Optimal. Leaf size=330 \[ -\frac {b f^2 \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^3}\right )}{3 d^3}-\frac {i e^{i a} f (c+d x)^2 \left (-\frac {i b}{(c+d x)^3}\right )^{2/3} (d e-c f) \Gamma \left (-\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )}{3 d^3}+\frac {i e^{-i a} f (c+d x)^2 \left (\frac {i b}{(c+d x)^3}\right )^{2/3} (d e-c f) \Gamma \left (-\frac {2}{3},\frac {i b}{(c+d x)^3}\right )}{3 d^3}-\frac {i e^{i a} (c+d x) \sqrt [3]{-\frac {i b}{(c+d x)^3}} (d e-c f)^2 \Gamma \left (-\frac {1}{3},-\frac {i b}{(c+d x)^3}\right )}{6 d^3}+\frac {i e^{-i a} (c+d x) \sqrt [3]{\frac {i b}{(c+d x)^3}} (d e-c f)^2 \Gamma \left (-\frac {1}{3},\frac {i b}{(c+d x)^3}\right )}{6 d^3}+\frac {b f^2 \sin (a) \text {Si}\left (\frac {b}{(c+d x)^3}\right )}{3 d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^3}\right )}{3 d^3} \]
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Rubi [A] time = 0.30, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3433, 3365, 2208, 3423, 2218, 3379, 3297, 3303, 3299, 3302} \[ -\frac {i e^{i a} f (c+d x)^2 \left (-\frac {i b}{(c+d x)^3}\right )^{2/3} (d e-c f) \text {Gamma}\left (-\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )}{3 d^3}+\frac {i e^{-i a} f (c+d x)^2 \left (\frac {i b}{(c+d x)^3}\right )^{2/3} (d e-c f) \text {Gamma}\left (-\frac {2}{3},\frac {i b}{(c+d x)^3}\right )}{3 d^3}-\frac {i e^{i a} (c+d x) \sqrt [3]{-\frac {i b}{(c+d x)^3}} (d e-c f)^2 \text {Gamma}\left (-\frac {1}{3},-\frac {i b}{(c+d x)^3}\right )}{6 d^3}+\frac {i e^{-i a} (c+d x) \sqrt [3]{\frac {i b}{(c+d x)^3}} (d e-c f)^2 \text {Gamma}\left (-\frac {1}{3},\frac {i b}{(c+d x)^3}\right )}{6 d^3}-\frac {b f^2 \cos (a) \text {CosIntegral}\left (\frac {b}{(c+d x)^3}\right )}{3 d^3}+\frac {b f^2 \sin (a) \text {Si}\left (\frac {b}{(c+d x)^3}\right )}{3 d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^3}\right )}{3 d^3} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 2218
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3365
Rule 3379
Rule 3423
Rule 3433
Rubi steps
\begin {align*} \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^3}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \left (d^2 e^2 \left (1+\frac {c f (-2 d e+c f)}{d^2 e^2}\right ) \sin \left (a+\frac {b}{x^3}\right )+2 d e f \left (1-\frac {c f}{d e}\right ) x \sin \left (a+\frac {b}{x^3}\right )+f^2 x^2 \sin \left (a+\frac {b}{x^3}\right )\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac {f^2 \operatorname {Subst}\left (\int x^2 \sin \left (a+\frac {b}{x^3}\right ) \, dx,x,c+d x\right )}{d^3}+\frac {(2 f (d e-c f)) \operatorname {Subst}\left (\int x \sin \left (a+\frac {b}{x^3}\right ) \, dx,x,c+d x\right )}{d^3}+\frac {(d e-c f)^2 \operatorname {Subst}\left (\int \sin \left (a+\frac {b}{x^3}\right ) \, dx,x,c+d x\right )}{d^3}\\ &=-\frac {f^2 \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,\frac {1}{(c+d x)^3}\right )}{3 d^3}+\frac {(i f (d e-c f)) \operatorname {Subst}\left (\int e^{-i a-\frac {i b}{x^3}} x \, dx,x,c+d x\right )}{d^3}-\frac {(i f (d e-c f)) \operatorname {Subst}\left (\int e^{i a+\frac {i b}{x^3}} x \, dx,x,c+d x\right )}{d^3}+\frac {\left (i (d e-c f)^2\right ) \operatorname {Subst}\left (\int e^{-i a-\frac {i b}{x^3}} \, dx,x,c+d x\right )}{2 d^3}-\frac {\left (i (d e-c f)^2\right ) \operatorname {Subst}\left (\int e^{i a+\frac {i b}{x^3}} \, dx,x,c+d x\right )}{2 d^3}\\ &=-\frac {i e^{i a} f (d e-c f) \left (-\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )}{3 d^3}+\frac {i e^{-i a} f (d e-c f) \left (\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},\frac {i b}{(c+d x)^3}\right )}{3 d^3}-\frac {i e^{i a} (d e-c f)^2 \sqrt [3]{-\frac {i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {i b}{(c+d x)^3}\right )}{6 d^3}+\frac {i e^{-i a} (d e-c f)^2 \sqrt [3]{\frac {i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},\frac {i b}{(c+d x)^3}\right )}{6 d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^3}\right )}{3 d^3}-\frac {\left (b f^2\right ) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,\frac {1}{(c+d x)^3}\right )}{3 d^3}\\ &=-\frac {i e^{i a} f (d e-c f) \left (-\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )}{3 d^3}+\frac {i e^{-i a} f (d e-c f) \left (\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},\frac {i b}{(c+d x)^3}\right )}{3 d^3}-\frac {i e^{i a} (d e-c f)^2 \sqrt [3]{-\frac {i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {i b}{(c+d x)^3}\right )}{6 d^3}+\frac {i e^{-i a} (d e-c f)^2 \sqrt [3]{\frac {i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},\frac {i b}{(c+d x)^3}\right )}{6 d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^3}\right )}{3 d^3}-\frac {\left (b f^2 \cos (a)\right ) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{(c+d x)^3}\right )}{3 d^3}+\frac {\left (b f^2 \sin (a)\right ) \operatorname {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{(c+d x)^3}\right )}{3 d^3}\\ &=-\frac {b f^2 \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^3}\right )}{3 d^3}-\frac {i e^{i a} f (d e-c f) \left (-\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )}{3 d^3}+\frac {i e^{-i a} f (d e-c f) \left (\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},\frac {i b}{(c+d x)^3}\right )}{3 d^3}-\frac {i e^{i a} (d e-c f)^2 \sqrt [3]{-\frac {i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {i b}{(c+d x)^3}\right )}{6 d^3}+\frac {i e^{-i a} (d e-c f)^2 \sqrt [3]{\frac {i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},\frac {i b}{(c+d x)^3}\right )}{6 d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^3}\right )}{3 d^3}+\frac {b f^2 \sin (a) \text {Si}\left (\frac {b}{(c+d x)^3}\right )}{3 d^3}\\ \end {align*}
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Mathematica [A] time = 2.65, size = 405, normalized size = 1.23 \[ \frac {\frac {3 b f (d e-c f) \left ((\cos (a)-i \sin (a)) \sqrt [3]{-\frac {i b}{(c+d x)^3}} \Gamma \left (\frac {1}{3},\frac {i b}{(c+d x)^3}\right )+(\cos (a)+i \sin (a)) \sqrt [3]{\frac {i b}{(c+d x)^3}} \Gamma \left (\frac {1}{3},-\frac {i b}{(c+d x)^3}\right )\right )}{2 (c+d x) \sqrt [3]{\frac {b^2}{(c+d x)^6}}}+\frac {3 b (d e-c f)^2 \left ((\cos (a)-i \sin (a)) \left (-\frac {i b}{(c+d x)^3}\right )^{2/3} \Gamma \left (\frac {2}{3},\frac {i b}{(c+d x)^3}\right )+(\cos (a)+i \sin (a)) \left (\frac {i b}{(c+d x)^3}\right )^{2/3} \Gamma \left (\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )\right )}{2 (c+d x)^2 \left (\frac {b^2}{(c+d x)^6}\right )^{2/3}}+\sin (a) (c+d x) \cos \left (\frac {b}{(c+d x)^3}\right ) \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )+\cos (a) (c+d x) \sin \left (\frac {b}{(c+d x)^3}\right ) \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )-b f^2 \left (\cos (a) \text {Ci}\left (\frac {b}{(c+d x)^3}\right )-\sin (a) \text {Si}\left (\frac {b}{(c+d x)^3}\right )\right )}{3 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.69, size = 488, normalized size = 1.48 \[ -\frac {b f^{2} {\rm Ei}\left (\frac {i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) e^{\left (i \, a\right )} + b f^{2} {\rm Ei}\left (-\frac {i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) e^{\left (-i \, a\right )} - {\left (-3 i \, d^{3} e f + 3 i \, c d^{2} f^{2}\right )} \left (\frac {i \, b}{d^{3}}\right )^{\frac {2}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {1}{3}, \frac {i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (3 i \, d^{3} e f - 3 i \, c d^{2} f^{2}\right )} \left (-\frac {i \, b}{d^{3}}\right )^{\frac {2}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (-3 i \, d^{3} e^{2} + 6 i \, c d^{2} e f - 3 i \, c^{2} d f^{2}\right )} \left (\frac {i \, b}{d^{3}}\right )^{\frac {1}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {2}{3}, \frac {i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (3 i \, d^{3} e^{2} - 6 i \, c d^{2} e f + 3 i \, c^{2} d f^{2}\right )} \left (-\frac {i \, b}{d^{3}}\right )^{\frac {1}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {2}{3}, -\frac {i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 2 \, {\left (d^{3} f^{2} x^{3} + 3 \, d^{3} e f x^{2} + 3 \, d^{3} e^{2} x + 3 \, c d^{2} e^{2} - 3 \, c^{2} d e f + c^{3} f^{2}\right )} \sin \left (\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{6 \, d^{3}} \]
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 \[ \int {\left (f x + e\right )}^{2} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{3}}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{3}}\right )\, dx \]
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 \[ \frac {1}{3} \, {\left (f^{2} x^{3} + 3 \, e f x^{2} + 3 \, e^{2} x\right )} \sin \left (\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + \int \frac {{\left (b d f^{2} x^{3} + 3 \, b d e f x^{2} + 3 \, b d e^{2} x\right )} \cos \left (\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{2 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} + \int \frac {{\left (b d f^{2} x^{3} + 3 \, b d e f x^{2} + 3 \, b d e^{2} x\right )} \cos \left (\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{2 \, {\left ({\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )} \cos \left (\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )^{2} + {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )} \sin \left (\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )^{2}\right )}}\,{d x} \]
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 \[ \int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^3}\right )\,{\left (e+f\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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