\(\int \frac {\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [362]

Optimal result

Integrand size = 21, antiderivative size = 32 \[ \int \frac {\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2785, 2686, 30, 8} \[ \int \frac {\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d} \]

[In]

Int[Cot[c + d*x]^3/(a + a*Sin[c + d*x]),x]

[Out]

Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2785

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cot (c+d x) \csc (c+d x) \, dx}{a}+\frac {\int \cot (c+d x) \csc ^2(c+d x) \, dx}{a} \\ & = \frac {\text {Subst}(\int 1 \, dx,x,\csc (c+d x))}{a d}-\frac {\text {Subst}(\int x \, dx,x,\csc (c+d x))}{a d} \\ & = \frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {(-2+\csc (c+d x)) \csc (c+d x)}{2 a d} \]

[In]

Integrate[Cot[c + d*x]^3/(a + a*Sin[c + d*x]),x]

[Out]

-1/2*((-2 + Csc[c + d*x])*Csc[c + d*x])/(a*d)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {\csc \left (d x +c \right )-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}}{d a}\)	\(25\)
default	\(\frac {\csc \left (d x +c \right )-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}}{d a}\)	\(25\)
risch	\(\frac {2 i \left (-i {\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}\right )}{a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}\)	\(56\)
parallelrisch	\(\frac {-1-\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\)	\(59\)
norman	\(\frac {-\frac {1}{8 a d}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\)	\(90\)

[In]

int(cos(d*x+c)^3*csc(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2 \, \sin \left (d x + c\right ) - 1}{2 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )}} \]

[In]

integrate(cos(d*x+c)^3*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**3*csc(d*x+c)**3/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 \, \sin \left (d x + c\right ) - 1}{2 \, a d \sin \left (d x + c\right )^{2}} \]

[In]

integrate(cos(d*x+c)^3*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 \, \sin \left (d x + c\right ) - 1}{2 \, a d \sin \left (d x + c\right )^{2}} \]

[In]

integrate(cos(d*x+c)^3*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.93 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin \left (c+d\,x\right )-\frac {1}{2}}{a\,d\,{\sin \left (c+d\,x\right )}^2} \]

[In]

int(cos(c + d*x)^3/(sin(c + d*x)^3*(a + a*sin(c + d*x))),x)

[Out]

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