3.146 \(\int \cos ^2(a+b x) \csc (2 a+2 b x) \, dx\)

Optimal. Leaf size=14 \[ \frac{\log (\sin (a+b x))}{2 b} \]

[Out]

Log[Sin[a + b*x]]/(2*b)

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Rubi [A]  time = 0.02515, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4287, 3475} \[ \frac{\log (\sin (a+b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^2*Csc[2*a + 2*b*x],x]

[Out]

Log[Sin[a + b*x]]/(2*b)

Rule 4287

Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[2^p/e^p, Int[(e*Cos
[a + b*x])^(m + p)*Sin[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
&& IntegerQ[p]

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 \cos ^2(a+b x) \csc (2 a+2 b x) \, dx &=\frac{1}{2} \int \cot (a+b x) \, dx\\ &=\frac{\log (\sin (a+b x))}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0146544, size = 22, normalized size = 1.57 \[ \frac{\log (\tan (a+b x))+\log (\cos (a+b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[Cos[a + b*x]] + Log[Tan[a + b*x]])/(2*b)

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Maple [A]  time = 0.014, size = 13, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( \sin \left ( bx+a \right ) \right ) }{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^2/sin(2*b*x+2*a),x)

[Out]

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

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Maxima [B]  time = 1.1804, size = 111, normalized size = 7.93 \begin{align*} \frac{\log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(log(cos(b*x)^2 + 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(a) + sin(a)^2) + log(cos(b*x)
^2 - 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(a) + sin(a)^2))/b

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Fricas [A]  time = 0.491007, size = 39, normalized size = 2.79 \begin{align*} \frac{\log \left (\frac{1}{2} \, \sin \left (b x + a\right )\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(1/2*sin(b*x + a))/b

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.28143, size = 76, normalized size = 5.43 \begin{align*} \frac{\log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 2 \, \log \left ({\left | -\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)) - 2*log(abs(-(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1)
))/b