\(\int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx\) [213]

Optimal result

Integrand size = 27, antiderivative size = 30 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {\csc ^4(c+d x) (a+a \sin (c+d x))^4}{4 a d} \]

[Out]

-1/4*csc(d*x+c)^4*(a+a*sin(d*x+c))^4/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} \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {\csc ^4(c+d x) (a \sin (c+d x)+a)^4}{4 a d} \]

[In]

Int[Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]

[Out]

-1/4*(Csc[c + d*x]^4*(a + a*Sin[c + d*x])^4)/(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^5 (a+x)^3}{x^5} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {a^4 \text {Subst}\left (\int \frac {(a+x)^3}{x^5} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {\csc ^4(c+d x) (a+a \sin (c+d x))^4}{4 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 (1+\csc (c+d x))^4}{4 d} \]

[In]

Integrate[Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]

[Out]

-1/4*(a^3*(1 + Csc[c + d*x])^4)/d

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63

method	result	size
derivativedivides	\(-\frac {a^{3} \left (\csc \left (d x +c \right )+1\right )^{4}}{4 d}\)	\(19\)
default	\(-\frac {a^{3} \left (\csc \left (d x +c \right )+1\right )^{4}}{4 d}\)	\(19\)
risch	\(-\frac {2 i a^{3} \left (3 i {\mathrm e}^{6 i \left (d x +c \right )}+{\mathrm e}^{7 i \left (d x +c \right )}-8 i {\mathrm e}^{4 i \left (d x +c \right )}-7 \,{\mathrm e}^{5 i \left (d x +c \right )}+3 i {\mathrm e}^{2 i \left (d x +c \right )}+7 \,{\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}\)	\(102\)
parallelrisch	\(-\frac {a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )+\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )+8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+28 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+28 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+56 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}\)	\(106\)
norman	\(\frac {-\frac {a^{3}}{64 d}-\frac {a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {31 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {5 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {31 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {11 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {31 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {5 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {31 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {37 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {37 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\)	\(263\)

[In]

int(cos(d*x+c)*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-1/4/d*a^3*(csc(d*x+c)+1)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (28) = 56\).

Time = 0.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.40 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {6 \, a^{3} \cos \left (d x + c\right )^{2} - 7 \, a^{3} + 4 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3}\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

[In]

integrate(cos(d*x+c)*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/4*(6*a^3*cos(d*x + c)^2 - 7*a^3 + 4*(a^3*cos(d*x + c)^2 - 2*a^3)*sin(d*x + c))/(d*cos(d*x + c)^4 - 2*d*cos(d
*x + c)^2 + d)

Sympy [F(-1)]

Timed out. \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)*csc(d*x+c)**5*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {4 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} + 4 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{4 \, d \sin \left (d x + c\right )^{4}} \]

[In]

integrate(cos(d*x+c)*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/4*(4*a^3*sin(d*x + c)^3 + 6*a^3*sin(d*x + c)^2 + 4*a^3*sin(d*x + c) + a^3)/(d*sin(d*x + c)^4)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {4 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} + 4 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{4 \, d \sin \left (d x + c\right )^{4}} \]

[In]

integrate(cos(d*x+c)*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/4*(4*a^3*sin(d*x + c)^3 + 6*a^3*sin(d*x + c)^2 + 4*a^3*sin(d*x + c) + a^3)/(d*sin(d*x + c)^4)

Mupad [B] (verification not implemented)

Time = 9.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {4\,a^3\,{\sin \left (c+d\,x\right )}^3+6\,a^3\,{\sin \left (c+d\,x\right )}^2+4\,a^3\,\sin \left (c+d\,x\right )+a^3}{4\,d\,{\sin \left (c+d\,x\right )}^4} \]

[In]

int((cos(c + d*x)*(a + a*sin(c + d*x))^3)/sin(c + d*x)^5,x)

[Out]

-(4*a^3*sin(c + d*x) + a^3 + 6*a^3*sin(c + d*x)^2 + 4*a^3*sin(c + d*x)^3)/(4*d*sin(c + d*x)^4)