Integrand size = 8, antiderivative size = 50 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {\cos (x)}{5 (a+a \sin (x))^3}-\frac {2 \cos (x)}{15 a (a+a \sin (x))^2}-\frac {2 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )} \] Output:
-1/5*cos(x)/(a+a*sin(x))^3-2/15*cos(x)/a/(a+a*sin(x))^2-2*cos(x)/(15*a^3+1 5*a^3*sin(x))
Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {5 \cos \left (\frac {3 x}{2}\right )-10 \sin \left (\frac {x}{2}\right )+\sin \left (\frac {5 x}{2}\right )}{15 a^3 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5} \] Input:
Integrate[(a + a*Sin[x])^(-3),x]
Output:
-1/15*(5*Cos[(3*x)/2] - 10*Sin[x/2] + Sin[(5*x)/2])/(a^3*(Cos[x/2] + Sin[x /2])^5)
Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3129, 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.
\(\displaystyle \int \frac {1}{(a \sin (x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a \sin (x)+a)^3}dx\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {2 \int \frac {1}{(\sin (x) a+a)^2}dx}{5 a}-\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \int \frac {1}{(\sin (x) a+a)^2}dx}{5 a}-\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {2 \left (\frac {\int \frac {1}{\sin (x) a+a}dx}{3 a}-\frac {\cos (x)}{3 (a \sin (x)+a)^2}\right )}{5 a}-\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\frac {\int \frac {1}{\sin (x) a+a}dx}{3 a}-\frac {\cos (x)}{3 (a \sin (x)+a)^2}\right )}{5 a}-\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle \frac {2 \left (-\frac {\cos (x)}{3 a (a \sin (x)+a)}-\frac {\cos (x)}{3 (a \sin (x)+a)^2}\right )}{5 a}-\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
Input:
Int[(a + a*Sin[x])^(-3),x]
Output:
-1/5*Cos[x]/(a + a*Sin[x])^3 + (2*(-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]
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.66
method | result | size |
risch | \(\frac {-\frac {4}{15}+\frac {8 \,{\mathrm e}^{2 i x}}{3}+\frac {4 i {\mathrm e}^{i x}}{3}}{\left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}\) | \(33\) |
parallelrisch | \(\frac {-\frac {14}{15}-2 \tan \left (\frac {x}{2}\right )^{4}-4 \tan \left (\frac {x}{2}\right )^{3}-\frac {16 \tan \left (\frac {x}{2}\right )^{2}}{3}-\frac {8 \tan \left (\frac {x}{2}\right )}{3}}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(46\) |
default | \(\frac {-\frac {16}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2}{\tan \left (\frac {x}{2}\right )+1}+\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}}{a^{3}}\) | \(57\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )^{4}}{a}-\frac {14}{15 a}-\frac {4 \tan \left (\frac {x}{2}\right )^{3}}{a}-\frac {8 \tan \left (\frac {x}{2}\right )}{3 a}-\frac {16 \tan \left (\frac {x}{2}\right )^{2}}{3 a}}{a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(61\) |
Input:
int(1/(a+a*sin(x))^3,x,method=_RETURNVERBOSE)
Output:
4/15*(-1+10*exp(2*I*x)+5*I*exp(I*x))/(exp(I*x)+I)^5/a^3
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (44) = 88\).
Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.84 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2 \, \cos \left (x\right )^{3} - 4 \, \cos \left (x\right )^{2} - {\left (2 \, \cos \left (x\right )^{2} + 6 \, \cos \left (x\right ) - 3\right )} \sin \left (x\right ) - 9 \, \cos \left (x\right ) - 3}{15 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \] Input:
integrate(1/(a+a*sin(x))^3,x, algorithm="fricas")
Output:
-1/15*(2*cos(x)^3 - 4*cos(x)^2 - (2*cos(x)^2 + 6*cos(x) - 3)*sin(x) - 9*co s(x) - 3)/(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. 348 vs. \(2 (49) = 98\).
Time = 0.61 (sec) , antiderivative size = 348, normalized size of antiderivative = 6.96 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=- \frac {30 \tan ^{4}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {60 \tan ^{3}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {80 \tan ^{2}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {40 \tan {\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {14}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} \] Input:
integrate(1/(a+a*sin(x))**3,x)
Output:
-30*tan(x/2)**4/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan( x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3) - 60*tan(x/2) **3/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 15 0*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3) - 80*tan(x/2)**2/(15*a**3 *tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 150*a**3*tan(x /2)**2 + 75*a**3*tan(x/2) + 15*a**3) - 40*tan(x/2)/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a** 3*tan(x/2) + 15*a**3) - 14/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 15 0*a**3*tan(x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3)
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (44) = 88\).
Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.56 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2 \, {\left (\frac {20 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {40 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 7\right )}}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {10 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} \] Input:
integrate(1/(a+a*sin(x))^3,x, algorithm="maxima")
Output:
-2/15*(20*sin(x)/(cos(x) + 1) + 40*sin(x)^2/(cos(x) + 1)^2 + 30*sin(x)^3/( cos(x) + 1)^3 + 15*sin(x)^4/(cos(x) + 1)^4 + 7)/(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 = 45, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 30 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 40 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 20 \, \tan \left (\frac {1}{2} \, x\right ) + 7\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \] Input:
integrate(1/(a+a*sin(x))^3,x, algorithm="giac")
Output:
-2/15*(15*tan(1/2*x)^4 + 30*tan(1/2*x)^3 + 40*tan(1/2*x)^2 + 20*tan(1/2*x) + 7)/(a^3*(tan(1/2*x) + 1)^5)
Time = 16.90 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2\,\left (15\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+30\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+40\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+20\,\mathrm {tan}\left (\frac {x}{2}\right )+7\right )}{15\,a^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \] Input:
int(1/(a + a*sin(x))^3,x)
Output:
-(2*(20*tan(x/2) + 40*tan(x/2)^2 + 30*tan(x/2)^3 + 15*tan(x/2)^4 + 7))/(15 *a^3*(tan(x/2) + 1)^5)
Time = 0.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=\frac {\frac {2 \tan \left (\frac {x}{2}\right )^{5}}{5}-\frac {4 \tan \left (\frac {x}{2}\right )^{2}}{3}-\frac {2 \tan \left (\frac {x}{2}\right )}{3}-\frac {8}{15}}{a^{3} \left (\tan \left (\frac {x}{2}\right )^{5}+5 \tan \left (\frac {x}{2}\right )^{4}+10 \tan \left (\frac {x}{2}\right )^{3}+10 \tan \left (\frac {x}{2}\right )^{2}+5 \tan \left (\frac {x}{2}\right )+1\right )} \] Input:
int(1/(a+a*sin(x))^3,x)
Output:
(2*(3*tan(x/2)**5 - 10*tan(x/2)**2 - 5*tan(x/2) - 4))/(15*a**3*(tan(x/2)** 5 + 5*tan(x/2)**4 + 10*tan(x/2)**3 + 10*tan(x/2)**2 + 5*tan(x/2) + 1))