Integrand size = 27, antiderivative size = 67 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\log (1+\sin (c+d x))}{a d}+\frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d}+\frac {\sin ^3(c+d x)}{3 a d} \] Output:
-ln(1+sin(d*x+c))/a/d+sin(d*x+c)/a/d-1/2*sin(d*x+c)^2/a/d+1/3*sin(d*x+c)^3 /a/d
Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-6 \log (1+\sin (c+d x))+6 \sin (c+d x)-3 \sin ^2(c+d x)+2 \sin ^3(c+d x)}{6 a d} \] Input:
Integrate[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
Output:
(-6*Log[1 + Sin[c + d*x]] + 6*Sin[c + d*x] - 3*Sin[c + d*x]^2 + 2*Sin[c + d*x]^3)/(6*a*d)
Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3312, 27, 49, 2009}
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.
\(\displaystyle \int \frac {\sin ^3(c+d x) \cos (c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)}{a \sin (c+d x)+a}dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {\int \frac {\sin ^3(c+d x)}{\sin (c+d x) a+a}d(a \sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {a^3 \sin ^3(c+d x)}{\sin (c+d x) a+a}d(a \sin (c+d x))}{a^4 d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (-\frac {a^3}{\sin (c+d x) a+a}+\sin ^2(c+d x) a^2-\sin (c+d x) a^2+a^2\right )d(a \sin (c+d x))}{a^4 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{3} a^3 \sin ^3(c+d x)-\frac {1}{2} a^3 \sin ^2(c+d x)+a^3 \sin (c+d x)-a^3 \log (a \sin (c+d x)+a)}{a^4 d}\) |
Input:
Int[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
Output:
(-(a^3*Log[a + a*Sin[c + d*x]]) + a^3*Sin[c + d*x] - (a^3*Sin[c + d*x]^2)/ 2 + (a^3*Sin[c + d*x]^3)/3)/(a^4*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (d x +c \right )^{3}}{3}-\frac {\sin \left (d x +c \right )^{2}}{2}+\sin \left (d x +c \right )-\ln \left (1+\sin \left (d x +c \right )\right )}{d a}\) | \(46\) |
default | \(\frac {\frac {\sin \left (d x +c \right )^{3}}{3}-\frac {\sin \left (d x +c \right )^{2}}{2}+\sin \left (d x +c \right )-\ln \left (1+\sin \left (d x +c \right )\right )}{d a}\) | \(46\) |
parallelrisch | \(\frac {12 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-3-\sin \left (3 d x +3 c \right )+15 \sin \left (d x +c \right )+3 \cos \left (2 d x +2 c \right )}{12 d a}\) | \(69\) |
risch | \(\frac {i x}{a}-\frac {5 i {\mathrm e}^{i \left (d x +c \right )}}{8 d a}+\frac {5 i {\mathrm e}^{-i \left (d x +c \right )}}{8 d a}+\frac {2 i c}{a d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}-\frac {\sin \left (3 d x +3 c \right )}{12 d a}+\frac {\cos \left (2 d x +2 c \right )}{4 a d}\) | \(110\) |
norman | \(\frac {-\frac {5}{3 a d}-\frac {20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d a}-\frac {20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d a}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d a}-\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d a}-\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{a d}-\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}\) | \(212\) |
Input:
int(cos(d*x+c)*sin(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d/a*(1/3*sin(d*x+c)^3-1/2*sin(d*x+c)^2+sin(d*x+c)-ln(1+sin(d*x+c)))
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{6 \, a d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
1/6*(3*cos(d*x + c)^2 - 2*(cos(d*x + c)^2 - 4)*sin(d*x + c) - 6*log(sin(d* x + c) + 1))/(a*d)
Time = 0.45 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.99 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\begin {cases} - \frac {\log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a d} + \frac {\sin ^{3}{\left (c + d x \right )}}{3 a d} + \frac {\sin {\left (c + d x \right )}}{a d} + \frac {\cos ^{2}{\left (c + d x \right )}}{2 a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{3}{\left (c \right )} \cos {\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)**3/(a+a*sin(d*x+c)),x)
Output:
Piecewise((-log(sin(c + d*x) + 1)/(a*d) + sin(c + d*x)**3/(3*a*d) + sin(c + d*x)/(a*d) + cos(c + d*x)**2/(2*a*d), Ne(d, 0)), (x*sin(c)**3*cos(c)/(a* sin(c) + a), True))
Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.79 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right )}{a} - \frac {6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a}}{6 \, d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
1/6*((2*sin(d*x + c)^3 - 3*sin(d*x + c)^2 + 6*sin(d*x + c))/a - 6*log(sin( d*x + c) + 1)/a)/d
Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.10 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a d} + \frac {2 \, a^{2} d^{2} \sin \left (d x + c\right )^{3} - 3 \, a^{2} d^{2} \sin \left (d x + c\right )^{2} + 6 \, a^{2} d^{2} \sin \left (d x + c\right )}{6 \, a^{3} d^{3}} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-log(abs(sin(d*x + c) + 1))/(a*d) + 1/6*(2*a^2*d^2*sin(d*x + c)^3 - 3*a^2* d^2*sin(d*x + c)^2 + 6*a^2*d^2*sin(d*x + c))/(a^3*d^3)
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{a}-\frac {\sin \left (c+d\,x\right )}{a}+\frac {{\sin \left (c+d\,x\right )}^2}{2\,a}-\frac {{\sin \left (c+d\,x\right )}^3}{3\,a}}{d} \] Input:
int((cos(c + d*x)*sin(c + d*x)^3)/(a + a*sin(c + d*x)),x)
Output:
-(log(sin(c + d*x) + 1)/a - sin(c + d*x)/a + sin(c + d*x)^2/(2*a) - sin(c + d*x)^3/(3*a))/d
Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \cos \left (d x +c \right )^{2}-6 \,\mathrm {log}\left (\sin \left (d x +c \right )+1\right )+2 \sin \left (d x +c \right )^{3}+6 \sin \left (d x +c \right )}{6 a d} \] Input:
int(cos(d*x+c)*sin(d*x+c)^3/(a+a*sin(d*x+c)),x)
Output:
(3*cos(c + d*x)**2 - 6*log(sin(c + d*x) + 1) + 2*sin(c + d*x)**3 + 6*sin(c + d*x))/(6*a*d)