3.95 \(\int \csc (x) \sin (3 x) \, dx\)

Optimal. Leaf size=8 \[ x+2 \sin (x) \cos (x) \]

[Out]

x+2*cos(x)*sin(x)

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Rubi [A]  time = 0.03, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {385, 203} \[ x+2 \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]*Sin[3*x],x]

[Out]

x + 2*Cos[x]*Sin[x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rubi steps

\begin {align*} \int \csc (x) \sin (3 x) \, dx &=\operatorname {Subst}\left (\int \frac {3-x^2}{\left (1+x^2\right )^2} \, dx,x,\tan (x)\right )\\ &=2 \cos (x) \sin (x)+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right )\\ &=x+2 \cos (x) \sin (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 6, normalized size = 0.75 \[ x+\sin (2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]*Sin[3*x],x]

[Out]

x + Sin[2*x]

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fricas [A]  time = 0.83, size = 8, normalized size = 1.00 \[ 2 \, \cos \relax (x) \sin \relax (x) + x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)*sin(3*x),x, algorithm="fricas")

[Out]

2*cos(x)*sin(x) + x

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giac [A]  time = 0.13, size = 6, normalized size = 0.75 \[ x + \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)*sin(3*x),x, algorithm="giac")

[Out]

x + sin(2*x)

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maple [A]  time = 0.10, size = 9, normalized size = 1.12 \[ x +2 \cos \relax (x ) \sin \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)*sin(3*x),x)

[Out]

x+2*cos(x)*sin(x)

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maxima [A]  time = 0.31, size = 6, normalized size = 0.75 \[ x + \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)*sin(3*x),x, algorithm="maxima")

[Out]

x + sin(2*x)

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mupad [B]  time = 2.25, size = 6, normalized size = 0.75 \[ x+\sin \left (2\,x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(3*x)/sin(x),x)

[Out]

x + sin(2*x)

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sympy [A]  time = 0.98, size = 5, normalized size = 0.62 \[ x + \sin {\left (2 x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)*sin(3*x),x)

[Out]

x + sin(2*x)

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