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

Optimal. Leaf size=57 \[ -\frac{\cot ^7(a+b x)}{7 b}+\frac{\cot ^5(a+b x)}{5 b}-\frac{\cot ^3(a+b x)}{3 b}+\frac{\cot (a+b x)}{b}+x \]

[Out]

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

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Rubi [A]  time = 0.0335139, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 8} \[ -\frac{\cot ^7(a+b x)}{7 b}+\frac{\cot ^5(a+b x)}{5 b}-\frac{\cot ^3(a+b x)}{3 b}+\frac{\cot (a+b x)}{b}+x \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x]^8,x]

[Out]

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

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cot ^8(a+b x) \, dx &=-\frac{\cot ^7(a+b x)}{7 b}-\int \cot ^6(a+b x) \, dx\\ &=\frac{\cot ^5(a+b x)}{5 b}-\frac{\cot ^7(a+b x)}{7 b}+\int \cot ^4(a+b x) \, dx\\ &=-\frac{\cot ^3(a+b x)}{3 b}+\frac{\cot ^5(a+b x)}{5 b}-\frac{\cot ^7(a+b x)}{7 b}-\int \cot ^2(a+b x) \, dx\\ &=\frac{\cot (a+b x)}{b}-\frac{\cot ^3(a+b x)}{3 b}+\frac{\cot ^5(a+b x)}{5 b}-\frac{\cot ^7(a+b x)}{7 b}+\int 1 \, dx\\ &=x+\frac{\cot (a+b x)}{b}-\frac{\cot ^3(a+b x)}{3 b}+\frac{\cot ^5(a+b x)}{5 b}-\frac{\cot ^7(a+b x)}{7 b}\\ \end{align*}

Mathematica [C]  time = 0.0091447, size = 33, normalized size = 0.58 \[ -\frac{\cot ^7(a+b x) \text{Hypergeometric2F1}\left (-\frac{7}{2},1,-\frac{5}{2},-\tan ^2(a+b x)\right )}{7 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x]^8,x]

[Out]

-(Cot[a + b*x]^7*Hypergeometric2F1[-7/2, 1, -5/2, -Tan[a + b*x]^2])/(7*b)

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Maple [A]  time = 0.013, size = 52, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{7}}{7}}+{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{5}}{5}}-{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{3}}{3}}+\cot \left ( bx+a \right ) -{\frac{\pi }{2}}+{\rm arccot} \left (\cot \left ( bx+a \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(-1/7*cot(b*x+a)^7+1/5*cot(b*x+a)^5-1/3*cot(b*x+a)^3+cot(b*x+a)-1/2*Pi+arccot(cot(b*x+a)))

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Maxima [A]  time = 1.60531, size = 73, normalized size = 1.28 \begin{align*} \frac{105 \, b x + 105 \, a + \frac{105 \, \tan \left (b x + a\right )^{6} - 35 \, \tan \left (b x + a\right )^{4} + 21 \, \tan \left (b x + a\right )^{2} - 15}{\tan \left (b x + a\right )^{7}}}{105 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(105*b*x + 105*a + (105*tan(b*x + a)^6 - 35*tan(b*x + a)^4 + 21*tan(b*x + a)^2 - 15)/tan(b*x + a)^7)/b

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Fricas [B]  time = 1.40016, size = 417, normalized size = 7.32 \begin{align*} \frac{176 \, \cos \left (2 \, b x + 2 \, a\right )^{4} - 108 \, \cos \left (2 \, b x + 2 \, a\right )^{3} + 20 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 105 \,{\left (b x \cos \left (2 \, b x + 2 \, a\right )^{3} - 3 \, b x \cos \left (2 \, b x + 2 \, a\right )^{2} + 3 \, b x \cos \left (2 \, b x + 2 \, a\right ) - b x\right )} \sin \left (2 \, b x + 2 \, a\right ) + 228 \, \cos \left (2 \, b x + 2 \, a\right ) - 76}{105 \,{\left (b \cos \left (2 \, b x + 2 \, a\right )^{3} - 3 \, b \cos \left (2 \, b x + 2 \, a\right )^{2} + 3 \, b \cos \left (2 \, b x + 2 \, a\right ) - b\right )} \sin \left (2 \, b x + 2 \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^8,x, algorithm="fricas")

[Out]

1/105*(176*cos(2*b*x + 2*a)^4 - 108*cos(2*b*x + 2*a)^3 + 20*cos(2*b*x + 2*a)^2 + 105*(b*x*cos(2*b*x + 2*a)^3 -
3*b*x*cos(2*b*x + 2*a)^2 + 3*b*x*cos(2*b*x + 2*a) - b*x)*sin(2*b*x + 2*a) + 228*cos(2*b*x + 2*a) - 76)/((b*co
s(2*b*x + 2*a)^3 - 3*b*cos(2*b*x + 2*a)^2 + 3*b*cos(2*b*x + 2*a) - b)*sin(2*b*x + 2*a))

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Sympy [A]  time = 0.947461, size = 51, normalized size = 0.89 \begin{align*} \begin{cases} x - \frac{\cot ^{7}{\left (a + b x \right )}}{7 b} + \frac{\cot ^{5}{\left (a + b x \right )}}{5 b} - \frac{\cot ^{3}{\left (a + b x \right )}}{3 b} + \frac{\cot{\left (a + b x \right )}}{b} & \text{for}\: b \neq 0 \\x \cot ^{8}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x - cot(a + b*x)**7/(7*b) + cot(a + b*x)**5/(5*b) - cot(a + b*x)**3/(3*b) + cot(a + b*x)/b, Ne(b, 0
)), (x*cot(a)**8, True))

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Giac [B]  time = 1.25256, size = 157, normalized size = 2.75 \begin{align*} \frac{15 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{7} - 189 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{5} + 1295 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{3} + 13440 \, b x + 13440 \, a + \frac{9765 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{6} - 1295 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{4} + 189 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{2} - 15}{\tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{7}} - 9765 \, \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}{13440 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13440*(15*tan(1/2*b*x + 1/2*a)^7 - 189*tan(1/2*b*x + 1/2*a)^5 + 1295*tan(1/2*b*x + 1/2*a)^3 + 13440*b*x + 13
440*a + (9765*tan(1/2*b*x + 1/2*a)^6 - 1295*tan(1/2*b*x + 1/2*a)^4 + 189*tan(1/2*b*x + 1/2*a)^2 - 15)/tan(1/2*
b*x + 1/2*a)^7 - 9765*tan(1/2*b*x + 1/2*a))/b