3.163 \(\int \cot ^3(a+b x) \csc (a+b x) \, dx\)

Optimal. Leaf size=26 \[ \frac{\csc (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b} \]

[Out]

Csc[a + b*x]/b - Csc[a + b*x]^3/(3*b)

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Rubi [A]  time = 0.0209366, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2606} \[ \frac{\csc (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x]^3*Csc[a + b*x],x]

[Out]

Csc[a + b*x]/b - Csc[a + b*x]^3/(3*b)

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])

Rubi steps

\begin{align*} \int \cot ^3(a+b x) \csc (a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (a+b x)\right )}{b}\\ &=\frac{\csc (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.0136318, size = 26, normalized size = 1. \[ \frac{\csc (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x]^3*Csc[a + b*x],x]

[Out]

Csc[a + b*x]/b - Csc[a + b*x]^3/(3*b)

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Maple [B]  time = 0.013, size = 60, normalized size = 2.3 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{3\, \left ( \sin \left ( bx+a \right ) \right ) ^{3}}}+{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{3\,\sin \left ( bx+a \right ) }}+{\frac{ \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^3/sin(b*x+a)^4,x)

[Out]

1/b*(-1/3*cos(b*x+a)^4/sin(b*x+a)^3+1/3*cos(b*x+a)^4/sin(b*x+a)+1/3*(2+cos(b*x+a)^2)*sin(b*x+a))

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Maxima [A]  time = 0.993042, size = 34, normalized size = 1.31 \begin{align*} \frac{3 \, \sin \left (b x + a\right )^{2} - 1}{3 \, b \sin \left (b x + a\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3/sin(b*x+a)^4,x, algorithm="maxima")

[Out]

1/3*(3*sin(b*x + a)^2 - 1)/(b*sin(b*x + a)^3)

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Fricas [A]  time = 2.09488, size = 89, normalized size = 3.42 \begin{align*} \frac{3 \, \cos \left (b x + a\right )^{2} - 2}{3 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3/sin(b*x+a)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.62873, size = 42, normalized size = 1.62 \begin{align*} \begin{cases} \frac{2}{3 b \sin{\left (a + b x \right )}} - \frac{\cos ^{2}{\left (a + b x \right )}}{3 b \sin ^{3}{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos ^{3}{\left (a \right )}}{\sin ^{4}{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**3/sin(b*x+a)**4,x)

[Out]

Piecewise((2/(3*b*sin(a + b*x)) - cos(a + b*x)**2/(3*b*sin(a + b*x)**3), Ne(b, 0)), (x*cos(a)**3/sin(a)**4, Tr
ue))

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Giac [A]  time = 1.12884, size = 34, normalized size = 1.31 \begin{align*} \frac{3 \, \sin \left (b x + a\right )^{2} - 1}{3 \, b \sin \left (b x + a\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3/sin(b*x+a)^4,x, algorithm="giac")

[Out]

1/3*(3*sin(b*x + a)^2 - 1)/(b*sin(b*x + a)^3)