Optimal. Leaf size=183 \[ \frac {2 i d^2 e^{i \left (a-\frac {b c}{d}\right )} \left (-\frac {i b (c+d x)}{d}\right )^{2/3} \Gamma \left (\frac {1}{3},-\frac {i b (c+d x)}{d}\right )}{9 b^3 (c+d x)^{2/3}}-\frac {2 i d^2 e^{-i \left (a-\frac {b c}{d}\right )} \left (\frac {i b (c+d x)}{d}\right )^{2/3} \Gamma \left (\frac {1}{3},\frac {i b (c+d x)}{d}\right )}{9 b^3 (c+d x)^{2/3}}+\frac {4 d \sqrt [3]{c+d x} \cos (a+b x)}{3 b^2}+\frac {(c+d x)^{4/3} \sin (a+b x)}{b} \]
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Rubi [A] time = 0.24, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3296, 3307, 2181} \[ \frac {2 i d^2 e^{i \left (a-\frac {b c}{d}\right )} \left (-\frac {i b (c+d x)}{d}\right )^{2/3} \text {Gamma}\left (\frac {1}{3},-\frac {i b (c+d x)}{d}\right )}{9 b^3 (c+d x)^{2/3}}-\frac {2 i d^2 e^{-i \left (a-\frac {b c}{d}\right )} \left (\frac {i b (c+d x)}{d}\right )^{2/3} \text {Gamma}\left (\frac {1}{3},\frac {i b (c+d x)}{d}\right )}{9 b^3 (c+d x)^{2/3}}+\frac {4 d \sqrt [3]{c+d x} \cos (a+b x)}{3 b^2}+\frac {(c+d x)^{4/3} \sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3296
Rule 3307
Rubi steps
\begin {align*} \int (c+d x)^{4/3} \cos (a+b x) \, dx &=\frac {(c+d x)^{4/3} \sin (a+b x)}{b}-\frac {(4 d) \int \sqrt [3]{c+d x} \sin (a+b x) \, dx}{3 b}\\ &=\frac {4 d \sqrt [3]{c+d x} \cos (a+b x)}{3 b^2}+\frac {(c+d x)^{4/3} \sin (a+b x)}{b}-\frac {\left (4 d^2\right ) \int \frac {\cos (a+b x)}{(c+d x)^{2/3}} \, dx}{9 b^2}\\ &=\frac {4 d \sqrt [3]{c+d x} \cos (a+b x)}{3 b^2}+\frac {(c+d x)^{4/3} \sin (a+b x)}{b}-\frac {\left (2 d^2\right ) \int \frac {e^{-i (a+b x)}}{(c+d x)^{2/3}} \, dx}{9 b^2}-\frac {\left (2 d^2\right ) \int \frac {e^{i (a+b x)}}{(c+d x)^{2/3}} \, dx}{9 b^2}\\ &=\frac {4 d \sqrt [3]{c+d x} \cos (a+b x)}{3 b^2}+\frac {2 i d^2 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 )}{9 b^3 (c+d x)^{2/3}}-\frac {2 i d^2 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 )}{9 b^3 (c+d x)^{2/3}}+\frac {(c+d x)^{4/3} \sin (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 122, normalized size = 0.67 \[ \frac {d \sqrt [3]{c+d x} e^{-\frac {i (a d+b c)}{d}} \left (\frac {e^{2 i a} \Gamma \left (\frac {7}{3},-\frac {i b (c+d x)}{d}\right )}{\sqrt [3]{-\frac {i b (c+d x)}{d}}}+\frac {e^{\frac {2 i b c}{d}} \Gamma \left (\frac {7}{3},\frac {i b (c+d x)}{d}\right )}{\sqrt [3]{\frac {i b (c+d x)}{d}}}\right )}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 132, normalized size = 0.72 \[ \frac {-2 i \, d^{2} \left (\frac {i \, b}{d}\right )^{\frac {2}{3}} e^{\left (\frac {i \, b c - i \, a d}{d}\right )} \Gamma \left (\frac {1}{3}, \frac {i \, b d x + i \, b c}{d}\right ) + 2 i \, d^{2} \left (-\frac {i \, b}{d}\right )^{\frac {2}{3}} e^{\left (\frac {-i \, b c + i \, a d}{d}\right )} \Gamma \left (\frac {1}{3}, \frac {-i \, b d x - i \, b c}{d}\right ) + 3 \, {\left (4 \, b d \cos \left (b x + a\right ) + 3 \, {\left (b^{2} d x + b^{2} c\right )} \sin \left (b x + a\right )\right )} {\left (d x + c\right )}^{\frac {1}{3}}}{9 \, b^{3}} \]
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 \[ \int {\left (d x + c\right )}^{\frac {4}{3}} \cos \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{\frac {4}{3}} \cos \left (b x +a \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 235, normalized size = 1.28 \[ \frac {9 \, {\left (d x + c\right )}^{\frac {4}{3}} b \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {1}{3}} d \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) + 12 \, {\left (d x + c\right )}^{\frac {1}{3}} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {1}{3}} d^{2} \cos \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) + {\left ({\left ({\left (\sqrt {3} - i\right )} \Gamma \left (\frac {1}{3}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} d^{2} \cos \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + {\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} d^{2} \sin \left (-\frac {b c - a d}{d}\right )\right )} {\left (d x + c\right )}^{\frac {1}{3}}}{9 \, b^{2} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {1}{3}} d} \]
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 \[ \int \cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^{4/3} \,d x \]
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 \[ \int \left (c + d x\right )^{\frac {4}{3}} \cos {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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