Optimal. Leaf size=153 \[ -\frac {3 \cos (a+b x)}{2 d (c+d x)^{2/3}}+\frac {3 e^{i \left (a-\frac {b c}{d}\right )} \left (-\frac {i b (c+d x)}{d}\right )^{2/3} \text {Gamma}\left (\frac {1}{3},-\frac {i b (c+d x)}{d}\right )}{4 d (c+d x)^{2/3}}+\frac {3 e^{-i \left (a-\frac {b c}{d}\right )} \left (\frac {i b (c+d x)}{d}\right )^{2/3} \text {Gamma}\left (\frac {1}{3},\frac {i b (c+d x)}{d}\right )}{4 d (c+d x)^{2/3}} \]
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Rubi [A]
time = 0.10, antiderivative size = 153, normalized size of antiderivative = 1.00, 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 = {3378, 3389,
2212} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3378
Rule 3389
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*}
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Mathematica [A]
time = 0.04, size = 121, normalized size = 0.79 \begin {gather*} -\frac {e^{-\frac {i (b c+a d)}{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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\cos \left (b x +a \right )}{\left (d x +c \right )^{\frac {5}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 138, normalized size = 0.90 \begin {gather*} \frac {{\left ({\left ({\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {2}{3}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + {\left (i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {2}{3}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) - {\left ({\left (\sqrt {3} - i\right )} \Gamma \left (-\frac {2}{3}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (-\frac {2}{3}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {2}{3}}}{4 \, {\left (d x + c\right )}^{\frac {2}{3}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.13, size = 117, normalized size = 0.76 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (a + b x \right )}}{\left (c + d x\right )^{\frac {5}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\cos \left (a+b\,x\right )}{{\left (c+d\,x\right )}^{5/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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