Integrand size = 14, antiderivative size = 115 \[ \int (c+d x)^3 \csc (x) \sin (3 x) \, dx=-\frac {3}{2} c d^2 x-\frac {3 d^3 x^2}{4}+\frac {(c+d x)^4}{4 d}-\frac {9}{8} d^3 \cos ^2(x)+\frac {9}{4} d (c+d x)^2 \cos ^2(x)-3 d^2 (c+d x) \cos (x) \sin (x)+2 (c+d x)^3 \cos (x) \sin (x)+\frac {3}{8} d^3 \sin ^2(x)-\frac {3}{4} d (c+d x)^2 \sin ^2(x) \]
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Time = 0.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4516, 3392, 32, 3391} \[ \int (c+d x)^3 \csc (x) \sin (3 x) \, dx=-\frac {3}{2} c d^2 x-3 d^2 \sin (x) \cos (x) (c+d x)+\frac {(c+d x)^4}{4 d}-\frac {3}{4} d \sin ^2(x) (c+d x)^2+\frac {9}{4} d \cos ^2(x) (c+d x)^2+2 \sin (x) \cos (x) (c+d x)^3-\frac {3 d^3 x^2}{4}+\frac {3}{8} d^3 \sin ^2(x)-\frac {9}{8} d^3 \cos ^2(x) \]
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Rule 32
Rule 3391
Rule 3392
Rule 4516
Rubi steps \begin{align*} \text {integral}& = \int \left (3 (c+d x)^3 \cos ^2(x)-(c+d x)^3 \sin ^2(x)\right ) \, dx \\ & = 3 \int (c+d x)^3 \cos ^2(x) \, dx-\int (c+d x)^3 \sin ^2(x) \, dx \\ & = \frac {9}{4} d (c+d x)^2 \cos ^2(x)+2 (c+d x)^3 \cos (x) \sin (x)-\frac {3}{4} d (c+d x)^2 \sin ^2(x)-\frac {1}{2} \int (c+d x)^3 \, dx+\frac {3}{2} \int (c+d x)^3 \, dx+\frac {1}{2} \left (3 d^2\right ) \int (c+d x) \sin ^2(x) \, dx-\frac {1}{2} \left (9 d^2\right ) \int (c+d x) \cos ^2(x) \, dx \\ & = \frac {(c+d x)^4}{4 d}-\frac {9}{8} d^3 \cos ^2(x)+\frac {9}{4} d (c+d x)^2 \cos ^2(x)-3 d^2 (c+d x) \cos (x) \sin (x)+2 (c+d x)^3 \cos (x) \sin (x)+\frac {3}{8} d^3 \sin ^2(x)-\frac {3}{4} d (c+d x)^2 \sin ^2(x)+\frac {1}{4} \left (3 d^2\right ) \int (c+d x) \, dx-\frac {1}{4} \left (9 d^2\right ) \int (c+d x) \, dx \\ & = -\frac {3}{2} c d^2 x-\frac {3 d^3 x^2}{4}+\frac {(c+d x)^4}{4 d}-\frac {9}{8} d^3 \cos ^2(x)+\frac {9}{4} d (c+d x)^2 \cos ^2(x)-3 d^2 (c+d x) \cos (x) \sin (x)+2 (c+d x)^3 \cos (x) \sin (x)+\frac {3}{8} d^3 \sin ^2(x)-\frac {3}{4} d (c+d x)^2 \sin ^2(x) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int (c+d x)^3 \csc (x) \sin (3 x) \, dx=\frac {1}{4} \left (x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+3 d \left (2 c^2+4 c d x+d^2 \left (-1+2 x^2\right )\right ) \cos (2 x)+2 \left (2 c^3+6 c^2 d x+d^3 x \left (-3+2 x^2\right )+3 c d^2 \left (-1+2 x^2\right )\right ) \sin (2 x)\right ) \]
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Time = 1.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\frac {d^{3} x^{4}}{4}+c \,d^{2} x^{3}+\frac {3 c^{2} d \,x^{2}}{2}+c^{3} x +\frac {c^{4}}{4 d}+\frac {3 d \left (2 x^{2} d^{2}+4 c d x +2 c^{2}-d^{2}\right ) \cos \left (2 x \right )}{4}+\frac {\left (2 d^{3} x^{3}+6 c \,d^{2} x^{2}+6 c^{2} d x -3 d^{3} x +2 c^{3}-3 c \,d^{2}\right ) \sin \left (2 x \right )}{2}\) | \(119\) |
default | \(4 d^{3} \left (x^{3} \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+\frac {3 x^{2} \cos \left (x \right )^{2}}{4}-\frac {3 x \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )}{2}+\frac {3 x^{2}}{8}+\frac {3 \sin \left (x \right )^{2}}{8}-\frac {3 x^{4}}{8}\right )+12 c \,d^{2} \left (x^{2} \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+\frac {x \cos \left (x \right )^{2}}{2}-\frac {\cos \left (x \right ) \sin \left (x \right )}{4}-\frac {x}{4}-\frac {x^{3}}{3}\right )+12 c^{2} d \left (x \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )-\frac {x^{2}}{4}-\frac {\sin \left (x \right )^{2}}{4}\right )+4 c^{3} \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )-\frac {d^{3} x^{4}}{4}-c \,d^{2} x^{3}-\frac {3 c^{2} d \,x^{2}}{2}-c^{3} x\) | \(179\) |
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Time = 0.25 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.10 \[ \int (c+d x)^3 \csc (x) \sin (3 x) \, dx=\frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, {\left (c^{2} d - d^{3}\right )} x^{2} + \frac {3}{2} \, {\left (2 \, d^{3} x^{2} + 4 \, c d^{2} x + 2 \, c^{2} d - d^{3}\right )} \cos \left (x\right )^{2} + {\left (2 \, d^{3} x^{3} + 6 \, c d^{2} x^{2} + 2 \, c^{3} - 3 \, c d^{2} + 3 \, {\left (2 \, c^{2} d - d^{3}\right )} x\right )} \cos \left (x\right ) \sin \left (x\right ) + {\left (c^{3} - 3 \, c d^{2}\right )} x \]
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Time = 7.78 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.21 \[ \int (c+d x)^3 \csc (x) \sin (3 x) \, dx=c^{3} \left (x + \sin {\left (2 x \right )}\right ) + 3 c^{2} d \left (- \frac {x^{2}}{2} + x \left (x + \sin {\left (2 x \right )}\right ) + \frac {\cos {\left (2 x \right )}}{2}\right ) + 3 c d^{2} \left (\frac {x^{3}}{3} + x^{2} \left (x + \sin {\left (2 x \right )}\right ) - 2 x \left (\frac {x^{2}}{2} - \frac {\cos {\left (2 x \right )}}{2}\right ) - \frac {\sin {\left (2 x \right )}}{2}\right ) + d^{3} \left (- \frac {x^{4}}{4} + x^{3} \left (x + \sin {\left (2 x \right )}\right ) - 3 x^{2} \left (\frac {x^{2}}{2} - \frac {\cos {\left (2 x \right )}}{2}\right ) + 3 x \left (\frac {x^{3}}{3} - \frac {\sin {\left (2 x \right )}}{2}\right ) - \frac {3 \cos {\left (2 x \right )}}{4}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.88 \[ \int (c+d x)^3 \csc (x) \sin (3 x) \, dx=\frac {3}{2} \, {\left (x^{2} + 2 \, x \sin \left (2 \, x\right ) + \cos \left (2 \, x\right )\right )} c^{2} d + \frac {1}{2} \, {\left (2 \, x^{3} + 6 \, x \cos \left (2 \, x\right ) + 3 \, {\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right )\right )} c d^{2} + \frac {1}{4} \, {\left (x^{4} + 3 \, {\left (2 \, x^{2} - 1\right )} \cos \left (2 \, x\right ) + 2 \, {\left (2 \, x^{3} - 3 \, x\right )} \sin \left (2 \, x\right )\right )} d^{3} + c^{3} {\left (x + \sin \left (2 \, x\right )\right )} \]
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Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int (c+d x)^3 \csc (x) \sin (3 x) \, dx=\frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} d x^{2} + c^{3} x + \frac {3}{4} \, {\left (2 \, d^{3} x^{2} + 4 \, c d^{2} x + 2 \, c^{2} d - d^{3}\right )} \cos \left (2 \, x\right ) + \frac {1}{2} \, {\left (2 \, d^{3} x^{3} + 6 \, c d^{2} x^{2} + 6 \, c^{2} d x - 3 \, d^{3} x + 2 \, c^{3} - 3 \, c d^{2}\right )} \sin \left (2 \, x\right ) \]
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Time = 0.44 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.18 \[ \int (c+d x)^3 \csc (x) \sin (3 x) \, dx=c^3\,\sin \left (2\,x\right )-\frac {3\,d^3\,\cos \left (2\,x\right )}{4}+c^3\,x+\frac {d^3\,x^4}{4}+\frac {3\,d^3\,x^2\,\cos \left (2\,x\right )}{2}+d^3\,x^3\,\sin \left (2\,x\right )+\frac {3\,c^2\,d\,x^2}{2}+c\,d^2\,x^3+\frac {3\,c^2\,d\,\cos \left (2\,x\right )}{2}-\frac {3\,c\,d^2\,\sin \left (2\,x\right )}{2}-\frac {3\,d^3\,x\,\sin \left (2\,x\right )}{2}+3\,c\,d^2\,x\,\cos \left (2\,x\right )+3\,c^2\,d\,x\,\sin \left (2\,x\right )+3\,c\,d^2\,x^2\,\sin \left (2\,x\right ) \]
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