Integrand size = 14, antiderivative size = 73 \[ \int (c+d x)^2 \csc (x) \sin (3 x) \, dx=-\frac {d^2 x}{2}+\frac {(c+d x)^3}{3 d}+\frac {3}{2} d (c+d x) \cos ^2(x)-d^2 \cos (x) \sin (x)+2 (c+d x)^2 \cos (x) \sin (x)-\frac {1}{2} d (c+d x) \sin ^2(x) \]
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Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4516, 3392, 32, 2715, 8} \[ \int (c+d x)^2 \csc (x) \sin (3 x) \, dx=\frac {(c+d x)^3}{3 d}-\frac {1}{2} d \sin ^2(x) (c+d x)+\frac {3}{2} d \cos ^2(x) (c+d x)+2 \sin (x) \cos (x) (c+d x)^2-\frac {d^2 x}{2}-d^2 \sin (x) \cos (x) \]
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Rule 8
Rule 32
Rule 2715
Rule 3392
Rule 4516
Rubi steps \begin{align*} \text {integral}& = \int \left (3 (c+d x)^2 \cos ^2(x)-(c+d x)^2 \sin ^2(x)\right ) \, dx \\ & = 3 \int (c+d x)^2 \cos ^2(x) \, dx-\int (c+d x)^2 \sin ^2(x) \, dx \\ & = \frac {3}{2} d (c+d x) \cos ^2(x)+2 (c+d x)^2 \cos (x) \sin (x)-\frac {1}{2} d (c+d x) \sin ^2(x)-\frac {1}{2} \int (c+d x)^2 \, dx+\frac {3}{2} \int (c+d x)^2 \, dx+\frac {1}{2} d^2 \int \sin ^2(x) \, dx-\frac {1}{2} \left (3 d^2\right ) \int \cos ^2(x) \, dx \\ & = \frac {(c+d x)^3}{3 d}+\frac {3}{2} d (c+d x) \cos ^2(x)-d^2 \cos (x) \sin (x)+2 (c+d x)^2 \cos (x) \sin (x)-\frac {1}{2} d (c+d x) \sin ^2(x)+\frac {1}{4} d^2 \int 1 \, dx-\frac {1}{4} \left (3 d^2\right ) \int 1 \, dx \\ & = -\frac {d^2 x}{2}+\frac {(c+d x)^3}{3 d}+\frac {3}{2} d (c+d x) \cos ^2(x)-d^2 \cos (x) \sin (x)+2 (c+d x)^2 \cos (x) \sin (x)-\frac {1}{2} d (c+d x) \sin ^2(x) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int (c+d x)^2 \csc (x) \sin (3 x) \, dx=c^2 x+c d x^2+\frac {d^2 x^3}{3}+d (c+d x) \cos (2 x)+\left (2 c^2+4 c d x+d^2 \left (-1+2 x^2\right )\right ) \cos (x) \sin (x) \]
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Time = 0.79 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {d^{2} x^{3}}{3}+d c \,x^{2}+c^{2} x +\frac {c^{3}}{3 d}+d \left (d x +c \right ) \cos \left (2 x \right )+\frac {\left (2 x^{2} d^{2}+4 c d x +2 c^{2}-d^{2}\right ) \sin \left (2 x \right )}{2}\) | \(70\) |
default | \(4 d^{2} \left (x^{2} \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+\frac {x \cos \left (x \right )^{2}}{2}-\frac {\cos \left (x \right ) \sin \left (x \right )}{4}-\frac {x}{4}-\frac {x^{3}}{3}\right )+8 c d \left (x \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )-\frac {x^{2}}{4}-\frac {\sin \left (x \right )^{2}}{4}\right )+4 c^{2} \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )-\frac {d^{2} x^{3}}{3}-d c \,x^{2}-c^{2} x\) | \(107\) |
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Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96 \[ \int (c+d x)^2 \csc (x) \sin (3 x) \, dx=\frac {1}{3} \, d^{2} x^{3} + c d x^{2} + 2 \, {\left (d^{2} x + c d\right )} \cos \left (x\right )^{2} + {\left (2 \, d^{2} x^{2} + 4 \, c d x + 2 \, c^{2} - d^{2}\right )} \cos \left (x\right ) \sin \left (x\right ) + {\left (c^{2} - d^{2}\right )} x \]
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Time = 3.89 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03 \[ \int (c+d x)^2 \csc (x) \sin (3 x) \, dx=c^{2} \left (x + \sin {\left (2 x \right )}\right ) + 2 c d \left (- \frac {x^{2}}{2} + x \left (x + \sin {\left (2 x \right )}\right ) + \frac {\cos {\left (2 x \right )}}{2}\right ) + d^{2} \left (\frac {x^{3}}{3} + x^{2} \left (x + \sin {\left (2 x \right )}\right ) - 2 x \left (\frac {x^{2}}{2} - \frac {\cos {\left (2 x \right )}}{2}\right ) - \frac {\sin {\left (2 x \right )}}{2}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int (c+d x)^2 \csc (x) \sin (3 x) \, dx={\left (x^{2} + 2 \, x \sin \left (2 \, x\right ) + \cos \left (2 \, x\right )\right )} c d + \frac {1}{6} \, {\left (2 \, x^{3} + 6 \, x \cos \left (2 \, x\right ) + 3 \, {\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right )\right )} d^{2} + c^{2} {\left (x + \sin \left (2 \, x\right )\right )} \]
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Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.88 \[ \int (c+d x)^2 \csc (x) \sin (3 x) \, dx=\frac {1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x + {\left (d^{2} x + c d\right )} \cos \left (2 \, x\right ) + \frac {1}{2} \, {\left (2 \, d^{2} x^{2} + 4 \, c d x + 2 \, c^{2} - d^{2}\right )} \sin \left (2 \, x\right ) \]
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Time = 25.72 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int (c+d x)^2 \csc (x) \sin (3 x) \, dx=c^2\,\sin \left (2\,x\right )-\frac {d^2\,\sin \left (2\,x\right )}{2}+c^2\,x+\frac {d^2\,x^3}{3}+d^2\,x^2\,\sin \left (2\,x\right )+c\,d\,\cos \left (2\,x\right )+d^2\,x\,\cos \left (2\,x\right )+c\,d\,x^2+2\,c\,d\,x\,\sin \left (2\,x\right ) \]
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