\(\int \frac {\cos (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx\) [577]

Optimal result

Integrand size = 20, antiderivative size = 105 \[ \int \frac {\cos (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=-\frac {\cos (a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}+\frac {6 \sin (a+b x)}{35 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {8 \cos (a+b x)}{35 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {16 \sin (a+b x)}{35 b \sqrt {\sin (2 a+2 b x)}} \] Output:

-1/7*cos(b*x+a)/b/sin(2*b*x+2*a)^(7/2)+6/35*sin(b*x+a)/b/sin(2*b*x+2*a)^(5 
/2)-8/35*cos(b*x+a)/b/sin(2*b*x+2*a)^(3/2)+16/35*sin(b*x+a)/b/sin(2*b*x+2* 
a)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64 \[ \int \frac {\cos (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\frac {(5-10 \cos (2 (a+b x))-4 \cos (4 (a+b x))+4 \cos (6 (a+b x))) \csc ^4(a+b x) \sec ^3(a+b x) \sqrt {\sin (2 (a+b x))}}{560 b} \] Input:

Integrate[Cos[a + b*x]/Sin[2*a + 2*b*x]^(9/2),x]
 

Output:

((5 - 10*Cos[2*(a + b*x)] - 4*Cos[4*(a + b*x)] + 4*Cos[6*(a + b*x)])*Csc[a 
 + b*x]^4*Sec[a + b*x]^3*Sqrt[Sin[2*(a + b*x)]])/(560*b)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 4791, 3042, 4792, 3042, 4791, 3042, 4780}

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.

\(\displaystyle \int \frac {\cos (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (a+b x)}{\sin (2 a+2 b x)^{9/2}}dx\)

\(\Big \downarrow \) 4791

\(\displaystyle \frac {6}{7} \int \frac {\sin (a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)}dx-\frac {\cos (a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6}{7} \int \frac {\sin (a+b x)}{\sin (2 a+2 b x)^{7/2}}dx-\frac {\cos (a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

\(\Big \downarrow \) 4792

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \int \frac {\cos (a+b x)}{\sin ^{\frac {5}{2}}(2 a+2 b x)}dx+\frac {\sin (a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\right )-\frac {\cos (a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \int \frac {\cos (a+b x)}{\sin (2 a+2 b x)^{5/2}}dx+\frac {\sin (a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\right )-\frac {\cos (a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

\(\Big \downarrow \) 4791

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\sin (a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)}dx-\frac {\cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}\right )+\frac {\sin (a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\right )-\frac {\cos (a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\sin (a+b x)}{\sin (2 a+2 b x)^{3/2}}dx-\frac {\cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}\right )+\frac {\sin (a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\right )-\frac {\cos (a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

\(\Big \downarrow \) 4780

\(\displaystyle \frac {6}{7} \left (\frac {\sin (a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}+\frac {4}{5} \left (\frac {2 \sin (a+b x)}{3 b \sqrt {\sin (2 a+2 b x)}}-\frac {\cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}\right )\right )-\frac {\cos (a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

Input:

Int[Cos[a + b*x]/Sin[2*a + 2*b*x]^(9/2),x]
 

Output:

(6*((4*(-1/3*Cos[a + b*x]/(b*Sin[2*a + 2*b*x]^(3/2)) + (2*Sin[a + b*x])/(3 
*b*Sqrt[Sin[2*a + 2*b*x]])))/5 + Sin[a + b*x]/(5*b*Sin[2*a + 2*b*x]^(5/2)) 
))/7 - Cos[a + b*x]/(7*b*Sin[2*a + 2*b*x]^(7/2))
 

Defintions of rubi rules used

rule 3042

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

rule 4780

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_.)*((g_.)*sin[(c_.) + (d_.)*(x_)])^( 
p_), x_Symbol] :> Simp[(e*Sin[a + b*x])^m*((g*Sin[c + d*x])^(p + 1)/(b*g*m) 
), x] /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b 
, 2] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]
 

rule 4791

Int[cos[(a_.) + (b_.)*(x_)]*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] 
 :> Simp[Cos[a + b*x]*((g*Sin[c + d*x])^(p + 1)/(2*b*g*(p + 1))), x] + Simp 
[(2*p + 3)/(2*g*(p + 1))   Int[Sin[a + b*x]*(g*Sin[c + d*x])^(p + 1), x], x 
] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] &&  !Int 
egerQ[p] && LtQ[p, -1] && IntegerQ[2*p]
 

rule 4792

Int[sin[(a_.) + (b_.)*(x_)]*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] 
 :> Simp[(-Sin[a + b*x])*((g*Sin[c + d*x])^(p + 1)/(2*b*g*(p + 1))), x] + S 
imp[(2*p + 3)/(2*g*(p + 1))   Int[Cos[a + b*x]*(g*Sin[c + d*x])^(p + 1), x] 
, x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] &&  ! 
IntegerQ[p] && LtQ[p, -1] && IntegerQ[2*p]
 

Maple [F(-1)]

Timed out.

\[\int \frac {\cos \left (b x +a \right )}{\sin \left (2 b x +2 a \right )^{\frac {9}{2}}}d x\]

Input:

int(cos(b*x+a)/sin(2*b*x+2*a)^(9/2),x)
 

Output:

int(cos(b*x+a)/sin(2*b*x+2*a)^(9/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.12 \[ \int \frac {\cos (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\frac {128 \, \cos \left (b x + a\right )^{7} - 256 \, \cos \left (b x + a\right )^{5} + 128 \, \cos \left (b x + a\right )^{3} + \sqrt {2} {\left (128 \, \cos \left (b x + a\right )^{6} - 224 \, \cos \left (b x + a\right )^{4} + 84 \, \cos \left (b x + a\right )^{2} + 7\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{560 \, {\left (b \cos \left (b x + a\right )^{7} - 2 \, b \cos \left (b x + a\right )^{5} + b \cos \left (b x + a\right )^{3}\right )}} \] Input:

integrate(cos(b*x+a)/sin(2*b*x+2*a)^(9/2),x, algorithm="fricas")
 

Output:

1/560*(128*cos(b*x + a)^7 - 256*cos(b*x + a)^5 + 128*cos(b*x + a)^3 + sqrt 
(2)*(128*cos(b*x + a)^6 - 224*cos(b*x + a)^4 + 84*cos(b*x + a)^2 + 7)*sqrt 
(cos(b*x + a)*sin(b*x + a)))/(b*cos(b*x + a)^7 - 2*b*cos(b*x + a)^5 + b*co 
s(b*x + a)^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\text {Timed out} \] Input:

integrate(cos(b*x+a)/sin(2*b*x+2*a)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\int { \frac {\cos \left (b x + a\right )}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate(cos(b*x+a)/sin(2*b*x+2*a)^(9/2),x, algorithm="maxima")
 

Output:

integrate(cos(b*x + a)/sin(2*b*x + 2*a)^(9/2), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cos (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\text {Timed out} \] Input:

integrate(cos(b*x+a)/sin(2*b*x+2*a)^(9/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 23.22 (sec) , antiderivative size = 350, normalized size of antiderivative = 3.33 \[ \int \frac {\cos (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}}{7\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}^4}+\frac {{\mathrm {e}}^{a\,3{}\mathrm {i}+b\,x\,3{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,16{}\mathrm {i}}{35\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left (\frac {1}{7\,b}-\frac {8\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}}{35\,b}\right )\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}}{{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )}^2\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}^2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left (\frac {16{}\mathrm {i}}{35\,b}+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,44{}\mathrm {i}}{35\,b}\right )\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}}{{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )}^3\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}^3} \] Input:

int(cos(a + b*x)/sin(2*a + 2*b*x)^(9/2),x)
 

Output:

(exp(a*3i + b*x*3i)*((exp(- a*2i - b*x*2i)*1i)/2 - (exp(a*2i + b*x*2i)*1i) 
/2)^(1/2)*16i)/(35*b*(exp(a*2i + b*x*2i) + 1)*(exp(a*2i + b*x*2i)*1i - 1i) 
) - (exp(a*1i + b*x*1i)*((exp(- a*2i - b*x*2i)*1i)/2 - (exp(a*2i + b*x*2i) 
*1i)/2)^(1/2))/(7*b*(exp(a*2i + b*x*2i)*1i - 1i)^4) - (exp(a*1i + b*x*1i)* 
(1/(7*b) - (8*exp(a*2i + b*x*2i))/(35*b))*((exp(- a*2i - b*x*2i)*1i)/2 - ( 
exp(a*2i + b*x*2i)*1i)/2)^(1/2))/((exp(a*2i + b*x*2i) + 1)^2*(exp(a*2i + b 
*x*2i)*1i - 1i)^2) + (exp(a*1i + b*x*1i)*(16i/(35*b) + (exp(a*2i + b*x*2i) 
*44i)/(35*b))*((exp(- a*2i - b*x*2i)*1i)/2 - (exp(a*2i + b*x*2i)*1i)/2)^(1 
/2))/((exp(a*2i + b*x*2i) + 1)^3*(exp(a*2i + b*x*2i)*1i - 1i)^3)
 

Reduce [F]

\[ \int \frac {\cos (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\int \frac {\sqrt {\sin \left (2 b x +2 a \right )}\, \cos \left (b x +a \right )}{\sin \left (2 b x +2 a \right )^{5}}d x \] Input:

int(cos(b*x+a)/sin(2*b*x+2*a)^(9/2),x)
 

Output:

int((sqrt(sin(2*a + 2*b*x))*cos(a + b*x))/sin(2*a + 2*b*x)**5,x)