3.7 \(\int \cot ^7(a+b x) \, dx\)

Optimal. Leaf size=58 \[ -\frac{\cot ^6(a+b x)}{6 b}+\frac{\cot ^4(a+b x)}{4 b}-\frac{\cot ^2(a+b x)}{2 b}-\frac{\log (\sin (a+b x))}{b} \]

[Out]

-Cot[a + b*x]^2/(2*b) + Cot[a + b*x]^4/(4*b) - Cot[a + b*x]^6/(6*b) - Log[Sin[a + b*x]]/b

________________________________________________________________________________________

Rubi [A]  time = 0.0297959, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 3475} \[ -\frac{\cot ^6(a+b x)}{6 b}+\frac{\cot ^4(a+b x)}{4 b}-\frac{\cot ^2(a+b x)}{2 b}-\frac{\log (\sin (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x]^7,x]

[Out]

-Cot[a + b*x]^2/(2*b) + Cot[a + b*x]^4/(4*b) - Cot[a + b*x]^6/(6*b) - Log[Sin[a + b*x]]/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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.295689, size = 56, normalized size = 0.97 \[ -\frac{2 \cot ^6(a+b x)-3 \cot ^4(a+b x)+6 \cot ^2(a+b x)+12 \log (\tan (a+b x))+12 \log (\cos (a+b x))}{12 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x]^7,x]

[Out]

-(6*Cot[a + b*x]^2 - 3*Cot[a + b*x]^4 + 2*Cot[a + b*x]^6 + 12*Log[Cos[a + b*x]] + 12*Log[Tan[a + b*x]])/(12*b)

________________________________________________________________________________________

Maple [A]  time = 0.014, size = 57, normalized size = 1. \begin{align*} -{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{6}}{6\,b}}+{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{4}}{4\,b}}-{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{2}}{2\,b}}+{\frac{\ln \left ( \left ( \cot \left ( bx+a \right ) \right ) ^{2}+1 \right ) }{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(b*x+a)^7,x)

[Out]

-1/6*cot(b*x+a)^6/b+1/4*cot(b*x+a)^4/b-1/2*cot(b*x+a)^2/b+1/2/b*ln(cot(b*x+a)^2+1)

________________________________________________________________________________________

Maxima [A]  time = 1.05457, size = 65, normalized size = 1.12 \begin{align*} -\frac{\frac{18 \, \sin \left (b x + a\right )^{4} - 9 \, \sin \left (b x + a\right )^{2} + 2}{\sin \left (b x + a\right )^{6}} + 6 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{12 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/12*((18*sin(b*x + a)^4 - 9*sin(b*x + a)^2 + 2)/sin(b*x + a)^6 + 6*log(sin(b*x + a)^2))/b

________________________________________________________________________________________

Fricas [B]  time = 1.34652, size = 319, normalized size = 5.5 \begin{align*} \frac{18 \, \cos \left (2 \, b x + 2 \, a\right )^{2} - 3 \,{\left (\cos \left (2 \, b x + 2 \, a\right )^{3} - 3 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 3 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac{1}{2}\right ) - 18 \, \cos \left (2 \, b x + 2 \, a\right ) + 8}{6 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^7,x, algorithm="fricas")

[Out]

1/6*(18*cos(2*b*x + 2*a)^2 - 3*(cos(2*b*x + 2*a)^3 - 3*cos(2*b*x + 2*a)^2 + 3*cos(2*b*x + 2*a) - 1)*log(-1/2*c
os(2*b*x + 2*a) + 1/2) - 18*cos(2*b*x + 2*a) + 8)/(b*cos(2*b*x + 2*a)^3 - 3*b*cos(2*b*x + 2*a)^2 + 3*b*cos(2*b
*x + 2*a) - b)

________________________________________________________________________________________

Sympy [A]  time = 1.59554, size = 85, normalized size = 1.47 \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: \left (a = 0 \vee a = - b x\right ) \wedge \left (a = - b x \vee b = 0\right ) \\x \cot ^{7}{\left (a \right )} & \text{for}\: b = 0 \\\frac{\log{\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} - \frac{\log{\left (\tan{\left (a + b x \right )} \right )}}{b} - \frac{1}{2 b \tan ^{2}{\left (a + b x \right )}} + \frac{1}{4 b \tan ^{4}{\left (a + b x \right )}} - \frac{1}{6 b \tan ^{6}{\left (a + b x \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)**7,x)

[Out]

Piecewise((zoo*x, (Eq(a, 0) | Eq(a, -b*x)) & (Eq(b, 0) | Eq(a, -b*x))), (x*cot(a)**7, Eq(b, 0)), (log(tan(a +
b*x)**2 + 1)/(2*b) - log(tan(a + b*x))/b - 1/(2*b*tan(a + b*x)**2) + 1/(4*b*tan(a + b*x)**4) - 1/(6*b*tan(a +
b*x)**6), True))

________________________________________________________________________________________

Giac [B]  time = 1.19852, size = 281, normalized size = 4.84 \begin{align*} \frac{\frac{{\left (\frac{12 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac{87 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{352 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + 1\right )}{\left (\cos \left (b x + a\right ) + 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{3}} + \frac{87 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac{12 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - 192 \, \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) + 384 \, \log \left ({\left | -\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{384 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^7,x, algorithm="giac")

[Out]

1/384*((12*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 87*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + 352*(cos(b*x
+ a) - 1)^3/(cos(b*x + a) + 1)^3 + 1)*(cos(b*x + a) + 1)^3/(cos(b*x + a) - 1)^3 + 87*(cos(b*x + a) - 1)/(cos(
b*x + a) + 1) + 12*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + (cos(b*x + a) - 1)^3/(cos(b*x + a) + 1)^3 - 192
*log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)) + 384*log(abs(-(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1)))
/b