3.18 \(\int (a+a \csc (c+d x)) (A-A \csc (c+d x)) \sin ^3(c+d x) \, dx\)

Optimal. Leaf size=17 \[ \frac{a A \cos ^3(c+d x)}{3 d} \]

[Out]

(a*A*Cos[c + d*x]^3)/(3*d)

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Rubi [A]  time = 0.0606128, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3962, 2565, 30} \[ \frac{a A \cos ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*A*Cos[c + d*x]^3)/(3*d)

Rule 3962

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)
]*(d_.) + (c_))^(n_.), x_Symbol] :> Dist[(-(a*c))^m, Int[ExpandTrig[(g*csc[e + f*x])^p*cot[e + f*x]^(2*m), (c
+ d*csc[e + f*x])^(n - m), x], x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2
- b^2, 0] && IntegersQ[m, n] && GeQ[n - m, 0] && GtQ[m*n, 0]

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0056628, size = 17, normalized size = 1. \[ \frac{a A \cos ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*A*Cos[c + d*x]^3)/(3*d)

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Maple [B]  time = 0.028, size = 35, normalized size = 2.1 \begin{align*}{\frac{1}{d} \left ( -{\frac{Aa \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) }{3}}+Aa\cos \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c)^3,x)

[Out]

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

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Maxima [B]  time = 0.981429, size = 49, normalized size = 2.88 \begin{align*} \frac{{\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} A a + 3 \, A a \cos \left (d x + c\right )}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c)^3,x, algorithm="maxima")

[Out]

1/3*((cos(d*x + c)^3 - 3*cos(d*x + c))*A*a + 3*A*a*cos(d*x + c))/d

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Fricas [A]  time = 0.472044, size = 35, normalized size = 2.06 \begin{align*} \frac{A a \cos \left (d x + c\right )^{3}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c)^3,x, algorithm="fricas")

[Out]

1/3*A*a*cos(d*x + c)^3/d

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Sympy [A]  time = 10.5602, size = 83, normalized size = 4.88 \begin{align*} \begin{cases} - \frac{2 A a \cot ^{3}{\left (c + d x \right )}}{3 d \csc ^{3}{\left (c + d x \right )}} + \frac{A a \cot{\left (c + d x \right )}}{d \csc{\left (c + d x \right )}} - \frac{A a \cot{\left (c + d x \right )}}{d \csc ^{3}{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x \left (- A \csc{\left (c \right )} + A\right ) \left (a \csc{\left (c \right )} + a\right )}{\csc ^{3}{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c)**3,x)

[Out]

Piecewise((-2*A*a*cot(c + d*x)**3/(3*d*csc(c + d*x)**3) + A*a*cot(c + d*x)/(d*csc(c + d*x)) - A*a*cot(c + d*x)
/(d*csc(c + d*x)**3), Ne(d, 0)), (x*(-A*csc(c) + A)*(a*csc(c) + a)/csc(c)**3, True))

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Giac [B]  time = 1.31462, size = 76, normalized size = 4.47 \begin{align*} -\frac{2 \,{\left (A a + \frac{3 \, A a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{3 \, d{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c)^3,x, algorithm="giac")

[Out]

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