Optimal. Leaf size=337 \[ \frac {2 b f^2 (d e-c f) (c+d x) \cos \left (\frac {b}{(c+d x)^2}\right )}{d^4}+\frac {b f^3 (c+d x)^2 \cos \left (\frac {b}{(c+d x)^2}\right )}{4 d^4}-\frac {3 b f (d e-c f)^2 \text {Ci}\left (\frac {b}{(c+d x)^2}\right )}{2 d^4}-\frac {\sqrt {b} (d e-c f)^3 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^4}+\frac {2 b^{3/2} f^2 (d e-c f) \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^4}+\frac {(d e-c f)^3 (c+d x) \sin \left (\frac {b}{(c+d x)^2}\right )}{d^4}+\frac {3 f (d e-c f)^2 (c+d x)^2 \sin \left (\frac {b}{(c+d x)^2}\right )}{2 d^4}+\frac {f^2 (d e-c f) (c+d x)^3 \sin \left (\frac {b}{(c+d x)^2}\right )}{d^4}+\frac {f^3 (c+d x)^4 \sin \left (\frac {b}{(c+d x)^2}\right )}{4 d^4}+\frac {b^2 f^3 \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{4 d^4} \]
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Rubi [A]
time = 0.29, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {3514, 3440,
3468, 3433, 3460, 3378, 3383, 3490, 3469, 3432, 3380} \begin {gather*} \frac {2 \sqrt {2 \pi } b^{3/2} f^2 (d e-c f) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^4}+\frac {b^2 f^3 \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{4 d^4}-\frac {3 b f (d e-c f)^2 \text {CosIntegral}\left (\frac {b}{(c+d x)^2}\right )}{2 d^4}+\frac {f^2 (c+d x)^3 (d e-c f) \sin \left (\frac {b}{(c+d x)^2}\right )}{d^4}+\frac {2 b f^2 (c+d x) (d e-c f) \cos \left (\frac {b}{(c+d x)^2}\right )}{d^4}-\frac {\sqrt {2 \pi } \sqrt {b} (d e-c f)^3 \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{c+d x}\right )}{d^4}+\frac {3 f (c+d x)^2 (d e-c f)^2 \sin \left (\frac {b}{(c+d x)^2}\right )}{2 d^4}+\frac {(c+d x) (d e-c f)^3 \sin \left (\frac {b}{(c+d x)^2}\right )}{d^4}+\frac {f^3 (c+d x)^4 \sin \left (\frac {b}{(c+d x)^2}\right )}{4 d^4}+\frac {b f^3 (c+d x)^2 \cos \left (\frac {b}{(c+d x)^2}\right )}{4 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3383
Rule 3432
Rule 3433
Rule 3440
Rule 3460
Rule 3468
Rule 3469
Rule 3490
Rule 3514
Rubi steps
\begin {align*} \int (e+f x)^3 \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx &=\frac {\text {Subst}\left (\int \left (d^3 e^3 \left (1-\frac {c f \left (3 d^2 e^2-3 c d e f+c^2 f^2\right )}{d^3 e^3}\right ) \sin \left (\frac {b}{x^2}\right )+3 d^2 e^2 f \left (1+\frac {c f (-2 d e+c f)}{d^2 e^2}\right ) x \sin \left (\frac {b}{x^2}\right )+3 d e f^2 \left (1-\frac {c f}{d e}\right ) x^2 \sin \left (\frac {b}{x^2}\right )+f^3 x^3 \sin \left (\frac {b}{x^2}\right )\right ) \, dx,x,c+d x\right )}{d^4}\\ &=\frac {f^3 \text {Subst}\left (\int x^3 \sin \left (\frac {b}{x^2}\right ) \, dx,x,c+d x\right )}{d^4}+\frac {\left (3 f^2 (d e-c f)\right ) \text {Subst}\left (\int x^2 \sin \left (\frac {b}{x^2}\right ) \, dx,x,c+d x\right )}{d^4}+\frac {\left (3 f (d e-c f)^2\right ) \text {Subst}\left (\int x \sin \left (\frac {b}{x^2}\right ) \, dx,x,c+d x\right )}{d^4}+\frac {(d e-c f)^3 \text {Subst}\left (\int \sin \left (\frac {b}{x^2}\right ) \, dx,x,c+d x\right )}{d^4}\\ &=-\frac {f^3 \text {Subst}\left (\int \frac {\sin (b x)}{x^3} \, dx,x,\frac {1}{(c+d x)^2}\right )}{2 d^4}-\frac {\left (3 f^2 (d e-c f)\right ) \text {Subst}\left (\int \frac {\sin \left (b x^2\right )}{x^4} \, dx,x,\frac {1}{c+d x}\right )}{d^4}-\frac {\left (3 f (d e-c f)^2\right ) \text {Subst}\left (\int \frac {\sin (b x)}{x^2} \, dx,x,\frac {1}{(c+d x)^2}\right )}{2 d^4}-\frac {(d e-c f)^3 \text {Subst}\left (\int \frac {\sin \left (b x^2\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d^4}\\ &=\frac {(d e-c f)^3 (c+d x) \sin \left (\frac {b}{(c+d x)^2}\right )}{d^4}+\frac {3 f (d e-c f)^2 (c+d x)^2 \sin \left (\frac {b}{(c+d x)^2}\right )}{2 d^4}+\frac {f^2 (d e-c f) (c+d x)^3 \sin \left (\frac {b}{(c+d x)^2}\right )}{d^4}+\frac {f^3 (c+d x)^4 \sin \left (\frac {b}{(c+d x)^2}\right )}{4 d^4}-\frac {\left (b f^3\right ) \text {Subst}\left (\int \frac {\cos (b x)}{x^2} \, dx,x,\frac {1}{(c+d x)^2}\right )}{4 d^4}-\frac {\left (2 b f^2 (d e-c f)\right ) \text {Subst}\left (\int \frac {\cos \left (b x^2\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d^4}-\frac {\left (3 b f (d e-c f)^2\right ) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{(c+d x)^2}\right )}{2 d^4}-\frac {\left (2 b (d e-c f)^3\right ) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{d^4}\\ &=\frac {2 b f^2 (d e-c f) (c+d x) \cos \left (\frac {b}{(c+d x)^2}\right )}{d^4}+\frac {b f^3 (c+d x)^2 \cos \left (\frac {b}{(c+d x)^2}\right )}{4 d^4}-\frac {3 b f (d e-c f)^2 \text {Ci}\left (\frac {b}{(c+d x)^2}\right )}{2 d^4}-\frac {\sqrt {b} (d e-c f)^3 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^4}+\frac {(d e-c f)^3 (c+d x) \sin \left (\frac {b}{(c+d x)^2}\right )}{d^4}+\frac {3 f (d e-c f)^2 (c+d x)^2 \sin \left (\frac {b}{(c+d x)^2}\right )}{2 d^4}+\frac {f^2 (d e-c f) (c+d x)^3 \sin \left (\frac {b}{(c+d x)^2}\right )}{d^4}+\frac {f^3 (c+d x)^4 \sin \left (\frac {b}{(c+d x)^2}\right )}{4 d^4}+\frac {\left (b^2 f^3\right ) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{(c+d x)^2}\right )}{4 d^4}+\frac {\left (4 b^2 f^2 (d e-c f)\right ) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{d^4}\\ &=\frac {2 b f^2 (d e-c f) (c+d x) \cos \left (\frac {b}{(c+d x)^2}\right )}{d^4}+\frac {b f^3 (c+d x)^2 \cos \left (\frac {b}{(c+d x)^2}\right )}{4 d^4}-\frac {3 b f (d e-c f)^2 \text {Ci}\left (\frac {b}{(c+d x)^2}\right )}{2 d^4}-\frac {\sqrt {b} (d e-c f)^3 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^4}+\frac {2 b^{3/2} f^2 (d e-c f) \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^4}+\frac {(d e-c f)^3 (c+d x) \sin \left (\frac {b}{(c+d x)^2}\right )}{d^4}+\frac {3 f (d e-c f)^2 (c+d x)^2 \sin \left (\frac {b}{(c+d x)^2}\right )}{2 d^4}+\frac {f^2 (d e-c f) (c+d x)^3 \sin \left (\frac {b}{(c+d x)^2}\right )}{d^4}+\frac {f^3 (c+d x)^4 \sin \left (\frac {b}{(c+d x)^2}\right )}{4 d^4}+\frac {b^2 f^3 \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{4 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 440, normalized size = 1.31 \begin {gather*} \frac {8 b c d e f^2 \cos \left (\frac {b}{(c+d x)^2}\right )-7 b c^2 f^3 \cos \left (\frac {b}{(c+d x)^2}\right )+8 b d^2 e f^2 x \cos \left (\frac {b}{(c+d x)^2}\right )-6 b c d f^3 x \cos \left (\frac {b}{(c+d x)^2}\right )+b d^2 f^3 x^2 \cos \left (\frac {b}{(c+d x)^2}\right )-6 b f (d e-c f)^2 \text {Ci}\left (\frac {b}{(c+d x)^2}\right )-4 \sqrt {b} (d e-c f)^3 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+8 b^{3/2} d e f^2 \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )-8 b^{3/2} c f^3 \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+4 c d^3 e^3 \sin \left (\frac {b}{(c+d x)^2}\right )-6 c^2 d^2 e^2 f \sin \left (\frac {b}{(c+d x)^2}\right )+4 c^3 d e f^2 \sin \left (\frac {b}{(c+d x)^2}\right )-c^4 f^3 \sin \left (\frac {b}{(c+d x)^2}\right )+4 d^4 e^3 x \sin \left (\frac {b}{(c+d x)^2}\right )+6 d^4 e^2 f x^2 \sin \left (\frac {b}{(c+d x)^2}\right )+4 d^4 e f^2 x^3 \sin \left (\frac {b}{(c+d x)^2}\right )+d^4 f^3 x^4 \sin \left (\frac {b}{(c+d x)^2}\right )+b^2 f^3 \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{4 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 365, normalized size = 1.08 Too large to display
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 448, normalized size = 1.33 \begin {gather*} \frac {b^{2} f^{3} \operatorname {Si}\left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 4 \, \sqrt {2} {\left (\pi c^{3} d f^{3} - 3 \, \pi c^{2} d^{2} f^{2} e + 3 \, \pi c d^{3} f e^{2} - \pi d^{4} e^{3}\right )} \sqrt {\frac {b}{\pi d^{2}}} \operatorname {C}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) - 8 \, \sqrt {2} {\left (\pi b c d f^{3} - \pi b d^{2} f^{2} e\right )} \sqrt {\frac {b}{\pi d^{2}}} \operatorname {S}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) + {\left (b d^{2} f^{3} x^{2} - 6 \, b c d f^{3} x - 7 \, b c^{2} f^{3} + 8 \, {\left (b d^{2} f^{2} x + b c d f^{2}\right )} e\right )} \cos \left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 3 \, {\left (b c^{2} f^{3} - 2 \, b c d f^{2} e + b d^{2} f e^{2}\right )} \operatorname {Ci}\left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 3 \, {\left (b c^{2} f^{3} - 2 \, b c d f^{2} e + b d^{2} f e^{2}\right )} \operatorname {Ci}\left (-\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + {\left (d^{4} f^{3} x^{4} - c^{4} f^{3} + 4 \, {\left (d^{4} x + c d^{3}\right )} e^{3} + 6 \, {\left (d^{4} f x^{2} - c^{2} d^{2} f\right )} e^{2} + 4 \, {\left (d^{4} f^{2} x^{3} + c^{3} d f^{2}\right )} e\right )} \sin \left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{4 \, d^{4}} \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 \left (e + f x\right )^{3} \sin {\left (\frac {b}{c^{2} + 2 c d x + d^{2} x^{2}} \right )}\, 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 = -0.00 \begin {gather*} \int \sin \left (\frac {b}{{\left (c+d\,x\right )}^2}\right )\,{\left (e+f\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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