3.6 \(\int \csc ^6(a+b x) \, dx\)

Optimal. Leaf size=42 \[ -\frac{\cot ^5(a+b x)}{5 b}-\frac{2 \cot ^3(a+b x)}{3 b}-\frac{\cot (a+b x)}{b} \]

[Out]

-(Cot[a + b*x]/b) - (2*Cot[a + b*x]^3)/(3*b) - Cot[a + b*x]^5/(5*b)

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Rubi [A]  time = 0.0139804, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3767} \[ -\frac{\cot ^5(a+b x)}{5 b}-\frac{2 \cot ^3(a+b x)}{3 b}-\frac{\cot (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^6,x]

[Out]

-(Cot[a + b*x]/b) - (2*Cot[a + b*x]^3)/(3*b) - Cot[a + b*x]^5/(5*b)

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin{align*} \int \csc ^6(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (a+b x)\right )}{b}\\ &=-\frac{\cot (a+b x)}{b}-\frac{2 \cot ^3(a+b x)}{3 b}-\frac{\cot ^5(a+b x)}{5 b}\\ \end{align*}

Mathematica [A]  time = 0.0162388, size = 56, normalized size = 1.33 \[ -\frac{8 \cot (a+b x)}{15 b}-\frac{\cot (a+b x) \csc ^4(a+b x)}{5 b}-\frac{4 \cot (a+b x) \csc ^2(a+b x)}{15 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^6,x]

[Out]

(-8*Cot[a + b*x])/(15*b) - (4*Cot[a + b*x]*Csc[a + b*x]^2)/(15*b) - (Cot[a + b*x]*Csc[a + b*x]^4)/(5*b)

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Maple [A]  time = 0.02, size = 33, normalized size = 0.8 \begin{align*}{\frac{\cot \left ( bx+a \right ) }{b} \left ( -{\frac{8}{15}}-{\frac{ \left ( \csc \left ( bx+a \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \csc \left ( bx+a \right ) \right ) ^{2}}{15}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^6,x)

[Out]

1/b*(-8/15-1/5*csc(b*x+a)^4-4/15*csc(b*x+a)^2)*cot(b*x+a)

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Maxima [A]  time = 1.01335, size = 47, normalized size = 1.12 \begin{align*} -\frac{15 \, \tan \left (b x + a\right )^{4} + 10 \, \tan \left (b x + a\right )^{2} + 3}{15 \, b \tan \left (b x + a\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^6,x, algorithm="maxima")

[Out]

-1/15*(15*tan(b*x + a)^4 + 10*tan(b*x + a)^2 + 3)/(b*tan(b*x + a)^5)

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Fricas [A]  time = 0.452196, size = 166, normalized size = 3.95 \begin{align*} -\frac{8 \, \cos \left (b x + a\right )^{5} - 20 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )}{15 \,{\left (b \cos \left (b x + a\right )^{4} - 2 \, 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(csc(b*x+a)^6,x, algorithm="fricas")

[Out]

-1/15*(8*cos(b*x + a)^5 - 20*cos(b*x + a)^3 + 15*cos(b*x + a))/((b*cos(b*x + a)^4 - 2*b*cos(b*x + a)^2 + b)*si
n(b*x + a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{6}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**6,x)

[Out]

Integral(csc(a + b*x)**6, x)

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Giac [A]  time = 1.26094, size = 47, normalized size = 1.12 \begin{align*} -\frac{15 \, \tan \left (b x + a\right )^{4} + 10 \, \tan \left (b x + a\right )^{2} + 3}{15 \, b \tan \left (b x + a\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^6,x, algorithm="giac")

[Out]

-1/15*(15*tan(b*x + a)^4 + 10*tan(b*x + a)^2 + 3)/(b*tan(b*x + a)^5)