3.2.64 \(\int \cot ^2(a+b x) \csc ^2(a+b x) \, dx\) [164]

Optimal. Leaf size=15 \[ -\frac {\cot ^3(a+b x)}{3 b} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2687, 30} \begin {gather*} -\frac {\cot ^3(a+b x)}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x]^2*Csc[a + b*x]^2,x]

[Out]

-1/3*Cot[a + b*x]^3/b

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rubi steps

\begin {align*} \int \cot ^2(a+b x) \csc ^2(a+b x) \, dx &=\frac {\text {Subst}\left (\int x^2 \, dx,x,-\cot (a+b x)\right )}{b}\\ &=-\frac {\cot ^3(a+b x)}{3 b}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -\frac {\cot ^3(a+b x)}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x]^2*Csc[a + b*x]^2,x]

[Out]

-1/3*Cot[a + b*x]^3/b

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Maple [A]
time = 0.03, size = 22, normalized size = 1.47

method	result	size
derivativedivides	\(-\frac {\cos ^{3}\left (b x +a \right )}{3 \sin \left (b x +a \right )^{3} b}\)	\(22\)
default	\(-\frac {\cos ^{3}\left (b x +a \right )}{3 \sin \left (b x +a \right )^{3} b}\)	\(22\)
risch	\(\frac {2 i \left (3 \,{\mathrm e}^{4 i \left (b x +a \right )}+1\right )}{3 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}}\)	\(33\)
norman	\(\frac {-\frac {1}{24 b}+\frac {\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}{8 b}-\frac {\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )}{8 b}+\frac {\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )}{24 b}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^2/sin(b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 13, normalized size = 0.87 \begin {gather*} -\frac {1}{3 \, b \tan \left (b x + a\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^2/sin(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/3/(b*tan(b*x + a)^3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (13) = 26\).
time = 0.35, size = 34, normalized size = 2.27 \begin {gather*} \frac {\cos \left (b x + a\right )^{3}}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^2/sin(b*x+a)^4,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (12) = 24\).
time = 0.78, size = 71, normalized size = 4.73 \begin {gather*} \begin {cases} \frac {\tan ^{3}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{24 b} - \frac {\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )}}{8 b} + \frac {1}{8 b \tan {\left (\frac {a}{2} + \frac {b x}{2} \right )}} - \frac {1}{24 b \tan ^{3}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{2}{\left (a \right )}}{\sin ^{4}{\left (a \right )}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**2/sin(b*x+a)**4,x)

[Out]

Piecewise((tan(a/2 + b*x/2)**3/(24*b) - tan(a/2 + b*x/2)/(8*b) + 1/(8*b*tan(a/2 + b*x/2)) - 1/(24*b*tan(a/2 +
b*x/2)**3), Ne(b, 0)), (x*cos(a)**2/sin(a)**4, True))

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Giac [A]
time = 4.40, size = 13, normalized size = 0.87 \begin {gather*} -\frac {1}{3 \, b \tan \left (b x + a\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^2/sin(b*x+a)^4,x, algorithm="giac")

[Out]

-1/3/(b*tan(b*x + a)^3)

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Mupad [B]
time = 0.38, size = 13, normalized size = 0.87 \begin {gather*} -\frac {{\mathrm {cot}\left (a+b\,x\right )}^3}{3\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^2/sin(a + b*x)^4,x)

[Out]

-cot(a + b*x)^3/(3*b)

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