3.8 \(\int \csc ^8(a+b x) \, dx\)

Optimal. Leaf size=55 \[ -\frac{\cot ^7(a+b x)}{7 b}-\frac{3 \cot ^5(a+b x)}{5 b}-\frac{\cot ^3(a+b x)}{b}-\frac{\cot (a+b x)}{b} \]

[Out]

-(Cot[a + b*x]/b) - Cot[a + b*x]^3/b - (3*Cot[a + b*x]^5)/(5*b) - Cot[a + b*x]^7/(7*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0165351, antiderivative size = 55, 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} \[ -\frac{\cot ^7(a+b x)}{7 b}-\frac{3 \cot ^5(a+b x)}{5 b}-\frac{\cot ^3(a+b x)}{b}-\frac{\cot (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^8,x]

[Out]

-(Cot[a + b*x]/b) - Cot[a + b*x]^3/b - (3*Cot[a + b*x]^5)/(5*b) - Cot[a + b*x]^7/(7*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 ^8(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (a+b x)\right )}{b}\\ &=-\frac{\cot (a+b x)}{b}-\frac{\cot ^3(a+b x)}{b}-\frac{3 \cot ^5(a+b x)}{5 b}-\frac{\cot ^7(a+b x)}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.0107059, size = 77, normalized size = 1.4 \[ -\frac{16 \cot (a+b x)}{35 b}-\frac{\cot (a+b x) \csc ^6(a+b x)}{7 b}-\frac{6 \cot (a+b x) \csc ^4(a+b x)}{35 b}-\frac{8 \cot (a+b x) \csc ^2(a+b x)}{35 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^8,x]

[Out]

(-16*Cot[a + b*x])/(35*b) - (8*Cot[a + b*x]*Csc[a + b*x]^2)/(35*b) - (6*Cot[a + b*x]*Csc[a + b*x]^4)/(35*b) -
(Cot[a + b*x]*Csc[a + b*x]^6)/(7*b)

________________________________________________________________________________________

Maple [A]  time = 0.02, size = 43, normalized size = 0.8 \begin{align*}{\frac{\cot \left ( bx+a \right ) }{b} \left ( -{\frac{16}{35}}-{\frac{ \left ( \csc \left ( bx+a \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \csc \left ( bx+a \right ) \right ) ^{4}}{35}}-{\frac{8\, \left ( \csc \left ( bx+a \right ) \right ) ^{2}}{35}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^8,x)

[Out]

1/b*(-16/35-1/7*csc(b*x+a)^6-6/35*csc(b*x+a)^4-8/35*csc(b*x+a)^2)*cot(b*x+a)

________________________________________________________________________________________

Maxima [A]  time = 1.01128, size = 61, normalized size = 1.11 \begin{align*} -\frac{35 \, \tan \left (b x + a\right )^{6} + 35 \, \tan \left (b x + a\right )^{4} + 21 \, \tan \left (b x + a\right )^{2} + 5}{35 \, b \tan \left (b x + a\right )^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^8,x, algorithm="maxima")

[Out]

-1/35*(35*tan(b*x + a)^6 + 35*tan(b*x + a)^4 + 21*tan(b*x + a)^2 + 5)/(b*tan(b*x + a)^7)

________________________________________________________________________________________

Fricas [A]  time = 0.460868, size = 223, normalized size = 4.05 \begin{align*} -\frac{16 \, \cos \left (b x + a\right )^{7} - 56 \, \cos \left (b x + a\right )^{5} + 70 \, \cos \left (b x + a\right )^{3} - 35 \, \cos \left (b x + a\right )}{35 \,{\left (b \cos \left (b x + a\right )^{6} - 3 \, b \cos \left (b x + a\right )^{4} + 3 \, 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)^8,x, algorithm="fricas")

[Out]

-1/35*(16*cos(b*x + a)^7 - 56*cos(b*x + a)^5 + 70*cos(b*x + a)^3 - 35*cos(b*x + a))/((b*cos(b*x + a)^6 - 3*b*c
os(b*x + a)^4 + 3*b*cos(b*x + a)^2 - b)*sin(b*x + a))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{8}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**8,x)

[Out]

Integral(csc(a + b*x)**8, x)

________________________________________________________________________________________

Giac [A]  time = 1.22946, size = 61, normalized size = 1.11 \begin{align*} -\frac{35 \, \tan \left (b x + a\right )^{6} + 35 \, \tan \left (b x + a\right )^{4} + 21 \, \tan \left (b x + a\right )^{2} + 5}{35 \, b \tan \left (b x + a\right )^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^8,x, algorithm="giac")

[Out]

-1/35*(35*tan(b*x + a)^6 + 35*tan(b*x + a)^4 + 21*tan(b*x + a)^2 + 5)/(b*tan(b*x + a)^7)