Integrand size = 11, antiderivative size = 50 \[ \int \frac {\sin (x)}{(a+a \sin (x))^3} \, dx=\frac {\cos (x)}{5 (a+a \sin (x))^3}-\frac {\cos (x)}{5 a (a+a \sin (x))^2}-\frac {\cos (x)}{5 \left (a^3+a^3 \sin (x)\right )} \] Output:
1/5*cos(x)/(a+a*sin(x))^3-1/5*cos(x)/a/(a+a*sin(x))^2-cos(x)/(5*a^3+5*a^3* sin(x))
Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.50 \[ \int \frac {\sin (x)}{(a+a \sin (x))^3} \, dx=-\frac {\cos (x) \left (1+3 \sin (x)+\sin ^2(x)\right )}{5 a^3 (1+\sin (x))^3} \] Input:
Integrate[Sin[x]/(a + a*Sin[x])^3,x]
Output:
-1/5*(Cos[x]*(1 + 3*Sin[x] + Sin[x]^2))/(a^3*(1 + Sin[x])^3)
Time = 0.33 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 3229, 3042, 3129, 3042, 3127}
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.
\(\displaystyle \int \frac {\sin (x)}{(a \sin (x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (x)}{(a \sin (x)+a)^3}dx\) |
\(\Big \downarrow \) 3229 |
\(\displaystyle \frac {3 \int \frac {1}{(\sin (x) a+a)^2}dx}{5 a}+\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \int \frac {1}{(\sin (x) a+a)^2}dx}{5 a}+\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {3 \left (\frac {\int \frac {1}{\sin (x) a+a}dx}{3 a}-\frac {\cos (x)}{3 (a \sin (x)+a)^2}\right )}{5 a}+\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \left (\frac {\int \frac {1}{\sin (x) a+a}dx}{3 a}-\frac {\cos (x)}{3 (a \sin (x)+a)^2}\right )}{5 a}+\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle \frac {\cos (x)}{5 (a \sin (x)+a)^3}+\frac {3 \left (-\frac {\cos (x)}{3 a (a \sin (x)+a)}-\frac {\cos (x)}{3 (a \sin (x)+a)^2}\right )}{5 a}\) |
Input:
Int[Sin[x]/(a + a*Sin[x])^3,x]
Output:
Cos[x]/(5*(a + a*Sin[x])^3) + (3*(-1/3*Cos[x]/(a + a*Sin[x])^2 - Cos[x]/(3 *a*(a + a*Sin[x]))))/(5*a)
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b ^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((a + b*Sin[e + f* x])^m/(a*f*(2*m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)) I nt[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.76
method | result | size |
parallelrisch | \(\frac {-\frac {2}{5}-2 \tan \left (\frac {x}{2}\right )^{3}-2 \tan \left (\frac {x}{2}\right )^{2}-2 \tan \left (\frac {x}{2}\right )}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(38\) |
risch | \(-\frac {2 i \left (5 i {\mathrm e}^{2 i x}+5 \,{\mathrm e}^{3 i x}-i-5 \,{\mathrm e}^{i x}\right )}{5 \left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}\) | \(42\) |
default | \(\frac {-\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}}{a^{3}}\) | \(45\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}-\frac {2 \tan \left (\frac {x}{2}\right )^{5}}{a}-\frac {4 \tan \left (\frac {x}{2}\right )^{3}}{a}-\frac {2 \tan \left (\frac {x}{2}\right )^{4}}{a}-\frac {2}{5 a}-\frac {12 \tan \left (\frac {x}{2}\right )^{2}}{5 a}}{\left (\tan \left (\frac {x}{2}\right )^{2}+1\right ) a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(82\) |
Input:
int(sin(x)/(a+a*sin(x))^3,x,method=_RETURNVERBOSE)
Output:
2/5*(-1-5*tan(1/2*x)^3-5*tan(1/2*x)^2-5*tan(1/2*x))/a^3/(tan(1/2*x)+1)^5
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.76 \[ \int \frac {\sin (x)}{(a+a \sin (x))^3} \, dx=-\frac {\cos \left (x\right )^{3} - 2 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} + 3 \, \cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) + 1}{5 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \] Input:
integrate(sin(x)/(a+a*sin(x))^3,x, algorithm="fricas")
Output:
-1/5*(cos(x)^3 - 2*cos(x)^2 - (cos(x)^2 + 3*cos(x) + 1)*sin(x) - 2*cos(x) + 1)/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))
Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (44) = 88\).
Time = 1.23 (sec) , antiderivative size = 277, normalized size of antiderivative = 5.54 \[ \int \frac {\sin (x)}{(a+a \sin (x))^3} \, dx=- \frac {10 \tan ^{3}{\left (\frac {x}{2} \right )}}{5 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan {\left (\frac {x}{2} \right )} + 5 a^{3}} - \frac {10 \tan ^{2}{\left (\frac {x}{2} \right )}}{5 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan {\left (\frac {x}{2} \right )} + 5 a^{3}} - \frac {10 \tan {\left (\frac {x}{2} \right )}}{5 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan {\left (\frac {x}{2} \right )} + 5 a^{3}} - \frac {2}{5 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan {\left (\frac {x}{2} \right )} + 5 a^{3}} \] Input:
integrate(sin(x)/(a+a*sin(x))**3,x)
Output:
-10*tan(x/2)**3/(5*a**3*tan(x/2)**5 + 25*a**3*tan(x/2)**4 + 50*a**3*tan(x/ 2)**3 + 50*a**3*tan(x/2)**2 + 25*a**3*tan(x/2) + 5*a**3) - 10*tan(x/2)**2/ (5*a**3*tan(x/2)**5 + 25*a**3*tan(x/2)**4 + 50*a**3*tan(x/2)**3 + 50*a**3* tan(x/2)**2 + 25*a**3*tan(x/2) + 5*a**3) - 10*tan(x/2)/(5*a**3*tan(x/2)**5 + 25*a**3*tan(x/2)**4 + 50*a**3*tan(x/2)**3 + 50*a**3*tan(x/2)**2 + 25*a* *3*tan(x/2) + 5*a**3) - 2/(5*a**3*tan(x/2)**5 + 25*a**3*tan(x/2)**4 + 50*a **3*tan(x/2)**3 + 50*a**3*tan(x/2)**2 + 25*a**3*tan(x/2) + 5*a**3)
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (44) = 88\).
Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.32 \[ \int \frac {\sin (x)}{(a+a \sin (x))^3} \, dx=-\frac {2 \, {\left (\frac {5 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + 1\right )}}{5 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {10 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} \] Input:
integrate(sin(x)/(a+a*sin(x))^3,x, algorithm="maxima")
Output:
-2/5*(5*sin(x)/(cos(x) + 1) + 5*sin(x)^2/(cos(x) + 1)^2 + 5*sin(x)^3/(cos( x) + 1)^3 + 1)/(a^3 + 5*a^3*sin(x)/(cos(x) + 1) + 10*a^3*sin(x)^2/(cos(x) + 1)^2 + 10*a^3*sin(x)^3/(cos(x) + 1)^3 + 5*a^3*sin(x)^4/(cos(x) + 1)^4 + a^3*sin(x)^5/(cos(x) + 1)^5)
Time = 0.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.74 \[ \int \frac {\sin (x)}{(a+a \sin (x))^3} \, dx=-\frac {2 \, {\left (5 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 5 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 5 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{5 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \] Input:
integrate(sin(x)/(a+a*sin(x))^3,x, algorithm="giac")
Output:
-2/5*(5*tan(1/2*x)^3 + 5*tan(1/2*x)^2 + 5*tan(1/2*x) + 1)/(a^3*(tan(1/2*x) + 1)^5)
Time = 17.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.74 \[ \int \frac {\sin (x)}{(a+a \sin (x))^3} \, dx=-\frac {2\,\left (5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+5\,\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}{5\,a^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \] Input:
int(sin(x)/(a + a*sin(x))^3,x)
Output:
-(2*(5*tan(x/2) + 5*tan(x/2)^2 + 5*tan(x/2)^3 + 1))/(5*a^3*(tan(x/2) + 1)^ 5)
Time = 0.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.12 \[ \int \frac {\sin (x)}{(a+a \sin (x))^3} \, dx=\frac {\sin \left (x \right ) \left (\cos \left (x \right )+2 \sin \left (x \right )^{2}+5 \sin \left (x \right )+1\right )}{5 a^{3} \left (\cos \left (x \right ) \sin \left (x \right )^{2}+2 \cos \left (x \right ) \sin \left (x \right )+\cos \left (x \right )-\sin \left (x \right )^{3}-3 \sin \left (x \right )^{2}-3 \sin \left (x \right )-1\right )} \] Input:
int(sin(x)/(a+a*sin(x))^3,x)
Output:
(sin(x)*(cos(x) + 2*sin(x)**2 + 5*sin(x) + 1))/(5*a**3*(cos(x)*sin(x)**2 + 2*cos(x)*sin(x) + cos(x) - sin(x)**3 - 3*sin(x)**2 - 3*sin(x) - 1))