3.203 \(\int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=55 \[ -\frac{a^2 \csc ^4(c+d x)}{4 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^2(c+d x)}{2 d} \]

[Out]

-(a^2*Csc[c + d*x]^2)/(2*d) - (2*a^2*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(4*d)

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Rubi [A]  time = 0.0663529, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ -\frac{a^2 \csc ^4(c+d x)}{4 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]

[Out]

-(a^2*Csc[c + d*x]^2)/(2*d) - (2*a^2*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(4*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 ^4(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^5 (a+x)^2}{x^5} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^4 \operatorname{Subst}\left (\int \frac{(a+x)^2}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^4 \operatorname{Subst}\left (\int \left (\frac{a^2}{x^5}+\frac{2 a}{x^4}+\frac{1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^4(c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.0290794, size = 55, normalized size = 1. \[ -\frac{a^2 \csc ^4(c+d x)}{4 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]

[Out]

-(a^2*Csc[c + d*x]^2)/(2*d) - (2*a^2*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(4*d)

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Maple [A]  time = 0.038, size = 39, normalized size = 0.7 \begin{align*}{\frac{{a}^{2}}{d} \left ( -{\frac{1}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{2}{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)^5*(a+a*sin(d*x+c))^2,x)

[Out]

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

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Maxima [A]  time = 1.23087, size = 58, normalized size = 1.05 \begin{align*} -\frac{6 \, a^{2} \sin \left (d x + c\right )^{2} + 8 \, a^{2} \sin \left (d x + c\right ) + 3 \, a^{2}}{12 \, d \sin \left (d x + c\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^5*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/12*(6*a^2*sin(d*x + c)^2 + 8*a^2*sin(d*x + c) + 3*a^2)/(d*sin(d*x + c)^4)

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Fricas [A]  time = 1.62856, size = 138, normalized size = 2.51 \begin{align*} \frac{6 \, a^{2} \cos \left (d x + c\right )^{2} - 8 \, a^{2} \sin \left (d x + c\right ) - 9 \, a^{2}}{12 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, 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)^5*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/12*(6*a^2*cos(d*x + c)^2 - 8*a^2*sin(d*x + c) - 9*a^2)/(d*cos(d*x + c)^4 - 2*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)**5*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.19858, size = 58, normalized size = 1.05 \begin{align*} -\frac{6 \, a^{2} \sin \left (d x + c\right )^{2} + 8 \, a^{2} \sin \left (d x + c\right ) + 3 \, a^{2}}{12 \, d \sin \left (d x + c\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^5*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/12*(6*a^2*sin(d*x + c)^2 + 8*a^2*sin(d*x + c) + 3*a^2)/(d*sin(d*x + c)^4)