\(\int (\csc (x)-\sin (x))^{7/2} \, dx\) [207]

Optimal result

Integrand size = 11, antiderivative size = 73 \[ \int (\csc (x)-\sin (x))^{7/2} \, dx=\frac {8}{7} \cos (x) \cot ^2(x) \sqrt {\cos (x) \cot (x)}+\frac {2}{7} \cos ^3(x) \cot ^2(x) \sqrt {\cos (x) \cot (x)}-\frac {64}{35} \cot (x) \sqrt {\cos (x) \cot (x)} \csc (x)+\frac {256}{35} \sqrt {\cos (x) \cot (x)} \sec (x) \] Output:

8/7*cos(x)*cot(x)^2*(cos(x)*cot(x))^(1/2)+2/7*cos(x)^3*cot(x)^2*(cos(x)*co 
t(x))^(1/2)-64/35*cot(x)*(cos(x)*cot(x))^(1/2)*csc(x)+256/35*(cos(x)*cot(x 
))^(1/2)*sec(x)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.51 \[ \int (\csc (x)-\sin (x))^{7/2} \, dx=-\frac {1}{70} \sqrt {\cos (x) \cot (x)} \left (-512+115 \cos ^2(x)+5 \cos (x) \cos (3 x)+28 \cot ^2(x)\right ) \sec (x) \] Input:

Integrate[(Csc[x] - Sin[x])^(7/2),x]
 

Output:

-1/70*(Sqrt[Cos[x]*Cot[x]]*(-512 + 115*Cos[x]^2 + 5*Cos[x]*Cos[3*x] + 28*C 
ot[x]^2)*Sec[x])
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.33, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.091, Rules used = {3042, 4897, 3042, 4900, 3042, 3078, 3042, 3078, 3042, 3074, 3042, 3069}

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.

\(\displaystyle \int (\csc (x)-\sin (x))^{7/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (\csc (x)-\sin (x))^{7/2}dx\)

\(\Big \downarrow \) 4897

\(\displaystyle \int (\cos (x) \cot (x))^{7/2}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (\cos (x) \cot (x))^{7/2}dx\)

\(\Big \downarrow \) 4900

\(\displaystyle \frac {\sqrt {\cos (x) \cot (x)} \int \cos ^{\frac {7}{2}}(x) \cot ^{\frac {7}{2}}(x)dx}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\cos (x) \cot (x)} \int \sin \left (x+\frac {\pi }{2}\right )^{7/2} \left (-\tan \left (x+\frac {\pi }{2}\right )\right )^{7/2}dx}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\)

\(\Big \downarrow \) 3078

\(\displaystyle \frac {\sqrt {\cos (x) \cot (x)} \left (\frac {12}{7} \int \cos ^{\frac {3}{2}}(x) \cot ^{\frac {7}{2}}(x)dx+\frac {2}{7} \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)\right )}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\cos (x) \cot (x)} \left (\frac {12}{7} \int \sin \left (x+\frac {\pi }{2}\right )^{3/2} \left (-\tan \left (x+\frac {\pi }{2}\right )\right )^{7/2}dx+\frac {2}{7} \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)\right )}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\)

\(\Big \downarrow \) 3078

\(\displaystyle \frac {\sqrt {\cos (x) \cot (x)} \left (\frac {12}{7} \left (\frac {8}{3} \int \frac {\cot ^{\frac {7}{2}}(x)}{\sqrt {\cos (x)}}dx+\frac {2}{3} \cos ^{\frac {3}{2}}(x) \cot ^{\frac {5}{2}}(x)\right )+\frac {2}{7} \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)\right )}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\cos (x) \cot (x)} \left (\frac {12}{7} \left (\frac {8}{3} \int \frac {\left (-\tan \left (x+\frac {\pi }{2}\right )\right )^{7/2}}{\sqrt {\sin \left (x+\frac {\pi }{2}\right )}}dx+\frac {2}{3} \cos ^{\frac {3}{2}}(x) \cot ^{\frac {5}{2}}(x)\right )+\frac {2}{7} \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)\right )}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\)

\(\Big \downarrow \) 3074

\(\displaystyle \frac {\sqrt {\cos (x) \cot (x)} \left (\frac {12}{7} \left (\frac {8}{3} \left (-\frac {4}{5} \int \frac {\cot ^{\frac {3}{2}}(x)}{\sqrt {\cos (x)}}dx-\frac {2 \cot ^{\frac {5}{2}}(x)}{5 \sqrt {\cos (x)}}\right )+\frac {2}{3} \cos ^{\frac {3}{2}}(x) \cot ^{\frac {5}{2}}(x)\right )+\frac {2}{7} \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)\right )}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\cos (x) \cot (x)} \left (\frac {12}{7} \left (\frac {8}{3} \left (-\frac {4}{5} \int \frac {\left (-\tan \left (x+\frac {\pi }{2}\right )\right )^{3/2}}{\sqrt {\sin \left (x+\frac {\pi }{2}\right )}}dx-\frac {2 \cot ^{\frac {5}{2}}(x)}{5 \sqrt {\cos (x)}}\right )+\frac {2}{3} \cos ^{\frac {3}{2}}(x) \cot ^{\frac {5}{2}}(x)\right )+\frac {2}{7} \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)\right )}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\)

\(\Big \downarrow \) 3069

\(\displaystyle \frac {\sqrt {\cos (x) \cot (x)} \left (\frac {2}{7} \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)+\frac {12}{7} \left (\frac {2}{3} \cos ^{\frac {3}{2}}(x) \cot ^{\frac {5}{2}}(x)+\frac {8}{3} \left (\frac {8 \sqrt {\cot (x)}}{5 \sqrt {\cos (x)}}-\frac {2 \cot ^{\frac {5}{2}}(x)}{5 \sqrt {\cos (x)}}\right )\right )\right )}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\)

Input:

Int[(Csc[x] - Sin[x])^(7/2),x]
 

Output:

(Sqrt[Cos[x]*Cot[x]]*((2*Cos[x]^(7/2)*Cot[x]^(5/2))/7 + (12*((2*Cos[x]^(3/ 
2)*Cot[x]^(5/2))/3 + (8*((8*Sqrt[Cot[x]])/(5*Sqrt[Cos[x]]) - (2*Cot[x]^(5/ 
2))/(5*Sqrt[Cos[x]])))/3))/7))/(Sqrt[Cos[x]]*Sqrt[Cot[x]])
 

Defintions of rubi rules used

rule 3042

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

rule 3069

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

rule 3074

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

rule 3078

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

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 [A] (verified)

Time = 1.69 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.49

method	result	size
default	\(\frac {2 \sec \left (x \right ) \csc \left (x \right )^{2} \left (5 \cos \left (x \right )^{6}+20 \cos \left (x \right )^{4}-160 \cos \left (x \right )^{2}+128\right ) \sqrt {\cos \left (x \right ) \cot \left (x \right )}}{35}\)	\(36\)

Input:

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

Output:

2/35*sec(x)*csc(x)^2*(5*cos(x)^6+20*cos(x)^4-160*cos(x)^2+128)*(cos(x)*cot 
(x))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.60 \[ \int (\csc (x)-\sin (x))^{7/2} \, dx=-\frac {2 \, {\left (5 \, \cos \left (x\right )^{6} + 20 \, \cos \left (x\right )^{4} - 160 \, \cos \left (x\right )^{2} + 128\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}}}{35 \, {\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )}} \] Input:

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

Output:

-2/35*(5*cos(x)^6 + 20*cos(x)^4 - 160*cos(x)^2 + 128)*sqrt(cos(x)^2/sin(x) 
)/(cos(x)^3 - cos(x))
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (57) = 114\).

Time = 0.24 (sec) , antiderivative size = 578, normalized size of antiderivative = 7.92 \[ \int (\csc (x)-\sin (x))^{7/2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/280*(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1)^(1/4)*(cos(x)^2 + sin(x)^2 - 2 
*cos(x) + 1)^(1/4)*(((5*cos(21/2*x) + 105*cos(17/2*x) - 2275*cos(13/2*x) + 
 5817*cos(9/2*x) - 5*cos(7/2*x) - 5817*cos(5/2*x) - 105*cos(3/2*x) + 2275* 
cos(1/2*x) - 5*sin(21/2*x) - 105*sin(17/2*x) + 2275*sin(13/2*x) - 5817*sin 
(9/2*x) - 5*sin(7/2*x) + 5817*sin(5/2*x) - 105*sin(3/2*x) - 2275*sin(1/2*x 
))*cos(7/2*arctan2(sin(x), cos(x) - 1)) + (5*cos(21/2*x) + 105*cos(17/2*x) 
 - 2275*cos(13/2*x) + 5817*cos(9/2*x) - 5*cos(7/2*x) - 5817*cos(5/2*x) - 1 
05*cos(3/2*x) + 2275*cos(1/2*x) + 5*sin(21/2*x) + 105*sin(17/2*x) - 2275*s 
in(13/2*x) + 5817*sin(9/2*x) + 5*sin(7/2*x) - 5817*sin(5/2*x) + 105*sin(3/ 
2*x) + 2275*sin(1/2*x))*sin(7/2*arctan2(sin(x), cos(x) - 1)))*cos(7/2*arct 
an2(sin(x), cos(x) + 1)) + ((5*cos(21/2*x) + 105*cos(17/2*x) - 2275*cos(13 
/2*x) + 5817*cos(9/2*x) - 5*cos(7/2*x) - 5817*cos(5/2*x) - 105*cos(3/2*x) 
+ 2275*cos(1/2*x) + 5*sin(21/2*x) + 105*sin(17/2*x) - 2275*sin(13/2*x) + 5 
817*sin(9/2*x) + 5*sin(7/2*x) - 5817*sin(5/2*x) + 105*sin(3/2*x) + 2275*si 
n(1/2*x))*cos(7/2*arctan2(sin(x), cos(x) - 1)) - (5*cos(21/2*x) + 105*cos( 
17/2*x) - 2275*cos(13/2*x) + 5817*cos(9/2*x) - 5*cos(7/2*x) - 5817*cos(5/2 
*x) - 105*cos(3/2*x) + 2275*cos(1/2*x) - 5*sin(21/2*x) - 105*sin(17/2*x) + 
 2275*sin(13/2*x) - 5817*sin(9/2*x) - 5*sin(7/2*x) + 5817*sin(5/2*x) - 105 
*sin(3/2*x) - 2275*sin(1/2*x))*sin(7/2*arctan2(sin(x), cos(x) - 1)))*sin(7 
/2*arctan2(sin(x), cos(x) + 1)))/(cos(x)^8 + sin(x)^8 + 4*(cos(x)^2 + 1...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int((1/sin(x) - sin(x))^(7/2),x)
 

Output:

int((1/sin(x) - sin(x))^(7/2), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int (\csc (x)-\sin (x))^{7/2} \, dx=3 \left (\int \sqrt {\csc \left (x \right )-\sin \left (x \right )}\, \csc \left (x \right ) \sin \left (x \right )^{2}d x \right )+\int \sqrt {\csc \left (x \right )-\sin \left (x \right )}\, \csc \left (x \right )^{3}d x -3 \left (\int \sqrt {\csc \left (x \right )-\sin \left (x \right )}\, \csc \left (x \right )^{2} \sin \left (x \right )d x \right )-\left (\int \sqrt {\csc \left (x \right )-\sin \left (x \right )}\, \sin \left (x \right )^{3}d x \right ) \] Input:

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

Output:

3*int(sqrt(csc(x) - sin(x))*csc(x)*sin(x)**2,x) + int(sqrt(csc(x) - sin(x) 
)*csc(x)**3,x) - 3*int(sqrt(csc(x) - sin(x))*csc(x)**2*sin(x),x) - int(sqr 
t(csc(x) - sin(x))*sin(x)**3,x)