Optimal. Leaf size=131 \[ \frac{3}{2} d^3 \sin ^2(x) (c+d x)-\frac{9}{2} d^3 \cos ^2(x) (c+d x)-6 d^2 \sin (x) \cos (x) (c+d x)^2+\frac{(c+d x)^5}{5 d}-d (c+d x)^3-d \sin ^2(x) (c+d x)^3+3 d \cos ^2(x) (c+d x)^3+2 \sin (x) \cos (x) (c+d x)^4+\frac{3 d^4 x}{2}+3 d^4 \sin (x) \cos (x) \]
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Rubi [A] time = 0.188099, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 14, 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} \[ \frac{3}{2} d^3 \sin ^2(x) (c+d x)-\frac{9}{2} d^3 \cos ^2(x) (c+d x)-6 d^2 \sin (x) \cos (x) (c+d x)^2+\frac{(c+d x)^5}{5 d}-d (c+d x)^3-d \sin ^2(x) (c+d x)^3+3 d \cos ^2(x) (c+d x)^3+2 \sin (x) \cos (x) (c+d x)^4+\frac{3 d^4 x}{2}+3 d^4 \sin (x) \cos (x) \]
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)^4 \csc (x) \sin (3 x) \, dx &=\int \left (3 (c+d x)^4 \cos ^2(x)-(c+d x)^4 \sin ^2(x)\right ) \, dx\\ &=3 \int (c+d x)^4 \cos ^2(x) \, dx-\int (c+d x)^4 \sin ^2(x) \, dx\\ &=3 d (c+d x)^3 \cos ^2(x)+2 (c+d x)^4 \cos (x) \sin (x)-d (c+d x)^3 \sin ^2(x)-\frac{1}{2} \int (c+d x)^4 \, dx+\frac{3}{2} \int (c+d x)^4 \, dx+\left (3 d^2\right ) \int (c+d x)^2 \sin ^2(x) \, dx-\left (9 d^2\right ) \int (c+d x)^2 \cos ^2(x) \, dx\\ &=\frac{(c+d x)^5}{5 d}-\frac{9}{2} d^3 (c+d x) \cos ^2(x)+3 d (c+d x)^3 \cos ^2(x)-6 d^2 (c+d x)^2 \cos (x) \sin (x)+2 (c+d x)^4 \cos (x) \sin (x)+\frac{3}{2} d^3 (c+d x) \sin ^2(x)-d (c+d x)^3 \sin ^2(x)+\frac{1}{2} \left (3 d^2\right ) \int (c+d x)^2 \, dx-\frac{1}{2} \left (9 d^2\right ) \int (c+d x)^2 \, dx-\frac{1}{2} \left (3 d^4\right ) \int \sin ^2(x) \, dx+\frac{1}{2} \left (9 d^4\right ) \int \cos ^2(x) \, dx\\ &=-d (c+d x)^3+\frac{(c+d x)^5}{5 d}-\frac{9}{2} d^3 (c+d x) \cos ^2(x)+3 d (c+d x)^3 \cos ^2(x)+3 d^4 \cos (x) \sin (x)-6 d^2 (c+d x)^2 \cos (x) \sin (x)+2 (c+d x)^4 \cos (x) \sin (x)+\frac{3}{2} d^3 (c+d x) \sin ^2(x)-d (c+d x)^3 \sin ^2(x)-\frac{1}{4} \left (3 d^4\right ) \int 1 \, dx+\frac{1}{4} \left (9 d^4\right ) \int 1 \, dx\\ &=\frac{3 d^4 x}{2}-d (c+d x)^3+\frac{(c+d x)^5}{5 d}-\frac{9}{2} d^3 (c+d x) \cos ^2(x)+3 d (c+d x)^3 \cos ^2(x)+3 d^4 \cos (x) \sin (x)-6 d^2 (c+d x)^2 \cos (x) \sin (x)+2 (c+d x)^4 \cos (x) \sin (x)+\frac{3}{2} d^3 (c+d x) \sin ^2(x)-d (c+d x)^3 \sin ^2(x)\\ \end{align*}
Mathematica [A] time = 0.218675, size = 154, normalized size = 1.18 \[ 2 c^2 d^2 x^3+\frac{1}{2} \sin (2 x) \left (6 c^2 d^2 \left (2 x^2-1\right )+8 c^3 d x+2 c^4+4 c d^3 x \left (2 x^2-3\right )+d^4 \left (2 x^4-6 x^2+3\right )\right )+d \cos (2 x) \left (6 c^2 d x+2 c^3+3 c d^2 \left (2 x^2-1\right )+d^3 x \left (2 x^2-3\right )\right )+2 c^3 d x^2+c^4 x+c d^3 x^4+\frac{d^4 x^5}{5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.066, size = 260, normalized size = 2. \begin{align*} 4\,{d}^{4} \left ({x}^{4} \left ( 1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) +{x}^{3} \left ( \cos \left ( x \right ) \right ) ^{2}-3\,{x}^{2} \left ( 1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) -3/2\,x \left ( \cos \left ( x \right ) \right ) ^{2}+3/4\,\cos \left ( x \right ) \sin \left ( x \right ) +3/4\,x+{x}^{3}-2/5\,{x}^{5} \right ) +16\,c{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 ) +24\,{c}^{2}{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 ) -{\frac{{d}^{4}{x}^{5}}{5}}+16\,{c}^{3}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 ) -c{d}^{3}{x}^{4}+4\,{c}^{4} \left ( 1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) -2\,{c}^{2}{d}^{2}{x}^{3}-2\,{c}^{3}d{x}^{2}-{c}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04852, size = 197, normalized size = 1.5 \begin{align*} 2 \,{\left (x^{2} + 2 \, x \sin \left (2 \, x\right ) + \cos \left (2 \, x\right )\right )} c^{3} d +{\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^{2} d^{2} +{\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 )} c d^{3} + \frac{1}{10} \,{\left (2 \, x^{5} + 10 \,{\left (2 \, x^{3} - 3 \, x\right )} \cos \left (2 \, x\right ) + 5 \,{\left (2 \, x^{4} - 6 \, x^{2} + 3\right )} \sin \left (2 \, x\right )\right )} d^{4} + c^{4}{\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 = 0.515347, size = 419, normalized size = 3.2 \begin{align*} \frac{1}{5} \, d^{4} x^{5} + c d^{3} x^{4} + 2 \,{\left (c^{2} d^{2} - d^{4}\right )} x^{3} + 2 \,{\left (c^{3} d - 3 \, c d^{3}\right )} x^{2} + 2 \,{\left (2 \, d^{4} x^{3} + 6 \, c d^{3} x^{2} + 2 \, c^{3} d - 3 \, c d^{3} + 3 \,{\left (2 \, c^{2} d^{2} - d^{4}\right )} x\right )} \cos \left (x\right )^{2} +{\left (2 \, d^{4} x^{4} + 8 \, c d^{3} x^{3} + 2 \, c^{4} - 6 \, c^{2} d^{2} + 3 \, d^{4} + 6 \,{\left (2 \, c^{2} d^{2} - d^{4}\right )} x^{2} + 4 \,{\left (2 \, c^{3} d - 3 \, c d^{3}\right )} x\right )} \cos \left (x\right ) \sin \left (x\right ) +{\left (c^{4} - 6 \, c^{2} d^{2} + 3 \, d^{4}\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13595, size = 225, normalized size = 1.72 \begin{align*} \frac{1}{5} \, d^{4} x^{5} + c d^{3} x^{4} + 2 \, c^{2} d^{2} x^{3} + 2 \, c^{3} d x^{2} + c^{4} x +{\left (2 \, d^{4} x^{3} + 6 \, c d^{3} x^{2} + 6 \, c^{2} d^{2} x - 3 \, d^{4} x + 2 \, c^{3} d - 3 \, c d^{3}\right )} \cos \left (2 \, x\right ) + \frac{1}{2} \,{\left (2 \, d^{4} x^{4} + 8 \, c d^{3} x^{3} + 12 \, c^{2} d^{2} x^{2} - 6 \, d^{4} x^{2} + 8 \, c^{3} d x - 12 \, c d^{3} x + 2 \, c^{4} - 6 \, c^{2} d^{2} + 3 \, d^{4}\right )} \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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