3.177 \(\int \cot ^3(a+b x) \csc ^2(a+b x) \, dx\)

Optimal. Leaf size=15 \[ -\frac{\cot ^4(a+b x)}{4 b} \]

[Out]

-Cot[a + b*x]^4/(4*b)

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Rubi [A]  time = 0.0276831, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2607, 30} \[ -\frac{\cot ^4(a+b x)}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x]^3*Csc[a + b*x]^2,x]

[Out]

-Cot[a + b*x]^4/(4*b)

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 \cot ^3(a+b x) \csc ^2(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,-\cot (a+b x)\right )}{b}\\ &=-\frac{\cot ^4(a+b x)}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.0068362, size = 15, normalized size = 1. \[ -\frac{\cot ^4(a+b x)}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x]^3*Csc[a + b*x]^2,x]

[Out]

-Cot[a + b*x]^4/(4*b)

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Maple [A]  time = 0.01, size = 22, normalized size = 1.5 \begin{align*} -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{4\, \left ( \sin \left ( bx+a \right ) \right ) ^{4}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^3/sin(b*x+a)^5,x)

[Out]

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

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Maxima [A]  time = 0.989208, size = 34, normalized size = 2.27 \begin{align*} \frac{2 \, \sin \left (b x + a\right )^{2} - 1}{4 \, b \sin \left (b x + a\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3/sin(b*x+a)^5,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.72945, size = 99, normalized size = 6.6 \begin{align*} -\frac{2 \, \cos \left (b x + a\right )^{2} - 1}{4 \,{\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3/sin(b*x+a)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.66975, size = 44, normalized size = 2.93 \begin{align*} \begin{cases} \frac{1}{4 b \sin ^{2}{\left (a + b x \right )}} - \frac{\cos ^{2}{\left (a + b x \right )}}{4 b \sin ^{4}{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos ^{3}{\left (a \right )}}{\sin ^{5}{\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)**5,x)

[Out]

Piecewise((1/(4*b*sin(a + b*x)**2) - cos(a + b*x)**2/(4*b*sin(a + b*x)**4), Ne(b, 0)), (x*cos(a)**3/sin(a)**5,
True))

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Giac [A]  time = 1.1294, size = 34, normalized size = 2.27 \begin{align*} \frac{2 \, \sin \left (b x + a\right )^{2} - 1}{4 \, b \sin \left (b x + a\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3/sin(b*x+a)^5,x, algorithm="giac")

[Out]

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