Optimal. Leaf size=235 \[ \frac {i e^{i a} (c+d x) (d e-c f) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{6 d^2 \sqrt [3]{-i b (c+d x)^3}}-\frac {i e^{-i a} (c+d x) (d e-c f) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{6 d^2 \sqrt [3]{i b (c+d x)^3}}+\frac {i e^{i a} f (c+d x)^2 \Gamma \left (\frac {2}{3},-i b (c+d x)^3\right )}{6 d^2 \left (-i b (c+d x)^3\right )^{2/3}}-\frac {i e^{-i a} f (c+d x)^2 \Gamma \left (\frac {2}{3},i b (c+d x)^3\right )}{6 d^2 \left (i b (c+d x)^3\right )^{2/3}} \]
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Rubi [A] time = 0.19, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3433, 3355, 2208, 3389, 2218} \[ \frac {i e^{i a} (c+d x) (d e-c f) \text {Gamma}\left (\frac {1}{3},-i b (c+d x)^3\right )}{6 d^2 \sqrt [3]{-i b (c+d x)^3}}-\frac {i e^{-i a} (c+d x) (d e-c f) \text {Gamma}\left (\frac {1}{3},i b (c+d x)^3\right )}{6 d^2 \sqrt [3]{i b (c+d x)^3}}+\frac {i e^{i a} f (c+d x)^2 \text {Gamma}\left (\frac {2}{3},-i b (c+d x)^3\right )}{6 d^2 \left (-i b (c+d x)^3\right )^{2/3}}-\frac {i e^{-i a} f (c+d x)^2 \text {Gamma}\left (\frac {2}{3},i b (c+d x)^3\right )}{6 d^2 \left (i b (c+d x)^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 2218
Rule 3355
Rule 3389
Rule 3433
Rubi steps
\begin {align*} \int (e+f x) \sin \left (a+b (c+d x)^3\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \left (d e \left (1-\frac {c f}{d e}\right ) \sin \left (a+b x^3\right )+f x \sin \left (a+b x^3\right )\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac {f \operatorname {Subst}\left (\int x \sin \left (a+b x^3\right ) \, dx,x,c+d x\right )}{d^2}+\frac {(d e-c f) \operatorname {Subst}\left (\int \sin \left (a+b x^3\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac {(i f) \operatorname {Subst}\left (\int e^{-i a-i b x^3} x \, dx,x,c+d x\right )}{2 d^2}-\frac {(i f) \operatorname {Subst}\left (\int e^{i a+i b x^3} x \, dx,x,c+d x\right )}{2 d^2}+\frac {(i (d e-c f)) \operatorname {Subst}\left (\int e^{-i a-i b x^3} \, dx,x,c+d x\right )}{2 d^2}-\frac {(i (d e-c f)) \operatorname {Subst}\left (\int e^{i a+i b x^3} \, dx,x,c+d x\right )}{2 d^2}\\ &=\frac {i e^{i a} (d e-c f) (c+d x) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{6 d^2 \sqrt [3]{-i b (c+d x)^3}}-\frac {i e^{-i a} (d e-c f) (c+d x) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{6 d^2 \sqrt [3]{i b (c+d x)^3}}+\frac {i e^{i a} f (c+d x)^2 \Gamma \left (\frac {2}{3},-i b (c+d x)^3\right )}{6 d^2 \left (-i b (c+d x)^3\right )^{2/3}}-\frac {i e^{-i a} f (c+d x)^2 \Gamma \left (\frac {2}{3},i b (c+d x)^3\right )}{6 d^2 \left (i b (c+d x)^3\right )^{2/3}}\\ \end {align*}
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Mathematica [F] time = 46.01, size = 0, normalized size = 0.00 \[ \int (e+f x) \sin \left (a+b (c+d x)^3\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.68, size = 225, normalized size = 0.96 \[ -\frac {\left (i \, b d^{3}\right )^{\frac {1}{3}} d f e^{\left (-i \, a\right )} \Gamma \left (\frac {2}{3}, i \, b d^{3} x^{3} + 3 i \, b c d^{2} x^{2} + 3 i \, b c^{2} d x + i \, b c^{3}\right ) + \left (-i \, b d^{3}\right )^{\frac {1}{3}} d f e^{\left (i \, a\right )} \Gamma \left (\frac {2}{3}, -i \, b d^{3} x^{3} - 3 i \, b c d^{2} x^{2} - 3 i \, b c^{2} d x - i \, b c^{3}\right ) + \left (i \, b d^{3}\right )^{\frac {2}{3}} {\left (d e - c f\right )} e^{\left (-i \, a\right )} \Gamma \left (\frac {1}{3}, i \, b d^{3} x^{3} + 3 i \, b c d^{2} x^{2} + 3 i \, b c^{2} d x + i \, b c^{3}\right ) + \left (-i \, b d^{3}\right )^{\frac {2}{3}} {\left (d e - c f\right )} e^{\left (i \, a\right )} \Gamma \left (\frac {1}{3}, -i \, b d^{3} x^{3} - 3 i \, b c d^{2} x^{2} - 3 i \, b c^{2} d x - i \, b c^{3}\right )}{6 \, b d^{4}} \]
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 \[ \int {\left (f x + e\right )} \sin \left ({\left (d x + c\right )}^{3} b + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right ) \sin \left (a +\left (d x +c \right )^{3} b \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (f x + e\right )} \sin \left ({\left (d x + c\right )}^{3} b + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sin \left (a+b\,{\left (c+d\,x\right )}^3\right )\,\left (e+f\,x\right ) \,d x \]
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 \[ \int \left (e + f x\right ) \sin {\left (a + b c^{3} + 3 b c^{2} d x + 3 b c d^{2} x^{2} + b d^{3} x^{3} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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