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

Optimal. Leaf size=33 \[ -\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0427132, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 43} \[ -\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*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 43

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])

Rubi steps

\begin{align*} \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^4 (a+x)}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{a+x}{x^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{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]  time = 0.0240431, size = 33, normalized size = 1. \[ -\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.034, size = 27, normalized size = 0.8 \begin{align*}{\frac{a}{d} \left ( -{\frac{1}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\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)^4*(a+a*sin(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.11486, size = 35, normalized size = 1.06 \begin{align*} -\frac{3 \, a \sin \left (d x + c\right ) + 2 \, a}{6 \, d \sin \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.54374, size = 92, normalized size = 2.79 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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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)**4*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.25737, size = 35, normalized size = 1.06 \begin{align*} -\frac{3 \, a \sin \left (d x + c\right ) + 2 \, a}{6 \, d \sin \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)