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]

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

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Rubi [A]  time = 0.07, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 30

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

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 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]

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*}

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Mathematica [A]  time = 0.02, size = 17, normalized size = 1.00 \[ \frac {a A \cot ^3(c+d x)}{3 d} \]

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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fricas [B]  time = 0.47, size = 36, normalized size = 2.12 \[ -\frac {A a \cos \left (d x + c\right )^{3}}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

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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giac [A]  time = 0.35, size = 15, normalized size = 0.88 \[ \frac {A a}{3 \, d \tan \left (d x + c\right )^{3}} \]

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)

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maple [B]  time = 1.00, size = 38, normalized size = 2.24 \[ \frac {-a A \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )-a A \cot \left (d x +c \right )}{d} \]

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

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maxima [B]  time = 0.32, size = 42, normalized size = 2.47 \[ -\frac {\frac {3 \, A a}{\tan \left (d x + c\right )} - \frac {{\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a}{\tan \left (d x + c\right )^{3}}}{3 \, d} \]

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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mupad [B]  time = 0.27, size = 15, normalized size = 0.88 \[ \frac {A\,a\,{\mathrm {cot}\left (c+d\,x\right )}^3}{3\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A*a*cot(c + d*x)^3)/(3*d)

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sympy [A]  time = 2.41, size = 54, normalized size = 3.18 \[ \begin {cases} \frac {- A a \left (- \frac {\cot ^{3}{\left (c + d x \right )}}{3} - \cot {\left (c + d x \right )}\right ) - A a \cot {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A \csc {\relax (c )} + A\right ) \left (- a \csc {\relax (c )} + a\right ) \csc ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]

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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