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

Optimal. Leaf size=15 \[ -\frac {\csc ^3(a+b x)}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2606, 30} \[ -\frac {\csc ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x]*Csc[a + b*x]^3,x]

[Out]

-Csc[a + b*x]^3/(3*b)

Rule 30

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

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

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ -\frac {\csc ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*Csc[a + b*x]^3/b

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fricas [A]  time = 0.42, size = 26, normalized size = 1.73 \[ \frac {1}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.57, size = 13, normalized size = 0.87 \[ -\frac {1}{3 \, b \sin \left (b x + a\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(b*sin(b*x + a)^3)

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maple [A]  time = 0.00, size = 14, normalized size = 0.93 \[ -\frac {1}{3 \sin \left (b x +a \right )^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.32, size = 13, normalized size = 0.87 \[ -\frac {1}{3 \, b \sin \left (b x + a\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(b*sin(b*x + a)^3)

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mupad [B]  time = 0.43, size = 13, normalized size = 0.87 \[ -\frac {1}{3\,b\,{\sin \left (a+b\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)/sin(a + b*x)^4,x)

[Out]

-1/(3*b*sin(a + b*x)^3)

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sympy [A]  time = 1.57, size = 24, normalized size = 1.60 \[ \begin {cases} - \frac {1}{3 b \sin ^{3}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos {\relax (a )}}{\sin ^{4}{\relax (a )}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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