Optimal. Leaf size=153 \[ \frac{3 e^{i \left (a-\frac{b c}{d}\right )} \left (-\frac{i b (c+d x)}{d}\right )^{2/3} \text{Gamma}\left (\frac{1}{3},-\frac{i b (c+d x)}{d}\right )}{4 d (c+d x)^{2/3}}+\frac{3 e^{-i \left (a-\frac{b c}{d}\right )} \left (\frac{i b (c+d x)}{d}\right )^{2/3} \text{Gamma}\left (\frac{1}{3},\frac{i b (c+d x)}{d}\right )}{4 d (c+d x)^{2/3}}-\frac{3 \cos (a+b x)}{2 d (c+d x)^{2/3}} \]
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Rubi [A] time = 0.154681, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3297, 3308, 2181} \[ \frac{3 e^{i \left (a-\frac{b c}{d}\right )} \left (-\frac{i b (c+d x)}{d}\right )^{2/3} \text{Gamma}\left (\frac{1}{3},-\frac{i b (c+d x)}{d}\right )}{4 d (c+d x)^{2/3}}+\frac{3 e^{-i \left (a-\frac{b c}{d}\right )} \left (\frac{i b (c+d x)}{d}\right )^{2/3} \text{Gamma}\left (\frac{1}{3},\frac{i b (c+d x)}{d}\right )}{4 d (c+d x)^{2/3}}-\frac{3 \cos (a+b x)}{2 d (c+d x)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int \frac{\cos (a+b x)}{(c+d x)^{5/3}} \, dx &=-\frac{3 \cos (a+b x)}{2 d (c+d x)^{2/3}}-\frac{(3 b) \int \frac{\sin (a+b x)}{(c+d x)^{2/3}} \, dx}{2 d}\\ &=-\frac{3 \cos (a+b x)}{2 d (c+d x)^{2/3}}-\frac{(3 i b) \int \frac{e^{-i (a+b x)}}{(c+d x)^{2/3}} \, dx}{4 d}+\frac{(3 i b) \int \frac{e^{i (a+b x)}}{(c+d x)^{2/3}} \, dx}{4 d}\\ &=-\frac{3 \cos (a+b x)}{2 d (c+d x)^{2/3}}+\frac{3 e^{i \left (a-\frac{b c}{d}\right )} \left (-\frac{i b (c+d x)}{d}\right )^{2/3} \Gamma \left (\frac{1}{3},-\frac{i b (c+d x)}{d}\right )}{4 d (c+d x)^{2/3}}+\frac{3 e^{-i \left (a-\frac{b c}{d}\right )} \left (\frac{i b (c+d x)}{d}\right )^{2/3} \Gamma \left (\frac{1}{3},\frac{i b (c+d x)}{d}\right )}{4 d (c+d x)^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0569638, size = 121, normalized size = 0.79 \[ -\frac{e^{-\frac{i (a d+b c)}{d}} \left (e^{2 i a} \left (-\frac{i b (c+d x)}{d}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{i b (c+d x)}{d}\right )+e^{\frac{2 i b c}{d}} \left (\frac{i b (c+d x)}{d}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},\frac{i b (c+d x)}{d}\right )\right )}{2 d (c+d x)^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.183, size = 0, normalized size = 0. \begin{align*} \int{\cos \left ( bx+a \right ) \left ( dx+c \right ) ^{-{\frac{5}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.55815, size = 632, normalized size = 4.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73329, size = 290, normalized size = 1.9 \begin{align*} \frac{3 \,{\left ({\left (d x + c\right )} \left (\frac{i \, b}{d}\right )^{\frac{2}{3}} e^{\left (\frac{i \, b c - i \, a d}{d}\right )} \Gamma \left (\frac{1}{3}, \frac{i \, b d x + i \, b c}{d}\right ) +{\left (d x + c\right )} \left (-\frac{i \, b}{d}\right )^{\frac{2}{3}} e^{\left (\frac{-i \, b c + i \, a d}{d}\right )} \Gamma \left (\frac{1}{3}, \frac{-i \, b d x - i \, b c}{d}\right ) - 2 \,{\left (d x + c\right )}^{\frac{1}{3}} \cos \left (b x + a\right )\right )}}{4 \,{\left (d^{2} x + c d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (a + b x \right )}}{\left (c + d x\right )^{\frac{5}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x + a\right )}{{\left (d x + c\right )}^{\frac{5}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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