\(\int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx\) [321]

Optimal result

Integrand size = 11, antiderivative size = 118 \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}-\frac {5 \arctan \left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{128 \sqrt {\cos (x) \cot (x)}}-\frac {5 \text {arctanh}\left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{128 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}} \]

[Out]

5/192*sec(x)/(cos(x)*cot(x))^(1/2)-5/48*sec(x)^3/(cos(x)*cot(x))^(1/2)-5/128*arctan((-sin(x))^(1/2))*cot(x)*(-
sin(x))^(1/2)/(cos(x)*cot(x))^(1/2)-5/128*arctanh((-sin(x))^(1/2))*cot(x)*(-sin(x))^(1/2)/(cos(x)*cot(x))^(1/2
)+1/6*sec(x)^3*tan(x)^2/(cos(x)*cot(x))^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {4482, 4485, 2677, 2679, 2681, 2644, 335, 218, 212, 209} \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=-\frac {5 \sqrt {-\sin (x)} \cot (x) \arctan \left (\sqrt {-\sin (x)}\right )}{128 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sqrt {-\sin (x)} \cot (x) \text {arctanh}\left (\sqrt {-\sin (x)}\right )}{128 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}+\frac {\tan ^2(x) \sec ^3(x)}{6 \sqrt {\cos (x) \cot (x)}} \]

[In]

Int[(Csc[x] - Sin[x])^(-7/2),x]

[Out]

(5*Sec[x])/(192*Sqrt[Cos[x]*Cot[x]]) - (5*Sec[x]^3)/(48*Sqrt[Cos[x]*Cot[x]]) - (5*ArcTan[Sqrt[-Sin[x]]]*Cot[x]
*Sqrt[-Sin[x]])/(128*Sqrt[Cos[x]*Cot[x]]) - (5*ArcTanh[Sqrt[-Sin[x]]]*Cot[x]*Sqrt[-Sin[x]])/(128*Sqrt[Cos[x]*C
ot[x]]) + (Sec[x]^3*Tan[x]^2)/(6*Sqrt[Cos[x]*Cot[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 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, 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}{(\cos (x) \cot (x))^{7/2}} \, dx \\ & = \frac {\left (\sqrt {\cos (x)} \sqrt {\cot (x)}\right ) \int \frac {1}{\cos ^{\frac {7}{2}}(x) \cot ^{\frac {7}{2}}(x)} \, dx}{\sqrt {\cos (x) \cot (x)}} \\ & = \frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}-\frac {\left (5 \sqrt {\cos (x)} \sqrt {\cot (x)}\right ) \int \frac {1}{\cos ^{\frac {7}{2}}(x) \cot ^{\frac {3}{2}}(x)} \, dx}{12 \sqrt {\cos (x) \cot (x)}} \\ & = -\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}+\frac {\left (5 \sqrt {\cos (x)} \sqrt {\cot (x)}\right ) \int \frac {\sqrt {\cot (x)}}{\cos ^{\frac {7}{2}}(x)} \, dx}{96 \sqrt {\cos (x) \cot (x)}} \\ & = \frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}+\frac {\left (5 \sqrt {\cos (x)} \sqrt {\cot (x)}\right ) \int \frac {\sqrt {\cot (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx}{128 \sqrt {\cos (x) \cot (x)}} \\ & = \frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}+\frac {\left (5 \cot (x) \sqrt {-\sin (x)}\right ) \int \frac {\sec (x)}{\sqrt {-\sin (x)}} \, dx}{128 \sqrt {\cos (x) \cot (x)}} \\ & = \frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}-\frac {\left (5 \cot (x) \sqrt {-\sin (x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-x^2\right )} \, dx,x,-\sin (x)\right )}{128 \sqrt {\cos (x) \cot (x)}} \\ & = \frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}-\frac {\left (5 \cot (x) \sqrt {-\sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\sqrt {-\sin (x)}\right )}{64 \sqrt {\cos (x) \cot (x)}} \\ & = \frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}-\frac {\left (5 \cot (x) \sqrt {-\sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {-\sin (x)}\right )}{128 \sqrt {\cos (x) \cot (x)}}-\frac {\left (5 \cot (x) \sqrt {-\sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-\sin (x)}\right )}{128 \sqrt {\cos (x) \cot (x)}} \\ & = \frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}-\frac {5 \arctan \left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{128 \sqrt {\cos (x) \cot (x)}}-\frac {5 \text {arctanh}\left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{128 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.63 \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\frac {15 \arctan \left (\sqrt [4]{\sin ^2(x)}\right ) \cos (x)+15 \text {arctanh}\left (\sqrt [4]{\sin ^2(x)}\right ) \cos (x)+2 \sec (x) \left (5-52 \sec ^2(x)+32 \sec ^4(x)\right ) \sqrt [4]{\sin ^2(x)}}{384 \sqrt {\cos (x) \cot (x)} \sqrt [4]{\sin ^2(x)}} \]

[In]

Integrate[(Csc[x] - Sin[x])^(-7/2),x]

[Out]

(15*ArcTan[(Sin[x]^2)^(1/4)]*Cos[x] + 15*ArcTanh[(Sin[x]^2)^(1/4)]*Cos[x] + 2*Sec[x]*(5 - 52*Sec[x]^2 + 32*Sec
[x]^4)*(Sin[x]^2)^(1/4))/(384*Sqrt[Cos[x]*Cot[x]]*(Sin[x]^2)^(1/4))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(185\) vs. \(2(90)=180\).

Time = 16.42 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.58

method	result	size
default	\(\frac {\tan \left (x \right )^{2} \sec \left (x \right )^{3} \left (15 \cos \left (x \right )^{6} \arctan \left (\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (\cot \left (x \right )+\csc \left (x \right )\right )\right )-15 \cos \left (x \right )^{6} \operatorname {arctanh}\left (\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (\cot \left (x \right )+\csc \left (x \right )\right )\right )-10 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )^{5}-10 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )^{4}+104 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )^{3}+104 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )^{2}-64 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )-64 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\right )}{384 \left (\cos \left (x \right )-1\right ) \left (\cos \left (x \right )+1\right )^{2} \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {\cot \left (x \right ) \cos \left (x \right )}}\)	\(186\)

[In]

int(1/(csc(x)-sin(x))^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/384*tan(x)^2*sec(x)^3*(15*cos(x)^6*arctan((sin(x)/(cos(x)+1)^2)^(1/2)*(cot(x)+csc(x)))-15*cos(x)^6*arctanh((
sin(x)/(cos(x)+1)^2)^(1/2)*(cot(x)+csc(x)))-10*(sin(x)/(cos(x)+1)^2)^(1/2)*cos(x)^5-10*(sin(x)/(cos(x)+1)^2)^(
1/2)*cos(x)^4+104*(sin(x)/(cos(x)+1)^2)^(1/2)*cos(x)^3+104*(sin(x)/(cos(x)+1)^2)^(1/2)*cos(x)^2-64*(sin(x)/(co
s(x)+1)^2)^(1/2)*cos(x)-64*(sin(x)/(cos(x)+1)^2)^(1/2))/(cos(x)-1)/(cos(x)+1)^2/(sin(x)/(cos(x)+1)^2)^(1/2)/(c
ot(x)*cos(x))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.42 \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=-\frac {30 \, \arctan \left (\frac {2 \, \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{\cos \left (x\right ) \sin \left (x\right ) - \cos \left (x\right )}\right ) \cos \left (x\right )^{7} - 15 \, \cos \left (x\right )^{7} \log \left (\frac {\cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} + 6 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) + 4 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} - 2 \, \cos \left (x\right ) + 4}{\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4}\right ) - 8 \, {\left (5 \, \cos \left (x\right )^{4} - 52 \, \cos \left (x\right )^{2} + 32\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{1536 \, \cos \left (x\right )^{7}} \]

[In]

integrate(1/(csc(x)-sin(x))^(7/2),x, algorithm="fricas")

[Out]

-1/1536*(30*arctan(2*sqrt(cos(x)^2/sin(x))*sin(x)/(cos(x)*sin(x) - cos(x)))*cos(x)^7 - 15*cos(x)^7*log((cos(x)
^3 - 5*cos(x)^2 - (cos(x)^2 + 6*cos(x) + 4)*sin(x) + 4*(cos(x)^2 - (cos(x) + 1)*sin(x) - 1)*sqrt(cos(x)^2/sin(
x)) - 2*cos(x) + 4)/(cos(x)^3 + 3*cos(x)^2 - (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)) - 8*(5*cos(x)^4
- 52*cos(x)^2 + 32)*sqrt(cos(x)^2/sin(x))*sin(x))/cos(x)^7

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(csc(x)-sin(x))**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(csc(x)-sin(x))^(7/2),x, algorithm="maxima")

[Out]

integrate((csc(x) - sin(x))^(-7/2), x)

Giac [F]

\[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(csc(x)-sin(x))^(7/2),x, algorithm="giac")

[Out]

integrate((csc(x) - sin(x))^(-7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (x\right )}-\sin \left (x\right )\right )}^{7/2}} \,d x \]

[In]

int(1/(1/sin(x) - sin(x))^(7/2),x)

[Out]

int(1/(1/sin(x) - sin(x))^(7/2), x)