3.379 \(\int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx\)

Optimal. Leaf size=71 \[ 3 \text{Unintegrable}\left (\frac{\csc (a+b x)}{c+d x},x\right )-\frac{4 \sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{d}-\frac{4 \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d} \]

[Out]

(-4*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d - (4*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d + 3*Uni
ntegrable[Csc[a + b*x]/(c + d*x), x]

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Rubi [A]  time = 0.2104, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Csc[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x),x]

[Out]

(-4*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d - (4*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d + 3*Def
er[Int][Csc[a + b*x]/(c + d*x), x]

Rubi steps

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

Mathematica [A]  time = 6.23243, size = 0, normalized size = 0. \[ \int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Csc[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x),x]

[Out]

Integrate[(Csc[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x), x]

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Maple [A]  time = 0.349, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \csc \left ( bx+a \right ) \right ) ^{2}\sin \left ( 3\,bx+3\,a \right ) }{dx+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^2*sin(3*b*x+3*a)/(d*x+c),x)

[Out]

int(csc(b*x+a)^2*sin(3*b*x+3*a)/(d*x+c),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (2 i \, E_{1}\left (\frac{i \, b d x + i \, b c}{d}\right ) - 2 i \, E_{1}\left (-\frac{i \, b d x + i \, b c}{d}\right )\right )} \cos \left (-\frac{b c - a d}{d}\right ) + 3 \, d \int \frac{\sin \left (b x + a\right )}{{\left (d x + c\right )}{\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x} + 3 \, d \int \frac{\sin \left (b x + a\right )}{{\left (d x + c\right )}{\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x} + 2 \,{\left (E_{1}\left (\frac{i \, b d x + i \, b c}{d}\right ) + E_{1}\left (-\frac{i \, b d x + i \, b c}{d}\right )\right )} \sin \left (-\frac{b c - a d}{d}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2*sin(3*b*x+3*a)/(d*x+c),x, algorithm="maxima")

[Out]

((2*I*exp_integral_e(1, (I*b*d*x + I*b*c)/d) - 2*I*exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d
) + 3*d*integrate(sin(b*x + a)/((d*x + c)*cos(b*x + a)^2 + (d*x + c)*sin(b*x + a)^2 + d*x + 2*(d*x + c)*cos(b*
x + a) + c), x) + 3*d*integrate(sin(b*x + a)/((d*x + c)*cos(b*x + a)^2 + (d*x + c)*sin(b*x + a)^2 + d*x - 2*(d
*x + c)*cos(b*x + a) + c), x) + 2*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*
c)/d))*sin(-(b*c - a*d)/d))/d

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2*sin(3*b*x+3*a)/(d*x+c),x, algorithm="fricas")

[Out]

integral(csc(b*x + a)^2*sin(3*b*x + 3*a)/(d*x + c), x)

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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(3*b*x+3*a)/(d*x+c),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2*sin(3*b*x+3*a)/(d*x+c),x, algorithm="giac")

[Out]

integrate(csc(b*x + a)^2*sin(3*b*x + 3*a)/(d*x + c), x)