\(\int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx\) [195]

Optimal result

Integrand size = 25, antiderivative size = 33 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2912, 12, 45} \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^2(c+d x)}{2 d} \]

[In]

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

[Out]

-1/2*(a*Csc[c + d*x]^2)/d - (a*Csc[c + d*x]^3)/(3*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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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)}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {a^3 \text {Subst}\left (\int \frac {a+x}{x^4} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^3 \text {Subst}\left (\int \left (\frac {a}{x^4}+\frac {1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{3 d} \]

[In]

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

[Out]

-1/2*(a*Csc[c + d*x]^2)/d - (a*Csc[c + d*x]^3)/(3*d)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(-\frac {a \left (\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(28\)
default	\(-\frac {a \left (\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(28\)
risch	\(\frac {2 a \left (4 i {\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{4 i \left (d x +c \right )}-3 \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}\)	\(55\)
parallelrisch	\(\frac {a \left (-\left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-2\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{24 d}\)	\(71\)
norman	\(\frac {-\frac {a}{24 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\)	\(135\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a \sin \left (d x + c\right ) + 2 \, a}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 \, a \sin \left (d x + c\right ) + 2 \, a}{6 \, d \sin \left (d x + c\right )^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 \, a \sin \left (d x + c\right ) + 2 \, a}{6 \, d \sin \left (d x + c\right )^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.88 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {\frac {5\,a\,\sin \left (c+d\,x\right )}{16}+\frac {a\,\left (\frac {3\,\sin \left (3\,c+3\,d\,x\right )}{16}+1\right )}{3}}{d\,{\sin \left (c+d\,x\right )}^3} \]

[In]

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

[Out]

-((5*a*sin(c + d*x))/16 + (a*((3*sin(3*c + 3*d*x))/16 + 1))/3)/(d*sin(c + d*x)^3)