\(\int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [550]

Optimal result

Integrand size = 27, antiderivative size = 31 \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\csc ^3(c+d x) (a-a \sin (c+d x))^3}{3 a^5 d} \]

[Out]

-1/3*csc(d*x+c)^3*(a-a*sin(d*x+c))^3/a^5/d

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2915, 12, 37} \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\csc ^3(c+d x) (a-a \sin (c+d x))^3}{3 a^5 d} \]

[In]

Int[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]

[Out]

-1/3*(Csc[c + d*x]^3*(a - a*Sin[c + d*x])^3)/(a^5*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 2915

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^4 (a-x)^2}{x^4} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {(a-x)^2}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = -\frac {\csc ^3(c+d x) (a-a \sin (c+d x))^3}{3 a^5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {(-1+\csc (c+d x))^3}{3 a^2 d} \]

[In]

Integrate[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]

[Out]

-1/3*(-1 + Csc[c + d*x])^3/(a^2*d)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61

method	result	size
derivativedivides	\(-\frac {\left (\csc \left (d x +c \right )-1\right )^{3}}{3 d \,a^{2}}\)	\(19\)
default	\(-\frac {\left (\csc \left (d x +c \right )-1\right )^{3}}{3 d \,a^{2}}\)	\(19\)
risch	\(-\frac {2 i \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}-10 \,{\mathrm e}^{3 i \left (d x +c \right )}-6 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+6 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}\)	\(81\)
parallelrisch	\(\frac {-\left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d \,a^{2}}\)	\(84\)
norman	\(\frac {-\frac {1}{24 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}-\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}+\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {7 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {7 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\)	\(222\)

[In]

int(cos(d*x+c)^5*csc(d*x+c)^4/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-1/3/d/a^2*(csc(d*x+c)-1)^3

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {3 \, \cos \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 4}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/3*(3*cos(d*x + c)^2 + 3*sin(d*x + c) - 4)/((a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**4/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {3 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 1}{3 \, a^{2} d \sin \left (d x + c\right )^{3}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/3*(3*sin(d*x + c)^2 - 3*sin(d*x + c) + 1)/(a^2*d*sin(d*x + c)^3)

Giac [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {3 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 1}{3 \, a^{2} d \sin \left (d x + c\right )^{3}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/3*(3*sin(d*x + c)^2 - 3*sin(d*x + c) + 1)/(a^2*d*sin(d*x + c)^3)

Mupad [B] (verification not implemented)

Time = 10.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {{\sin \left (c+d\,x\right )}^2-\sin \left (c+d\,x\right )+\frac {1}{3}}{a^2\,d\,{\sin \left (c+d\,x\right )}^3} \]

[In]

int(cos(c + d*x)^5/(sin(c + d*x)^4*(a + a*sin(c + d*x))^2),x)

[Out]

-(sin(c + d*x)^2 - sin(c + d*x) + 1/3)/(a^2*d*sin(c + d*x)^3)