3.8 \(\int \csc ^2(c+d x) (a-a \csc (c+d x)) (A+A \csc (c+d x)) \, dx\)

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

[Out]

(a*A*Cot[c + d*x]^3)/(3*d)

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Rubi [A]  time = 0.0724796, 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, 2607, 30} \[ \frac{a A \cot ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^2*(a - a*Csc[c + d*x])*(A + A*Csc[c + d*x]),x]

[Out]

(a*A*Cot[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 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

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

Mathematica [A]  time = 0.0163463, size = 17, normalized size = 1. \[ \frac{a A \cot ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^2*(a - a*Csc[c + d*x])*(A + A*Csc[c + d*x]),x]

[Out]

(a*A*Cot[c + d*x]^3)/(3*d)

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Maple [B]  time = 0.02, size = 38, normalized size = 2.2 \begin{align*}{\frac{1}{d} \left ( -Aa\cot \left ( dx+c \right ) -Aa \left ( -{\frac{2}{3}}-{\frac{ \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cot \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^2*(a-a*csc(d*x+c))*(A+A*csc(d*x+c)),x)

[Out]

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

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Maxima [B]  time = 1.05654, size = 57, normalized size = 3.35 \begin{align*} -\frac{\frac{3 \, A a}{\tan \left (d x + c\right )} - \frac{{\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a}{\tan \left (d x + c\right )^{3}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2*(a-a*csc(d*x+c))*(A+A*csc(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.455756, size = 85, normalized size = 5. \begin{align*} -\frac{A a \cos \left (d x + c\right )^{3}}{3 \,{\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(csc(d*x+c)^2*(a-a*csc(d*x+c))*(A+A*csc(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.1531, size = 54, normalized size = 3.18 \begin{align*} \begin{cases} - \frac{A a \left (- \frac{\cot ^{3}{\left (c + d x \right )}}{3} - \cot{\left (c + d x \right )}\right ) + A a \cot{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (A \csc{\left (c \right )} + A\right ) \left (- a \csc{\left (c \right )} + a\right ) \csc ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**2*(a-a*csc(d*x+c))*(A+A*csc(d*x+c)),x)

[Out]

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

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Giac [A]  time = 1.36653, size = 20, normalized size = 1.18 \begin{align*} \frac{A a}{3 \, d \tan \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2*(a-a*csc(d*x+c))*(A+A*csc(d*x+c)),x, algorithm="giac")

[Out]

1/3*A*a/(d*tan(d*x + c)^3)