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]

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

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Rubi [A]  time = 0.0417927, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, 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 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]

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]

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 = 0.0191811, 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)

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Maple [A]  time = 0.033, size = 27, normalized size = 0.9 \begin{align*}{\frac{a}{d} \left ( - \left ( \sin \left ( dx+c \right ) \right ) ^{-1}-{\frac{1}{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \right ) } \end{align*}

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]

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

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Maxima [A]  time = 1.24401, size = 32, normalized size = 1.07 \begin{align*} -\frac{2 \, a \sin \left (d x + c\right ) + a}{2 \, d \sin \left (d x + c\right )^{2}} \end{align*}

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)

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Fricas [A]  time = 1.5656, size = 69, normalized size = 2.3 \begin{align*} \frac{2 \, a \sin \left (d x + c\right ) + a}{2 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )}} \end{align*}

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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [A]  time = 1.29335, size = 32, normalized size = 1.07 \begin{align*} -\frac{2 \, a \sin \left (d x + c\right ) + a}{2 \, d \sin \left (d x + c\right )^{2}} \end{align*}

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)