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

Optimal. Leaf size=114 \[ 3 \text{Unintegrable}\left (\frac{\csc (a+b x)}{(c+d x)^3},x\right )+\frac{2 b^2 \sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{d^3}+\frac{2 b^2 \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d^3}+\frac{2 b \cos (a+b x)}{d^2 (c+d x)}+\frac{2 \sin (a+b x)}{d (c+d x)^2} \]

[Out]

(2*b*Cos[a + b*x])/(d^2*(c + d*x)) + (2*b^2*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d^3 + (2*Sin[a + b*x]
)/(d*(c + d*x)^2) + (2*b^2*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d^3 + 3*Unintegrable[Csc[a + b*x]/(c +
d*x)^3, x]

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Rubi [A]  time = 0.328324, 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 = {} \[ \int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(2*b*Cos[a + b*x])/(d^2*(c + d*x)) + (2*b^2*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d^3 + (2*Sin[a + b*x]
)/(d*(c + d*x)^2) + (2*b^2*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d^3 + 3*Defer[Int][Csc[a + b*x]/(c + d
*x)^3, x]

Rubi steps

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

Mathematica [A]  time = 7.24875, size = 0, normalized size = 0. \[ \int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.71, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \csc \left ( bx+a \right ) \right ) ^{2}\sin \left ( 3\,bx+3\,a \right ) }{ \left ( dx+c \right ) ^{3}}}\, 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)^3,x)

[Out]

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

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Maxima [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)^3,x, algorithm="maxima")

[Out]

Timed out

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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^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}, 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)^3,x, algorithm="fricas")

[Out]

integral(csc(b*x + a)^2*sin(3*b*x + 3*a)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), 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)**3,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 )}{{\left (d x + c\right )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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