Integrand size = 24, antiderivative size = 38 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {x}{a^2}-\frac {\tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d} \] Output:
x/a^2-tan(d*x+c)/a^2/d+1/3*tan(d*x+c)^3/a^2/d
Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {\arctan (\tan (c+d x))}{d}-\frac {\tan (c+d x)}{d}+\frac {\tan ^3(c+d x)}{3 d}}{a^2} \] Input:
Integrate[Sin[c + d*x]^4/(a - a*Sin[c + d*x]^2)^2,x]
Output:
(ArcTan[Tan[c + d*x]]/d - Tan[c + d*x]/d + Tan[c + d*x]^3/(3*d))/a^2
Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {3042, 3654, 3042, 3954, 3042, 3954, 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.
\(\displaystyle \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x)^4}{\left (a-a \sin (c+d x)^2\right )^2}dx\) |
\(\Big \downarrow \) 3654 |
\(\displaystyle \frac {\int \tan ^4(c+d x)dx}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \tan (c+d x)^4dx}{a^2}\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle \frac {\frac {\tan ^3(c+d x)}{3 d}-\int \tan ^2(c+d x)dx}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\tan ^3(c+d x)}{3 d}-\int \tan (c+d x)^2dx}{a^2}\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle \frac {\int 1dx+\frac {\tan ^3(c+d x)}{3 d}-\frac {\tan (c+d x)}{d}}{a^2}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\frac {\tan ^3(c+d x)}{3 d}-\frac {\tan (c+d x)}{d}+x}{a^2}\) |
Input:
Int[Sin[c + d*x]^4/(a - a*Sin[c + d*x]^2)^2,x]
Output:
(x - Tan[c + d*x]/d + Tan[c + d*x]^3/(3*d))/a^2
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[ a^p Int[ActivateTrig[u*cos[e + f*x]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Time = 0.67 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )}{d \,a^{2}}\) | \(34\) |
default | \(\frac {\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )}{d \,a^{2}}\) | \(34\) |
risch | \(\frac {x}{a^{2}}-\frac {4 i \left (3 \,{\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}+2\right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(53\) |
parallelrisch | \(\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} x d -9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} x d +6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x d -20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-3 d x +6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\) | \(122\) |
norman | \(\frac {\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{a}+\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{a}-\frac {x}{a}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}-\frac {38 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 a d}-\frac {24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{a d}-\frac {38 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 a d}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 a d}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{a d}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a}+\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3}}\) | \(290\) |
Input:
int(sin(d*x+c)^4/(a-a*sin(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/d/a^2*(1/3*tan(d*x+c)^3-tan(d*x+c)+arctan(tan(d*x+c)))
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {3 \, d x \cos \left (d x + c\right )^{3} - {\left (4 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right )}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \] Input:
integrate(sin(d*x+c)^4/(a-a*sin(d*x+c)^2)^2,x, algorithm="fricas")
Output:
1/3*(3*d*x*cos(d*x + c)^3 - (4*cos(d*x + c)^2 - 1)*sin(d*x + c))/(a^2*d*co s(d*x + c)^3)
Leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (31) = 62\).
Time = 7.24 (sec) , antiderivative size = 551, normalized size of antiderivative = 14.50 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\begin {cases} \frac {3 d x \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} - \frac {9 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} + \frac {9 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} - \frac {3 d x}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} + \frac {6 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} - \frac {20 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} + \frac {6 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{4}{\left (c \right )}}{\left (- a \sin ^{2}{\left (c \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \] Input:
integrate(sin(d*x+c)**4/(a-a*sin(d*x+c)**2)**2,x)
Output:
Piecewise((3*d*x*tan(c/2 + d*x/2)**6/(3*a**2*d*tan(c/2 + d*x/2)**6 - 9*a** 2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**2 - 3*a**2*d) - 9*d*x *tan(c/2 + d*x/2)**4/(3*a**2*d*tan(c/2 + d*x/2)**6 - 9*a**2*d*tan(c/2 + d* x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**2 - 3*a**2*d) + 9*d*x*tan(c/2 + d*x/2 )**2/(3*a**2*d*tan(c/2 + d*x/2)**6 - 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2 *d*tan(c/2 + d*x/2)**2 - 3*a**2*d) - 3*d*x/(3*a**2*d*tan(c/2 + d*x/2)**6 - 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**2 - 3*a**2*d) + 6*tan(c/2 + d*x/2)**5/(3*a**2*d*tan(c/2 + d*x/2)**6 - 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**2 - 3*a**2*d) - 20*tan(c/2 + d*x/2) **3/(3*a**2*d*tan(c/2 + d*x/2)**6 - 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2* d*tan(c/2 + d*x/2)**2 - 3*a**2*d) + 6*tan(c/2 + d*x/2)/(3*a**2*d*tan(c/2 + d*x/2)**6 - 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**2 - 3*a**2*d), Ne(d, 0)), (x*sin(c)**4/(-a*sin(c)**2 + a)**2, True))
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {\tan \left (d x + c\right )^{3} - 3 \, \tan \left (d x + c\right )}{a^{2}} + \frac {3 \, {\left (d x + c\right )}}{a^{2}}}{3 \, d} \] Input:
integrate(sin(d*x+c)^4/(a-a*sin(d*x+c)^2)^2,x, algorithm="maxima")
Output:
1/3*((tan(d*x + c)^3 - 3*tan(d*x + c))/a^2 + 3*(d*x + c)/a^2)/d
Time = 0.43 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} + \frac {a^{4} \tan \left (d x + c\right )^{3} - 3 \, a^{4} \tan \left (d x + c\right )}{a^{6}}}{3 \, d} \] Input:
integrate(sin(d*x+c)^4/(a-a*sin(d*x+c)^2)^2,x, algorithm="giac")
Output:
1/3*(3*(d*x + c)/a^2 + (a^4*tan(d*x + c)^3 - 3*a^4*tan(d*x + c))/a^6)/d
Time = 36.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {x}{a^2}-\frac {\mathrm {tan}\left (c+d\,x\right )-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3}}{a^2\,d} \] Input:
int(sin(c + d*x)^4/(a - a*sin(c + d*x)^2)^2,x)
Output:
x/a^2 - (tan(c + d*x) - tan(c + d*x)^3/3)/(a^2*d)
Time = 0.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.97 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {3 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} d x -3 \cos \left (d x +c \right ) d x -4 \sin \left (d x +c \right )^{3}+3 \sin \left (d x +c \right )}{3 \cos \left (d x +c \right ) a^{2} d \left (\sin \left (d x +c \right )^{2}-1\right )} \] Input:
int(sin(d*x+c)^4/(a-a*sin(d*x+c)^2)^2,x)
Output:
(3*cos(c + d*x)*sin(c + d*x)**2*d*x - 3*cos(c + d*x)*d*x - 4*sin(c + d*x)* *3 + 3*sin(c + d*x))/(3*cos(c + d*x)*a**2*d*(sin(c + d*x)**2 - 1))