Integrand size = 14, antiderivative size = 85 \[ \int \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {6 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {6 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {3 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]
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Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3443, 3377, 2717} \[ \int \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {6 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {6 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {3 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]
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Rule 2717
Rule 3377
Rule 3443
Rubi steps \begin{align*} \text {integral}& = \frac {3 \text {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 \text {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 \text {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*}
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {6 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )+3 \left (-2+b^2 (c+d x)^{2/3}\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d} \]
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Time = 0.91 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.54
method | result | size |
derivativedivides | \(\frac {3 a^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 a \left (\cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )+3 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{b^{3} d}\) | \(131\) |
default | \(\frac {3 a^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 a \left (\cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )+3 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{b^{3} d}\) | \(131\) |
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Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.67 \[ \int \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \, {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} b \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{3} d} \]
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Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.11 \[ \int \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\begin {cases} x \cos {\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\x \cos {\left (a + b \sqrt [3]{c} \right )} & \text {for}\: d = 0 \\\frac {3 \left (c + d x\right )^{\frac {2}{3}} \sin {\left (a + b \sqrt [3]{c + d x} \right )}}{b d} + \frac {6 \sqrt [3]{c + d x} \cos {\left (a + b \sqrt [3]{c + d x} \right )}}{b^{2} d} - \frac {6 \sin {\left (a + b \sqrt [3]{c + d x} \right )}}{b^{3} d} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39 \[ \int \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \, {\left (a^{2} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 2 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a + 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{3} d} \]
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Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \, {\left (\frac {2 \, {\left (d x + c\right )}^{\frac {1}{3}} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b} + \frac {{\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a + a^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{2}}\right )}}{b d} \]
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Time = 14.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.80 \[ \int \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {6\,b\,\cos \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c+d\,x\right )}^{1/3}-6\,\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )+3\,b^2\,\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c+d\,x\right )}^{2/3}}{b^3\,d} \]
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