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 = 0.141352, antiderivative size = 115, normalized size of antiderivative = 1., 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 4431
Rule 3311
Rule 32
Rule 3310
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*}
Mathematica [A] time = 0.157289, size = 109, normalized size = 0.95 \[ \frac{1}{4} \left (x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )+2 \sin (2 x) \left (6 c^2 d x+2 c^3+3 c d^2 \left (2 x^2-1\right )+d^3 x \left (2 x^2-3\right )\right )+3 d \cos (2 x) \left (2 c^2+4 c d x+d^2 \left (2 x^2-1\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 179, normalized size = 1.6 \begin{align*} 4\,{d}^{3} \left ({x}^{3} \left ( 1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) +3/4\,{x}^{2} \left ( \cos \left ( x \right ) \right ) ^{2}-3/2\,x \left ( 1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) +3/8\,{x}^{2}+3/8\, \left ( \sin \left ( x \right ) \right ) ^{2}-3/8\,{x}^{4} \right ) +12\,{d}^{2}c \left ({x}^{2} \left ( 1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) +1/2\,x \left ( \cos \left ( x \right ) \right ) ^{2}-1/4\,\cos \left ( x \right ) \sin \left ( x \right ) -x/4-1/3\,{x}^{3} \right ) +12\,{c}^{2}d \left ( x \left ( 1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) -1/4\,{x}^{2}-1/4\, \left ( \sin \left ( x \right ) \right ) ^{2} \right ) -{\frac{{d}^{3}{x}^{4}}{4}}+4\,{c}^{3} \left ( 1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) -c{d}^{2}{x}^{3}-{\frac{3\,{c}^{2}d{x}^{2}}{2}}-{c}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02836, size = 136, normalized size = 1.18 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.496439, size = 278, normalized size = 2.42 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 146.651, size = 289, normalized size = 2.51 \begin{align*} c^{3} x + c^{3} \sin{\left (2 x \right )} - 3 c^{2} d x^{2} \sin ^{2}{\left (x \right )} - 3 c^{2} d x^{2} \cos ^{2}{\left (x \right )} + \frac{9 c^{2} d x^{2}}{2} + 6 c^{2} d x \sin{\left (x \right )} \cos{\left (x \right )} - 3 c^{2} d \sin ^{2}{\left (x \right )} - 2 c d^{2} x^{3} \sin ^{2}{\left (x \right )} - 2 c d^{2} x^{3} \cos ^{2}{\left (x \right )} + 3 c d^{2} x^{3} + 6 c d^{2} x^{2} \sin{\left (x \right )} \cos{\left (x \right )} - 3 c d^{2} x \sin ^{2}{\left (x \right )} + 3 c d^{2} x \cos ^{2}{\left (x \right )} - 3 c d^{2} \sin{\left (x \right )} \cos{\left (x \right )} - \frac{d^{3} x^{4} \sin ^{2}{\left (x \right )}}{2} - \frac{d^{3} x^{4} \cos ^{2}{\left (x \right )}}{2} + \frac{3 d^{3} x^{4}}{4} + 2 d^{3} x^{3} \sin{\left (x \right )} \cos{\left (x \right )} - \frac{3 d^{3} x^{2} \sin ^{2}{\left (x \right )}}{2} + \frac{3 d^{3} x^{2} \cos ^{2}{\left (x \right )}}{2} - 3 d^{3} x \sin{\left (x \right )} \cos{\left (x \right )} + \frac{3 d^{3} \sin ^{2}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14367, size = 151, normalized size = 1.31 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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