3.69 \(\int \frac{\csc ^2(c+d x)}{a+a \sec (c+d x)} \, dx\)

Optimal. Leaf size=37 \[ \frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^3(c+d x)}{3 a d} \]

[Out]

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

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Rubi [A]  time = 0.1261, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3872, 2839, 2606, 30, 2607} \[ \frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^3(c+d x)}{3 a d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^2/(a + a*Sec[c + d*x]),x]

[Out]

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

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 2606

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

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])

Rubi steps

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

Mathematica [A]  time = 0.209556, size = 66, normalized size = 1.78 \[ \frac{\csc (c) (2 \sin (c+d x)+\sin (2 (c+d x))+2 \sin (c+2 d x)-6 \sin (c)+4 \sin (d x)) \csc (2 (c+d x))}{6 a d (\sec (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^2/(a + a*Sec[c + d*x]),x]

[Out]

(Csc[c]*Csc[2*(c + d*x)]*(-6*Sin[c] + 4*Sin[d*x] + 2*Sin[c + d*x] + Sin[2*(c + d*x)] + 2*Sin[c + 2*d*x]))/(6*a
*d*(1 + Sec[c + d*x]))

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Maple [A]  time = 0.049, size = 36, normalized size = 1. \begin{align*}{\frac{1}{4\,da} \left ( -{\frac{1}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}- \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^2/(a+a*sec(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 0.996431, size = 66, normalized size = 1.78 \begin{align*} -\frac{\frac{3 \,{\left (\cos \left (d x + c\right ) + 1\right )}}{a \sin \left (d x + c\right )} + \frac{\sin \left (d x + c\right )^{3}}{a{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*(cos(d*x + c) + 1)/(a*sin(d*x + c)) + sin(d*x + c)^3/(a*(cos(d*x + c) + 1)^3))/d

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Fricas [A]  time = 1.5547, size = 111, normalized size = 3. \begin{align*} -\frac{\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1}{3 \,{\left (a d \cos \left (d x + c\right ) + a 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*sec(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\csc ^{2}{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**2/(a+a*sec(d*x+c)),x)

[Out]

Integral(csc(c + d*x)**2/(sec(c + d*x) + 1), x)/a

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Giac [A]  time = 1.30797, size = 50, normalized size = 1.35 \begin{align*} -\frac{\frac{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}{a} + \frac{3}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(tan(1/2*d*x + 1/2*c)^3/a + 3/(a*tan(1/2*d*x + 1/2*c)))/d