Optimal. Leaf size=73 \[ \frac{(c+d x)^3}{3 d}-\frac{1}{2} d \sin ^2(x) (c+d x)+\frac{3}{2} d \cos ^2(x) (c+d x)+2 \sin (x) \cos (x) (c+d x)^2-\frac{d^2 x}{2}-d^2 \sin (x) \cos (x) \]
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Rubi [A] time = 0.102692, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {4431, 3311, 32, 2635, 8} \[ \frac{(c+d x)^3}{3 d}-\frac{1}{2} d \sin ^2(x) (c+d x)+\frac{3}{2} d \cos ^2(x) (c+d x)+2 \sin (x) \cos (x) (c+d x)^2-\frac{d^2 x}{2}-d^2 \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 4431
Rule 3311
Rule 32
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int (c+d x)^2 \csc (x) \sin (3 x) \, dx &=\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*}
Mathematica [A] time = 0.110072, size = 60, normalized size = 0.82 \[ \sin (x) \cos (x) \left (2 c^2+4 c d x+d^2 \left (2 x^2-1\right )\right )+c^2 x+c d x^2+d \cos (2 x) (c+d x)+\frac{d^2 x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 107, normalized size = 1.5 \begin{align*} 4\,{d}^{2} \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 ) +8\,cd \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 ) +4\,{c}^{2} \left ( 1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) -{\frac{{d}^{2}{x}^{3}}{3}}-cd{x}^{2}-{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995402, size = 81, normalized size = 1.11 \begin{align*}{\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.512624, size = 159, normalized size = 2.18 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 21.0139, size = 155, normalized size = 2.12 \begin{align*} c^{2} x + c^{2} \sin{\left (2 x \right )} - 2 c d x^{2} \sin ^{2}{\left (x \right )} - 2 c d x^{2} \cos ^{2}{\left (x \right )} + 3 c d x^{2} + 4 c d x \sin{\left (x \right )} \cos{\left (x \right )} - 2 c d \sin ^{2}{\left (x \right )} - \frac{2 d^{2} x^{3} \sin ^{2}{\left (x \right )}}{3} - \frac{2 d^{2} x^{3} \cos ^{2}{\left (x \right )}}{3} + d^{2} x^{3} + 2 d^{2} x^{2} \sin{\left (x \right )} \cos{\left (x \right )} - d^{2} x \sin ^{2}{\left (x \right )} + d^{2} x \cos ^{2}{\left (x \right )} - d^{2} \sin{\left (x \right )} \cos{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12762, size = 86, normalized size = 1.18 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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