Integrand size = 13, antiderivative size = 53 \[ \int \frac {\sin ^4(x)}{a+a \sin (x)} \, dx=-\frac {3 x}{2 a}-\frac {4 \cos (x)}{a}+\frac {4 \cos ^3(x)}{3 a}+\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin ^3(x)}{a+a \sin (x)} \] Output:
-3/2*x/a-4*cos(x)/a+4/3*cos(x)^3/a+3/2*cos(x)*sin(x)/a+cos(x)*sin(x)^3/(a+ a*sin(x))
Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.91 \[ \int \frac {\sin ^4(x)}{a+a \sin (x)} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-3 (7+12 x) \cos \left (\frac {x}{2}\right )-18 \cos \left (\frac {3 x}{2}\right )-2 \cos \left (\frac {5 x}{2}\right )+\cos \left (\frac {7 x}{2}\right )+69 \sin \left (\frac {x}{2}\right )-36 x \sin \left (\frac {x}{2}\right )-18 \sin \left (\frac {3 x}{2}\right )+2 \sin \left (\frac {5 x}{2}\right )+\sin \left (\frac {7 x}{2}\right )\right )}{24 a (1+\sin (x))} \] Input:
Integrate[Sin[x]^4/(a + a*Sin[x]),x]
Output:
((Cos[x/2] + Sin[x/2])*(-3*(7 + 12*x)*Cos[x/2] - 18*Cos[(3*x)/2] - 2*Cos[( 5*x)/2] + Cos[(7*x)/2] + 69*Sin[x/2] - 36*x*Sin[x/2] - 18*Sin[(3*x)/2] + 2 *Sin[(5*x)/2] + Sin[(7*x)/2]))/(24*a*(1 + Sin[x]))
Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {3042, 3246, 3042, 3227, 3042, 3113, 2009, 3115, 24}
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.
\(\displaystyle \int \frac {\sin ^4(x)}{a \sin (x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (x)^4}{a \sin (x)+a}dx\) |
\(\Big \downarrow \) 3246 |
\(\displaystyle \frac {\sin ^3(x) \cos (x)}{a \sin (x)+a}-\frac {\int \sin ^2(x) (3 a-4 a \sin (x))dx}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin ^3(x) \cos (x)}{a \sin (x)+a}-\frac {\int \sin (x)^2 (3 a-4 a \sin (x))dx}{a^2}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle \frac {\sin ^3(x) \cos (x)}{a \sin (x)+a}-\frac {3 a \int \sin ^2(x)dx-4 a \int \sin ^3(x)dx}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin ^3(x) \cos (x)}{a \sin (x)+a}-\frac {3 a \int \sin (x)^2dx-4 a \int \sin (x)^3dx}{a^2}\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle \frac {\sin ^3(x) \cos (x)}{a \sin (x)+a}-\frac {3 a \int \sin (x)^2dx+4 a \int \left (1-\cos ^2(x)\right )d\cos (x)}{a^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sin ^3(x) \cos (x)}{a \sin (x)+a}-\frac {3 a \int \sin (x)^2dx+4 a \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )}{a^2}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\sin ^3(x) \cos (x)}{a \sin (x)+a}-\frac {3 a \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )+4 a \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )}{a^2}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\sin ^3(x) \cos (x)}{a \sin (x)+a}-\frac {4 a \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )+3 a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2}\) |
Input:
Int[Sin[x]^4/(a + a*Sin[x]),x]
Output:
(Cos[x]*Sin[x]^3)/(a + a*Sin[x]) - (4*a*(Cos[x] - Cos[x]^3/3) + 3*a*(x/2 - (Cos[x]*Sin[x])/2))/a^2
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(a + b*Sin[e + f*x]))), x] - Simp[d/(a*b) Int[(c + d* Sin[e + f*x])^(n - 2)*Simp[b*d*(n - 1) - a*c*n + (b*c*(n - 1) - a*d*n)*Sin[ e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] & & EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && (IntegerQ[2*n] || EqQ[c, 0])
Time = 0.43 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.75
method | result | size |
parallelrisch | \(\frac {-20 \cos \left (2 x \right )+\cos \left (4 x \right )+3 \sin \left (3 x \right )+\left (-36 x -64\right ) \cos \left (x \right )+27 \sin \left (x \right )-45}{24 a \cos \left (x \right )}\) | \(40\) |
risch | \(-\frac {3 x}{2 a}-\frac {7 \,{\mathrm e}^{i x}}{8 a}-\frac {7 \,{\mathrm e}^{-i x}}{8 a}-\frac {2}{\left ({\mathrm e}^{i x}+i\right ) a}+\frac {\cos \left (3 x \right )}{12 a}+\frac {\sin \left (2 x \right )}{4 a}\) | \(61\) |
default | \(\frac {-\frac {2 \left (\frac {\tan \left (\frac {x}{2}\right )^{5}}{2}+\tan \left (\frac {x}{2}\right )^{4}+4 \tan \left (\frac {x}{2}\right )^{2}-\frac {\tan \left (\frac {x}{2}\right )}{2}+\frac {5}{3}\right )}{\left (\tan \left (\frac {x}{2}\right )^{2}+1\right )^{3}}-3 \arctan \left (\tan \left (\frac {x}{2}\right )\right )-\frac {2}{\tan \left (\frac {x}{2}\right )+1}}{a}\) | \(66\) |
norman | \(\frac {-\frac {3 \tan \left (\frac {x}{2}\right )^{8}}{a}-\frac {11 \tan \left (\frac {x}{2}\right )^{6}}{a}-\frac {3 \tan \left (\frac {x}{2}\right )^{7}}{a}-\frac {11 \tan \left (\frac {x}{2}\right )^{5}}{a}-\frac {3 x}{2 a}-\frac {16}{3 a}-\frac {3 x \tan \left (\frac {x}{2}\right )}{2 a}-\frac {6 x \tan \left (\frac {x}{2}\right )^{2}}{a}-\frac {6 x \tan \left (\frac {x}{2}\right )^{3}}{a}-\frac {9 x \tan \left (\frac {x}{2}\right )^{4}}{a}-\frac {9 x \tan \left (\frac {x}{2}\right )^{5}}{a}-\frac {6 x \tan \left (\frac {x}{2}\right )^{6}}{a}-\frac {6 x \tan \left (\frac {x}{2}\right )^{7}}{a}-\frac {3 x \tan \left (\frac {x}{2}\right )^{8}}{2 a}-\frac {3 x \tan \left (\frac {x}{2}\right )^{9}}{2 a}-\frac {21 \tan \left (\frac {x}{2}\right )^{4}}{a}-\frac {7 \tan \left (\frac {x}{2}\right )}{3 a}-\frac {55 \tan \left (\frac {x}{2}\right )^{2}}{3 a}-\frac {31 \tan \left (\frac {x}{2}\right )^{3}}{3 a}}{\left (\tan \left (\frac {x}{2}\right )^{2}+1\right )^{4} \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(224\) |
Input:
int(sin(x)^4/(a+a*sin(x)),x,method=_RETURNVERBOSE)
Output:
1/24*(-20*cos(2*x)+cos(4*x)+3*sin(3*x)+(-36*x-64)*cos(x)+27*sin(x)-45)/a/c os(x)
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.32 \[ \int \frac {\sin ^4(x)}{a+a \sin (x)} \, dx=\frac {2 \, \cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, {\left (3 \, x + 5\right )} \cos \left (x\right ) - 12 \, \cos \left (x\right )^{2} + {\left (2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - 9 \, x - 9 \, \cos \left (x\right ) + 6\right )} \sin \left (x\right ) - 9 \, x - 6}{6 \, {\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )}} \] Input:
integrate(sin(x)^4/(a+a*sin(x)),x, algorithm="fricas")
Output:
1/6*(2*cos(x)^4 - cos(x)^3 - 3*(3*x + 5)*cos(x) - 12*cos(x)^2 + (2*cos(x)^ 3 + 3*cos(x)^2 - 9*x - 9*cos(x) + 6)*sin(x) - 9*x - 6)/(a*cos(x) + a*sin(x ) + a)
Leaf count of result is larger than twice the leaf count of optimal. 1221 vs. \(2 (49) = 98\).
Time = 1.88 (sec) , antiderivative size = 1221, normalized size of antiderivative = 23.04 \[ \int \frac {\sin ^4(x)}{a+a \sin (x)} \, dx=\text {Too large to display} \] Input:
integrate(sin(x)**4/(a+a*sin(x)),x)
Output:
-9*x*tan(x/2)**7/(6*a*tan(x/2)**7 + 6*a*tan(x/2)**6 + 18*a*tan(x/2)**5 + 1 8*a*tan(x/2)**4 + 18*a*tan(x/2)**3 + 18*a*tan(x/2)**2 + 6*a*tan(x/2) + 6*a ) - 9*x*tan(x/2)**6/(6*a*tan(x/2)**7 + 6*a*tan(x/2)**6 + 18*a*tan(x/2)**5 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**3 + 18*a*tan(x/2)**2 + 6*a*tan(x/2) + 6*a) - 27*x*tan(x/2)**5/(6*a*tan(x/2)**7 + 6*a*tan(x/2)**6 + 18*a*tan(x/2) **5 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**3 + 18*a*tan(x/2)**2 + 6*a*tan(x/2 ) + 6*a) - 27*x*tan(x/2)**4/(6*a*tan(x/2)**7 + 6*a*tan(x/2)**6 + 18*a*tan( x/2)**5 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**3 + 18*a*tan(x/2)**2 + 6*a*tan (x/2) + 6*a) - 27*x*tan(x/2)**3/(6*a*tan(x/2)**7 + 6*a*tan(x/2)**6 + 18*a* tan(x/2)**5 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**3 + 18*a*tan(x/2)**2 + 6*a *tan(x/2) + 6*a) - 27*x*tan(x/2)**2/(6*a*tan(x/2)**7 + 6*a*tan(x/2)**6 + 1 8*a*tan(x/2)**5 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**3 + 18*a*tan(x/2)**2 + 6*a*tan(x/2) + 6*a) - 9*x*tan(x/2)/(6*a*tan(x/2)**7 + 6*a*tan(x/2)**6 + 1 8*a*tan(x/2)**5 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**3 + 18*a*tan(x/2)**2 + 6*a*tan(x/2) + 6*a) - 9*x/(6*a*tan(x/2)**7 + 6*a*tan(x/2)**6 + 18*a*tan(x /2)**5 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**3 + 18*a*tan(x/2)**2 + 6*a*tan( x/2) + 6*a) - 18*tan(x/2)**6/(6*a*tan(x/2)**7 + 6*a*tan(x/2)**6 + 18*a*tan (x/2)**5 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**3 + 18*a*tan(x/2)**2 + 6*a*ta n(x/2) + 6*a) - 18*tan(x/2)**5/(6*a*tan(x/2)**7 + 6*a*tan(x/2)**6 + 18*a*t an(x/2)**5 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**3 + 18*a*tan(x/2)**2 + 6...
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (47) = 94\).
Time = 0.11 (sec) , antiderivative size = 180, normalized size of antiderivative = 3.40 \[ \int \frac {\sin ^4(x)}{a+a \sin (x)} \, dx=-\frac {\frac {7 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {39 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {24 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {24 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {9 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {9 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 16}{3 \, {\left (a + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {3 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, a \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {3 \, a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {a \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {a \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}} - \frac {3 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \] Input:
integrate(sin(x)^4/(a+a*sin(x)),x, algorithm="maxima")
Output:
-1/3*(7*sin(x)/(cos(x) + 1) + 39*sin(x)^2/(cos(x) + 1)^2 + 24*sin(x)^3/(co s(x) + 1)^3 + 24*sin(x)^4/(cos(x) + 1)^4 + 9*sin(x)^5/(cos(x) + 1)^5 + 9*s in(x)^6/(cos(x) + 1)^6 + 16)/(a + a*sin(x)/(cos(x) + 1) + 3*a*sin(x)^2/(co s(x) + 1)^2 + 3*a*sin(x)^3/(cos(x) + 1)^3 + 3*a*sin(x)^4/(cos(x) + 1)^4 + 3*a*sin(x)^5/(cos(x) + 1)^5 + a*sin(x)^6/(cos(x) + 1)^6 + a*sin(x)^7/(cos( x) + 1)^7) - 3*arctan(sin(x)/(cos(x) + 1))/a
Time = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.26 \[ \int \frac {\sin ^4(x)}{a+a \sin (x)} \, dx=-\frac {3 \, x}{2 \, a} - \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} - \frac {3 \, \tan \left (\frac {1}{2} \, x\right )^{5} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 24 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, x\right ) + 10}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3} a} \] Input:
integrate(sin(x)^4/(a+a*sin(x)),x, algorithm="giac")
Output:
-3/2*x/a - 2/(a*(tan(1/2*x) + 1)) - 1/3*(3*tan(1/2*x)^5 + 6*tan(1/2*x)^4 + 24*tan(1/2*x)^2 - 3*tan(1/2*x) + 10)/((tan(1/2*x)^2 + 1)^3*a)
Time = 17.84 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.47 \[ \int \frac {\sin ^4(x)}{a+a \sin (x)} \, dx=-\frac {3\,x}{2\,a}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+13\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {7\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {16}{3}}{a\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^3\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \] Input:
int(sin(x)^4/(a + a*sin(x)),x)
Output:
- (3*x)/(2*a) - ((7*tan(x/2))/3 + 13*tan(x/2)^2 + 8*tan(x/2)^3 + 8*tan(x/2 )^4 + 3*tan(x/2)^5 + 3*tan(x/2)^6 + 16/3)/(a*(tan(x/2)^2 + 1)^3*(tan(x/2) + 1))
Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.47 \[ \int \frac {\sin ^4(x)}{a+a \sin (x)} \, dx=\frac {2 \cos \left (x \right ) \sin \left (x \right )^{3}-\cos \left (x \right ) \sin \left (x \right )^{2}+7 \cos \left (x \right ) \sin \left (x \right )-9 \cos \left (x \right ) x +18 \cos \left (x \right )+2 \sin \left (x \right )^{4}-3 \sin \left (x \right )^{3}+8 \sin \left (x \right )^{2}+9 \sin \left (x \right ) x +7 \sin \left (x \right )+9 x -18}{6 a \left (\cos \left (x \right )-\sin \left (x \right )-1\right )} \] Input:
int(sin(x)^4/(a+a*sin(x)),x)
Output:
(2*cos(x)*sin(x)**3 - cos(x)*sin(x)**2 + 7*cos(x)*sin(x) - 9*cos(x)*x + 18 *cos(x) + 2*sin(x)**4 - 3*sin(x)**3 + 8*sin(x)**2 + 9*sin(x)*x + 7*sin(x) + 9*x - 18)/(6*a*(cos(x) - sin(x) - 1))