3.4 \(\int \csc ^4(a+b x) \, dx\)

Optimal. Leaf size=27 \[ -\frac{\cot ^3(a+b x)}{3 b}-\frac{\cot (a+b x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.0113776, antiderivative size = 27, 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} \[ -\frac{\cot ^3(a+b x)}{3 b}-\frac{\cot (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^4,x]

[Out]

-(Cot[a + b*x]/b) - Cot[a + b*x]^3/(3*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 ^4(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (a+b x)\right )}{b}\\ &=-\frac{\cot (a+b x)}{b}-\frac{\cot ^3(a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.012798, size = 35, normalized size = 1.3 \[ -\frac{2 \cot (a+b x)}{3 b}-\frac{\cot (a+b x) \csc ^2(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^4,x]

[Out]

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

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Maple [A]  time = 0.018, size = 23, normalized size = 0.9 \begin{align*}{\frac{\cot \left ( bx+a \right ) }{b} \left ( -{\frac{2}{3}}-{\frac{ \left ( \csc \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^4,x)

[Out]

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

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Maxima [A]  time = 1.05666, size = 34, normalized size = 1.26 \begin{align*} -\frac{3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.450437, size = 108, normalized size = 4. \begin{align*} -\frac{2 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{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(csc(b*x+a)^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{4}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**4,x)

[Out]

Integral(csc(a + b*x)**4, x)

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Giac [A]  time = 1.2539, size = 34, normalized size = 1.26 \begin{align*} -\frac{3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^4,x, algorithm="giac")

[Out]

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