3.3 \(\int \cot ^3(a+b x) \, dx\)

Optimal. Leaf size=28 \[ -\frac {\cot ^2(a+b x)}{2 b}-\frac {\log (\sin (a+b x))}{b} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3473, 3475} \[ -\frac {\cot ^2(a+b x)}{2 b}-\frac {\log (\sin (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x]^3,x]

[Out]

-Cot[a + b*x]^2/(2*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 ^3(a+b x) \, dx &=-\frac {\cot ^2(a+b x)}{2 b}-\int \cot (a+b x) \, dx\\ &=-\frac {\cot ^2(a+b x)}{2 b}-\frac {\log (\sin (a+b x))}{b}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 34, normalized size = 1.21 \[ -\frac {\cot ^2(a+b x)+2 \log (\tan (a+b x))+2 \log (\cos (a+b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x]^3,x]

[Out]

-1/2*(Cot[a + b*x]^2 + 2*Log[Cos[a + b*x]] + 2*Log[Tan[a + b*x]])/b

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fricas [A]  time = 0.54, size = 47, normalized size = 1.68 \[ -\frac {{\left (\cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) - 2}{2 \, {\left (b \cos \left (2 \, b x + 2 \, a\right ) - b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/2*((cos(2*b*x + 2*a) - 1)*log(-1/2*cos(2*b*x + 2*a) + 1/2) - 2)/(b*cos(2*b*x + 2*a) - b)

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giac [B]  time = 0.34, size = 118, normalized size = 4.21 \[ \frac {\frac {{\left (\frac {4 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} + \frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 4 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) + 8 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^3,x, algorithm="giac")

[Out]

1/8*((4*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1)*(cos(b*x + a) + 1)/(cos(b*x + a) - 1) + (cos(b*x + a) - 1)/
(cos(b*x + a) + 1) - 4*log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)) + 8*log(abs(-(cos(b*x + a) - 1)/(cos(
b*x + a) + 1) + 1)))/b

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maple [A]  time = 0.03, size = 31, normalized size = 1.11 \[ -\frac {\cot ^{2}\left (b x +a \right )}{2 b}+\frac {\ln \left (\cot ^{2}\left (b x +a \right )+1\right )}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(b*x+a)^3,x)

[Out]

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

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maxima [A]  time = 0.36, size = 23, normalized size = 0.82 \[ -\frac {\frac {1}{\sin \left (b x + a\right )^{2}} + \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.03, size = 78, normalized size = 2.79 \[ x\,1{}\mathrm {i}-\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}-1\right )}{b}+\frac {2}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}+\frac {2}{b\,\left (1+{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + b*x)^3,x)

[Out]

x*1i - log(exp(a*2i)*exp(b*x*2i) - 1)/b + 2/(b*(exp(a*2i + b*x*2i) - 1)) + 2/(b*(exp(a*4i + b*x*4i) - 2*exp(a*
2i + b*x*2i) + 1))

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sympy [A]  time = 0.43, size = 58, normalized size = 2.07 \[ \begin {cases} \tilde {\infty } x & \text {for}\: \left (a = 0 \vee a = - b x\right ) \wedge \left (a = - b x \vee b = 0\right ) \\x \cot ^{3}{\relax (a )} & \text {for}\: b = 0 \\\frac {\log {\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} - \frac {\log {\left (\tan {\left (a + b x \right )} \right )}}{b} - \frac {1}{2 b \tan ^{2}{\left (a + b x \right )}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)**3,x)

[Out]

Piecewise((zoo*x, (Eq(a, 0) | Eq(a, -b*x)) & (Eq(b, 0) | Eq(a, -b*x))), (x*cot(a)**3, 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), True))

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