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

Optimal. Leaf size=11 \[ \frac{2 \sin (a+b x)}{b} \]

[Out]

(2*Sin[a + b*x])/b

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Rubi [A]  time = 0.0165524, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4288, 2637} \[ \frac{2 \sin (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]*Sin[2*a + 2*b*x],x]

[Out]

(2*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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [B]  time = 0.0081312, size = 23, normalized size = 2.09 \[ 2 \left (\frac{\sin (a) \cos (b x)}{b}+\frac{\cos (a) \sin (b x)}{b}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.018, size = 12, normalized size = 1.1 \begin{align*} 2\,{\frac{\sin \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07669, size = 15, normalized size = 1.36 \begin{align*} \frac{2 \, \sin \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sin(b*x + a)/b

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Fricas [A]  time = 0.46841, size = 24, normalized size = 2.18 \begin{align*} \frac{2 \, \sin \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.30253, size = 15, normalized size = 1.36 \begin{align*} \frac{2 \, \sin \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sin(b*x + a)/b