3.364 \(\int (c+d x) \csc (x) \sin (3 x) \, dx\)

Optimal. Leaf size=41 \[ 2 \sin (x) \cos (x) (c+d x)+c x+\frac{d x^2}{2}-\frac{1}{4} d \sin ^2(x)+\frac{3}{4} d \cos ^2(x) \]

[Out]

c*x + (d*x^2)/2 + (3*d*Cos[x]^2)/4 + 2*(c + d*x)*Cos[x]*Sin[x] - (d*Sin[x]^2)/4

________________________________________________________________________________________

Rubi [A]  time = 0.0563706, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4431, 3310} \[ 2 \sin (x) \cos (x) (c+d x)+c x+\frac{d x^2}{2}-\frac{1}{4} d \sin ^2(x)+\frac{3}{4} d \cos ^2(x) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)*Csc[x]*Sin[3*x],x]

[Out]

c*x + (d*x^2)/2 + (3*d*Cos[x]^2)/4 + 2*(c + d*x)*Cos[x]*Sin[x] - (d*Sin[x]^2)/4

Rule 4431

Int[((e_.) + (f_.)*(x_))^(m_.)*(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int
[ExpandTrigExpand[(e + f*x)^m*G[c + d*x]^q, F, c + d*x, p, b/d, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && M
emberQ[{Sin, Cos}, F] && MemberQ[{Sec, Csc}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d,
1]

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rubi steps

\begin{align*} \int (c+d x) \csc (x) \sin (3 x) \, dx &=\int \left (3 (c+d x) \cos ^2(x)-(c+d x) \sin ^2(x)\right ) \, dx\\ &=3 \int (c+d x) \cos ^2(x) \, dx-\int (c+d x) \sin ^2(x) \, dx\\ &=\frac{3}{4} d \cos ^2(x)+2 (c+d x) \cos (x) \sin (x)-\frac{1}{4} d \sin ^2(x)-\frac{1}{2} \int (c+d x) \, dx+\frac{3}{2} \int (c+d x) \, dx\\ &=c x+\frac{d x^2}{2}+\frac{3}{4} d \cos ^2(x)+2 (c+d x) \cos (x) \sin (x)-\frac{1}{4} d \sin ^2(x)\\ \end{align*}

Mathematica [A]  time = 0.0198871, size = 34, normalized size = 0.83 \[ c x+c \sin (2 x)+\frac{d x^2}{2}+d x \sin (2 x)+\frac{1}{2} d \cos (2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)*Csc[x]*Sin[3*x],x]

[Out]

c*x + (d*x^2)/2 + (d*Cos[2*x])/2 + c*Sin[2*x] + d*x*Sin[2*x]

________________________________________________________________________________________

Maple [A]  time = 0.048, size = 52, normalized size = 1.3 \begin{align*} 4\,d \left ( x \left ( 1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) -1/4\,{x}^{2}-1/4\, \left ( \sin \left ( x \right ) \right ) ^{2} \right ) +4\,c \left ( 1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) -{\frac{d{x}^{2}}{2}}-cx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)*csc(x)*sin(3*x),x)

[Out]

4*d*(x*(1/2*cos(x)*sin(x)+1/2*x)-1/4*x^2-1/4*sin(x)^2)+4*c*(1/2*cos(x)*sin(x)+1/2*x)-1/2*d*x^2-c*x

________________________________________________________________________________________

Maxima [A]  time = 1.02177, size = 36, normalized size = 0.88 \begin{align*} \frac{1}{2} \,{\left (x^{2} + 2 \, x \sin \left (2 \, x\right ) + \cos \left (2 \, x\right )\right )} d + c{\left (x + \sin \left (2 \, x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^2 + 2*x*sin(2*x) + cos(2*x))*d + c*(x + sin(2*x))

________________________________________________________________________________________

Fricas [A]  time = 0.492165, size = 78, normalized size = 1.9 \begin{align*} \frac{1}{2} \, d x^{2} + d \cos \left (x\right )^{2} + 2 \,{\left (d x + c\right )} \cos \left (x\right ) \sin \left (x\right ) + c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*d*x^2 + d*cos(x)^2 + 2*(d*x + c)*cos(x)*sin(x) + c*x

________________________________________________________________________________________

Sympy [A]  time = 14.3026, size = 56, normalized size = 1.37 \begin{align*} c x + c \sin{\left (2 x \right )} - d x^{2} \sin ^{2}{\left (x \right )} - d x^{2} \cos ^{2}{\left (x \right )} + \frac{3 d x^{2}}{2} + 2 d x \sin{\left (x \right )} \cos{\left (x \right )} - d \sin ^{2}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*csc(x)*sin(3*x),x)

[Out]

c*x + c*sin(2*x) - d*x**2*sin(x)**2 - d*x**2*cos(x)**2 + 3*d*x**2/2 + 2*d*x*sin(x)*cos(x) - d*sin(x)**2

________________________________________________________________________________________

Giac [A]  time = 1.0992, size = 36, normalized size = 0.88 \begin{align*} \frac{1}{2} \, d x^{2} + c x + \frac{1}{2} \, d \cos \left (2 \, x\right ) +{\left (d x + c\right )} \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*d*x^2 + c*x + 1/2*d*cos(2*x) + (d*x + c)*sin(2*x)