3.194 \(\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=30 \[ -\frac {\csc ^2(c+d x) (a \sin (c+d x)+a)^2}{2 a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 37} \[ -\frac {\csc ^2(c+d x) (a \sin (c+d x)+a)^2}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2)/(2*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 2833

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*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 29, normalized size = 0.97 \[ -\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 29, normalized size = 0.97 \[ \frac {2 \, a \sin \left (d x + c\right ) + a}{2 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.13, size = 24, normalized size = 0.80 \[ -\frac {2 \, a \sin \left (d x + c\right ) + a}{2 \, d \sin \left (d x + c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.13, size = 27, normalized size = 0.90 \[ \frac {a \left (-\frac {1}{\sin \left (d x +c \right )}-\frac {1}{2 \sin \left (d x +c \right )^{2}}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.30, size = 24, normalized size = 0.80 \[ -\frac {2 \, a \sin \left (d x + c\right ) + a}{2 \, d \sin \left (d x + c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 8.52, size = 25, normalized size = 0.83 \[ -\frac {\frac {a}{2}+a\,\sin \left (c+d\,x\right )}{d\,{\sin \left (c+d\,x\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \cos {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________