Optimal. Leaf size=115 \[ -\frac {3}{2} c d^2 x-3 d^2 \sin (x) \cos (x) (c+d x)+\frac {(c+d x)^4}{4 d}-\frac {3}{4} d \sin ^2(x) (c+d x)^2+\frac {9}{4} d \cos ^2(x) (c+d x)^2+2 \sin (x) \cos (x) (c+d x)^3-\frac {3 d^3 x^2}{4}+\frac {3}{8} d^3 \sin ^2(x)-\frac {9}{8} d^3 \cos ^2(x) \]
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Rubi [A] time = 0.14, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4431, 3311, 32, 3310} \[ -\frac {3}{2} c d^2 x-3 d^2 \sin (x) \cos (x) (c+d x)+\frac {(c+d x)^4}{4 d}-\frac {3}{4} d \sin ^2(x) (c+d x)^2+\frac {9}{4} d \cos ^2(x) (c+d x)^2+2 \sin (x) \cos (x) (c+d x)^3-\frac {3 d^3 x^2}{4}+\frac {3}{8} d^3 \sin ^2(x)-\frac {9}{8} d^3 \cos ^2(x) \]
Antiderivative was successfully verified.
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Rule 32
Rule 3310
Rule 3311
Rule 4431
Rubi steps
\begin {align*} \int (c+d x)^3 \csc (x) \sin (3 x) \, dx &=\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*}
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Mathematica [A] time = 0.16, size = 109, normalized size = 0.95 \[ \frac {1}{4} \left (3 d \cos (2 x) \left (2 c^2+4 c d x+d^2 \left (2 x^2-1\right )\right )+2 \sin (2 x) \left (2 c^3+6 c^2 d x+3 c d^2 \left (2 x^2-1\right )+d^3 x \left (2 x^2-3\right )\right )+x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 127, normalized size = 1.10 \[ \frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, {\left (c^{2} d - d^{3}\right )} x^{2} + \frac {3}{2} \, {\left (2 \, d^{3} x^{2} + 4 \, c d^{2} x + 2 \, c^{2} d - d^{3}\right )} \cos \relax (x)^{2} + {\left (2 \, d^{3} x^{3} + 6 \, c d^{2} x^{2} + 2 \, c^{3} - 3 \, c d^{2} + 3 \, {\left (2 \, c^{2} d - d^{3}\right )} x\right )} \cos \relax (x) \sin \relax (x) + {\left (c^{3} - 3 \, c d^{2}\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 112, normalized size = 0.97 \[ \frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} d x^{2} + c^{3} x + \frac {3}{4} \, {\left (2 \, d^{3} x^{2} + 4 \, c d^{2} x + 2 \, c^{2} d - d^{3}\right )} \cos \left (2 \, x\right ) + \frac {1}{2} \, {\left (2 \, d^{3} x^{3} + 6 \, c d^{2} x^{2} + 6 \, c^{2} d x - 3 \, d^{3} x + 2 \, c^{3} - 3 \, c d^{2}\right )} \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 179, normalized size = 1.56 \[ 4 d^{3} \left (x^{3} \left (\frac {\cos \relax (x ) \sin \relax (x )}{2}+\frac {x}{2}\right )+\frac {3 x^{2} \left (\cos ^{2}\relax (x )\right )}{4}-\frac {3 x \left (\frac {\cos \relax (x ) \sin \relax (x )}{2}+\frac {x}{2}\right )}{2}+\frac {3 x^{2}}{8}+\frac {3 \left (\sin ^{2}\relax (x )\right )}{8}-\frac {3 x^{4}}{8}\right )+12 c \,d^{2} \left (x^{2} \left (\frac {\cos \relax (x ) \sin \relax (x )}{2}+\frac {x}{2}\right )+\frac {x \left (\cos ^{2}\relax (x )\right )}{2}-\frac {\cos \relax (x ) \sin \relax (x )}{4}-\frac {x}{4}-\frac {x^{3}}{3}\right )+12 c^{2} d \left (x \left (\frac {\cos \relax (x ) \sin \relax (x )}{2}+\frac {x}{2}\right )-\frac {x^{2}}{4}-\frac {\left (\sin ^{2}\relax (x )\right )}{4}\right )-\frac {d^{3} x^{4}}{4}+4 c^{3} \left (\frac {\cos \relax (x ) \sin \relax (x )}{2}+\frac {x}{2}\right )-c \,d^{2} x^{3}-\frac {3 c^{2} d \,x^{2}}{2}-c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 101, normalized size = 0.88 \[ \frac {3}{2} \, {\left (x^{2} + 2 \, x \sin \left (2 \, x\right ) + \cos \left (2 \, x\right )\right )} c^{2} d + \frac {1}{2} \, {\left (2 \, x^{3} + 6 \, x \cos \left (2 \, x\right ) + 3 \, {\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right )\right )} c d^{2} + \frac {1}{4} \, {\left (x^{4} + 3 \, {\left (2 \, x^{2} - 1\right )} \cos \left (2 \, x\right ) + 2 \, {\left (2 \, x^{3} - 3 \, x\right )} \sin \left (2 \, x\right )\right )} d^{3} + c^{3} {\left (x + \sin \left (2 \, x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 136, normalized size = 1.18 \[ c^3\,\sin \left (2\,x\right )-\frac {3\,d^3\,\cos \left (2\,x\right )}{4}+c^3\,x+\frac {d^3\,x^4}{4}+\frac {3\,d^3\,x^2\,\cos \left (2\,x\right )}{2}+d^3\,x^3\,\sin \left (2\,x\right )+\frac {3\,c^2\,d\,x^2}{2}+c\,d^2\,x^3+\frac {3\,c^2\,d\,\cos \left (2\,x\right )}{2}-\frac {3\,c\,d^2\,\sin \left (2\,x\right )}{2}-\frac {3\,d^3\,x\,\sin \left (2\,x\right )}{2}+3\,c\,d^2\,x\,\cos \left (2\,x\right )+3\,c^2\,d\,x\,\sin \left (2\,x\right )+3\,c\,d^2\,x^2\,\sin \left (2\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 13.05, size = 289, normalized size = 2.51 \[ c^{3} x + c^{3} \sin {\left (2 x \right )} - 3 c^{2} d x^{2} \sin ^{2}{\relax (x )} - 3 c^{2} d x^{2} \cos ^{2}{\relax (x )} + \frac {9 c^{2} d x^{2}}{2} + 6 c^{2} d x \sin {\relax (x )} \cos {\relax (x )} + 3 c^{2} d \cos ^{2}{\relax (x )} - 2 c d^{2} x^{3} \sin ^{2}{\relax (x )} - 2 c d^{2} x^{3} \cos ^{2}{\relax (x )} + 3 c d^{2} x^{3} + 6 c d^{2} x^{2} \sin {\relax (x )} \cos {\relax (x )} - 3 c d^{2} x \sin ^{2}{\relax (x )} + 3 c d^{2} x \cos ^{2}{\relax (x )} - 3 c d^{2} \sin {\relax (x )} \cos {\relax (x )} - \frac {d^{3} x^{4} \sin ^{2}{\relax (x )}}{2} - \frac {d^{3} x^{4} \cos ^{2}{\relax (x )}}{2} + \frac {3 d^{3} x^{4}}{4} + 2 d^{3} x^{3} \sin {\relax (x )} \cos {\relax (x )} - \frac {3 d^{3} x^{2} \sin ^{2}{\relax (x )}}{2} + \frac {3 d^{3} x^{2} \cos ^{2}{\relax (x )}}{2} - 3 d^{3} x \sin {\relax (x )} \cos {\relax (x )} - \frac {3 d^{3} \cos ^{2}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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