\(\int \frac {\sin (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx\) [490]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64 \[ \int \frac {\sin (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\frac {(-5-10 \cos (2 (a+b x))+4 \cos (4 (a+b x))+4 \cos (6 (a+b x))) \csc ^3(a+b x) \sec ^4(a+b x) \sqrt {\sin (2 (a+b x))}}{560 b} \] Input:

Integrate[Sin[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]^3*Sec[a + b*x]^4*Sqrt[Sin[2*(a + b*x)]])/(560*b)
 

Rubi [A] (verified)

Time = 0.51 (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, 4792, 3042, 4791, 3042, 4792, 3042, 4779}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4792

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4791

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4792

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4779

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

Input:

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

Output:

(6*((4*(Sin[a + b*x]/(3*b*Sin[2*a + 2*b*x]^(3/2)) - (2*Cos[a + b*x])/(3*b* 
Sqrt[Sin[2*a + 2*b*x]])))/5 - Cos[a + b*x]/(5*b*Sin[2*a + 2*b*x]^(5/2))))/ 
7 + Sin[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 4779

Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_.)*((g_.)*sin[(c_.) + (d_.)*(x_)])^( 
p_), x_Symbol] :> Simp[(-(e*Cos[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.

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.08 \[ \int \frac {\sin (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=-\frac {\sqrt {2} {\left (128 \, \cos \left (b x + a\right )^{6} - 160 \, \cos \left (b x + a\right )^{4} + 20 \, \cos \left (b x + a\right )^{2} + 5\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 128 \, {\left (\cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} \sin \left (b x + a\right )}{560 \, {\left (b \cos \left (b x + a\right )^{6} - b \cos \left (b x + a\right )^{4}\right )} \sin \left (b x + a\right )} \] Input:

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

Output:

-1/560*(sqrt(2)*(128*cos(b*x + a)^6 - 160*cos(b*x + a)^4 + 20*cos(b*x + a) 
^2 + 5)*sqrt(cos(b*x + a)*sin(b*x + a)) + 128*(cos(b*x + a)^6 - cos(b*x + 
a)^4)*sin(b*x + a))/((b*cos(b*x + a)^6 - b*cos(b*x + a)^4)*sin(b*x + a))
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 24.47 (sec) , antiderivative size = 351, normalized size of antiderivative = 3.34 \[ \int \frac {\sin (a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,1{}\mathrm {i}}{7\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^4}+\frac {16\,{\mathrm {e}}^{a\,3{}\mathrm {i}+b\,x\,3{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}}{35\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left (\frac {1{}\mathrm {i}}{7\,b}+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,8{}\mathrm {i}}{35\,b}\right )\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}}{{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^2\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^2}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left (\frac {16}{35\,b}-\frac {44\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}}{35\,b}\right )\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}}{{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^3\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^3} \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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