\(\int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx\) [202]

Optimal result

Integrand size = 27, antiderivative size = 30 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {\csc ^3(c+d x) (a+a \sin (c+d x))^3}{3 a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 30, 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 = {2912, 12, 37} \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {\csc ^3(c+d x) (a \sin (c+d x)+a)^3}{3 a d} \]

[In]

Int[Cot[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]

[Out]

-1/3*(Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(a*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 2912

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[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]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 (1+\csc (c+d x))^3}{3 d} \]

[In]

Integrate[Cot[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]

[Out]

-1/3*(a^2*(1 + Csc[c + d*x])^3)/d

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63

method	result	size
derivativedivides	\(-\frac {a^{2} \left (\csc \left (d x +c \right )+1\right )^{3}}{3 d}\)	\(19\)
default	\(-\frac {a^{2} \left (\csc \left (d x +c \right )+1\right )^{3}}{3 d}\)	\(19\)
parallelrisch	\(-\frac {a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+15 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}\)	\(80\)
risch	\(-\frac {2 i a^{2} \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}-10 \,{\mathrm e}^{3 i \left (d x +c \right )}+6 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}-6 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}\)	\(81\)
norman	\(\frac {-\frac {a^{2}}{24 d}-\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {17 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {23 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {23 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {17 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\)	\(187\)

[In]

int(cos(d*x+c)*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 = 0.33 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {3 \, a^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2} \sin \left (d x + c\right ) - 4 \, a^{2}}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=a^{2} \left (\int \cos {\left (c + d x \right )} \csc ^{4}{\left (c + d x \right )}\, dx + \int 2 \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{4}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{4}{\left (c + d x \right )}\, dx\right ) \]

[In]

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

[Out]

a**2*(Integral(cos(c + d*x)*csc(c + d*x)**4, x) + Integral(2*sin(c + d*x)*cos(c + d*x)*csc(c + d*x)**4, x) + I
ntegral(sin(c + d*x)**2*cos(c + d*x)*csc(c + d*x)**4, x))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {3 \, a^{2} \sin \left (d x + c\right )^{2} + 3 \, a^{2} \sin \left (d x + c\right ) + a^{2}}{3 \, d \sin \left (d x + c\right )^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {3 \, a^{2} \sin \left (d x + c\right )^{2} + 3 \, a^{2} \sin \left (d x + c\right ) + a^{2}}{3 \, d \sin \left (d x + c\right )^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2\,{\sin \left (c+d\,x\right )}^2+a^2\,\sin \left (c+d\,x\right )+\frac {a^2}{3}}{d\,{\sin \left (c+d\,x\right )}^3} \]

[In]

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

[Out]

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