac {\left (3 \sqrt {\sin (x)} \sqrt {\tan (x)}\right ) \int \frac {1}{\sin ^{\frac {5}{2}}(x) \sqrt {\tan (x)}} \, dx}{8 \sqrt {\sin (x) \tan (x)}} \\ & = \frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}-\frac {\left (3 \sqrt {\sin (x)} \sqrt {\tan (x)}\right ) \int \frac {1}{\sqrt {\sin (x)} \sqrt {\tan (x)}} \, dx}{32 \sqrt {\sin (x) \tan (x)}} \\ & = \frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}-\frac {(3 \sin (x)) \int \sqrt {\cos (x)} \csc (x) \, dx}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ & = \frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}+\frac {(3 \sin (x)) \text {Subst}\left (\int \frac {\sqrt {x}}{1-x^2} \, dx,x,\cos (x)\right )}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ & = \frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}+\frac {(3 \sin (x)) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\sqrt {\cos (x)}\right )}{16 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ & = \frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}+\frac {(3 \sin (x)) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\cos (x)}\right )}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}-\frac {(3 \sin (x)) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\cos (x)}\right )}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ & = \frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}-\frac {3 \arctan \left (\sqrt {\cos (x)}\right ) \sin (x)}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}+\frac {3 \text {arctanh}\left (\sqrt {\cos (x)}\right ) \sin (x)}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=-\frac {\cot (x) \left (3 \arctan \left (\sqrt [4]{\cos ^2(x)}\right ) \cos (x)-3 \text {arctanh}\left (\sqrt [4]{\cos ^2(x)}\right ) \cos (x)+\cos ^2(x)^{3/4} (5+3 \cos (2 x)) \cot (x) \csc ^3(x)\right ) \sqrt {\sin (x) \tan (x)}}{32 \cos ^2(x)^{3/4}} \]

[In]

Integrate[(-Cos[x] + Sec[x])^(-5/2),x]

[Out]

-1/32*(Cot[x]*(3*ArcTan[(Cos[x]^2)^(1/4)]*Cos[x] - 3*ArcTanh[(Cos[x]^2)^(1/4)]*Cos[x] + (Cos[x]^2)^(3/4)*(5 +
3*Cos[2*x])*Cot[x]*Csc[x]^3)*Sqrt[Sin[x]*Tan[x]])/(Cos[x]^2)^(3/4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(194\) vs. \(2(67)=134\).

Time = 1.23 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.14

method	result	size
default	\(-\frac {12 \cot \left (x \right )^{3} \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-3 \cot \left (x \right ) \arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )-3 \cot \left (x \right ) \ln \left (\frac {2 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )+1}\right )+3 \csc \left (x \right ) \arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )+3 \csc \left (x \right ) \ln \left (\frac {2 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )+1}\right )+4 \csc \left (x \right )^{2} \cot \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}{64 \sqrt {\sin \left (x \right ) \tan \left (x \right )}\, \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\)	\(195\)

[In]

int(1/(-cos(x)+sec(x))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/64/(sin(x)*tan(x))^(1/2)/(-cos(x)/(cos(x)+1)^2)^(1/2)*(12*cot(x)^3*(-cos(x)/(cos(x)+1)^2)^(1/2)-3*cot(x)*ar
ctan(1/2/(-cos(x)/(cos(x)+1)^2)^(1/2))-3*cot(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))+3*csc(x)*arctan(1/2/(-cos(x)/(cos(x)+1)^2)^(1/2))+3*csc(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))+4*csc(x)^2*cot(x)*(-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. 147 vs. \(2 (67) = 134\).

Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=\frac {3 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \arctan \left (\frac {2 \, \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) + 3 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac {{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) + 2 \, \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) - 4 \, {\left (3 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2}\right )} \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}}}{64 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )} \]

[In]

integrate(1/(-cos(x)+sec(x))^(5/2),x, algorithm="fricas")

[Out]

1/64*(3*(cos(x)^4 - 2*cos(x)^2 + 1)*arctan(2*sqrt(-(cos(x)^2 - 1)/cos(x))*cos(x)/((cos(x) - 1)*sin(x)))*sin(x)
+ 3*(cos(x)^4 - 2*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*(3*cos(x)^4 + cos(x)^2)*sqrt(-(cos(x)^2 - 1)/cos(x)))/((cos(x)^4 - 2*cos(x)^2 + 1)*sin(
x))

Sympy [F]

\[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=\int \frac {1}{\left (- \cos {\left (x \right )} + \sec {\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/(-cos(x)+sec(x))**(5/2),x)

[Out]

Integral((-cos(x) + sec(x))**(-5/2), x)

Maxima [F]

\[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=\int { \frac {1}{{\left (-\cos \left (x\right ) + \sec \left (x\right )\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(-cos(x)+sec(x))^(5/2),x, algorithm="maxima")

[Out]

integrate((-cos(x) + sec(x))^(-5/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (67) = 134\).

Time = 0.38 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=\frac {{\left (\frac {4 \, {\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2}} + 1\right )} \tan \left (\frac {1}{2} \, x\right )^{4}}{256 \, {\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}^{2}} - \frac {1}{64} \, \sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 2\right )} - \frac {\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1}{64 \, \tan \left (\frac {1}{2} \, x\right )^{2}} - \frac {{\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}^{2}}{256 \, \tan \left (\frac {1}{2} \, x\right )^{4}} - \frac {3}{64} \, \arcsin \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right ) + \frac {3}{64} \, \log \left (-\frac {\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1}{\tan \left (\frac {1}{2} \, x\right )^{2}}\right ) \]

[In]

integrate(1/(-cos(x)+sec(x))^(5/2),x, algorithm="giac")

[Out]

1/256*(4*(sqrt(-tan(1/2*x)^4 + 1) - 1)/tan(1/2*x)^2 + 1)*tan(1/2*x)^4/(sqrt(-tan(1/2*x)^4 + 1) - 1)^2 - 1/64*s
qrt(-tan(1/2*x)^4 + 1)*(tan(1/2*x)^2 - 2) - 1/64*(sqrt(-tan(1/2*x)^4 + 1) - 1)/tan(1/2*x)^2 - 1/256*(sqrt(-tan
(1/2*x)^4 + 1) - 1)^2/tan(1/2*x)^4 - 3/64*arcsin(tan(1/2*x)^2) + 3/64*log(-(sqrt(-tan(1/2*x)^4 + 1) - 1)/tan(1
/2*x)^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=\int \frac {1}{{\left (\frac {1}{\cos \left (x\right )}-\cos \left (x\right )\right )}^{5/2}} \,d x \]

[In]

int(1/(1/cos(x) - cos(x))^(5/2),x)

[Out]

int(1/(1/cos(x) - cos(x))^(5/2), x)