Optimal. Leaf size=85 \[ -\frac{6 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac{6 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac{3 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]
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Rubi [A] time = 0.0551264, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3362, 3296, 2637} \[ -\frac{6 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac{6 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac{3 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 3362
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac{3 \operatorname{Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac{3 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac{6 \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d}\\ &=\frac{6 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac{3 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac{6 \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d}\\ &=\frac{6 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac{6 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac{3 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d}\\ \end{align*}
Mathematica [A] time = 0.111096, size = 65, normalized size = 0.76 \[ \frac{3 \left (b^2 (c+d x)^{2/3}-2\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )+6 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 131, normalized size = 1.5 \begin{align*} 3\,{\frac{ \left ( a+b\sqrt [3]{dx+c} \right ) ^{2}\sin \left ( a+b\sqrt [3]{dx+c} \right ) -2\,\sin \left ( a+b\sqrt [3]{dx+c} \right ) +2\, \left ( a+b\sqrt [3]{dx+c} \right ) \cos \left ( a+b\sqrt [3]{dx+c} \right ) -2\,a \left ( \cos \left ( a+b\sqrt [3]{dx+c} \right ) + \left ( a+b\sqrt [3]{dx+c} \right ) \sin \left ( a+b\sqrt [3]{dx+c} \right ) \right ) +{a}^{2}\sin \left ( a+b\sqrt [3]{dx+c} \right ) }{d{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11794, size = 159, normalized size = 1.87 \begin{align*} \frac{3 \,{\left (a^{2} \sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right ) - 2 \,{\left ({\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )} \sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right ) + \cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )\right )} a + 2 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )} \cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right ) +{\left ({\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )\right )}}{b^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62542, size = 155, normalized size = 1.82 \begin{align*} \frac{3 \,{\left (2 \,{\left (d x + c\right )}^{\frac{1}{3}} b \cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right ) +{\left ({\left (d x + c\right )}^{\frac{2}{3}} b^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )\right )}}{b^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.67818, size = 94, normalized size = 1.11 \begin{align*} \begin{cases} x \cos{\left (a \right )} & \text{for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\x \cos{\left (a + b \sqrt [3]{c} \right )} & \text{for}\: d = 0 \\\frac{3 \left (c + d x\right )^{\frac{2}{3}} \sin{\left (a + b \sqrt [3]{c + d x} \right )}}{b d} + \frac{6 \sqrt [3]{c + d x} \cos{\left (a + b \sqrt [3]{c + d x} \right )}}{b^{2} d} - \frac{6 \sin{\left (a + b \sqrt [3]{c + d x} \right )}}{b^{3} d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26277, size = 109, normalized size = 1.28 \begin{align*} \frac{3 \,{\left (\frac{2 \,{\left (d x + c\right )}^{\frac{1}{3}} \cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{b} + \frac{{\left ({\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}^{2} - 2 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )} a + a^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{b^{2}}\right )}}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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