3.52 \(\int \csc ^2(a+b x) \sin (2 a+2 b x) \, dx\)

Optimal. Leaf size=12 \[ \frac{2 \log (\sin (a+b x))}{b} \]

[Out]

(2*Log[Sin[a + b*x]])/b

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Rubi [A]  time = 0.0201196, antiderivative size = 12, 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 = {4288, 3475} \[ \frac{2 \log (\sin (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Log[Sin[a + b*x]])/b

Rule 4288

Int[((f_.)*sin[(a_.) + (b_.)*(x_)])^(n_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[2^p/f^p, Int[Cos[a
+ b*x]^p*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, f, n}, 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 \csc ^2(a+b x) \sin (2 a+2 b x) \, dx &=2 \int \cot (a+b x) \, dx\\ &=\frac{2 \log (\sin (a+b x))}{b}\\ \end{align*}

Mathematica [A]  time = 0.0151928, size = 20, normalized size = 1.67 \[ \frac{2 (\log (\tan (a+b x))+\log (\cos (a+b x)))}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.02, size = 13, normalized size = 1.1 \begin{align*} 2\,{\frac{\ln \left ( \sin \left ( bx+a \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.2166, size = 109, normalized size = 9.08 \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 )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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 = 0.483796, size = 36, normalized size = 3. \begin{align*} \frac{2 \, \log \left (\frac{1}{2} \, \sin \left (b x + a\right )\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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(csc(b*x+a)**2*sin(2*b*x+2*a),x)

[Out]

Timed out

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Giac [B]  time = 1.30846, size = 74, normalized size = 6.17 \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 )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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