Optimal. Leaf size=120 \[ -\frac {b f \text {Ci}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2}-\frac {\sqrt {2 \pi } \sqrt {b} (d e-c f) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^2}+\frac {(c+d x) (d e-c f) \sin \left (\frac {b}{(c+d x)^2}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (\frac {b}{(c+d x)^2}\right )}{2 d^2} \]
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Rubi [A] time = 0.13, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3433, 3359, 3387, 3352, 3379, 3297, 3302} \[ -\frac {b f \text {CosIntegral}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2}-\frac {\sqrt {2 \pi } \sqrt {b} (d e-c f) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{c+d x}\right )}{d^2}+\frac {(c+d x) (d e-c f) \sin \left (\frac {b}{(c+d x)^2}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (\frac {b}{(c+d x)^2}\right )}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3302
Rule 3352
Rule 3359
Rule 3379
Rule 3387
Rule 3433
Rubi steps
\begin {align*} \int (e+f x) \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \left (d e \left (1-\frac {c f}{d e}\right ) \sin \left (\frac {b}{x^2}\right )+f x \sin \left (\frac {b}{x^2}\right )\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac {f \operatorname {Subst}\left (\int x \sin \left (\frac {b}{x^2}\right ) \, dx,x,c+d x\right )}{d^2}+\frac {(d e-c f) \operatorname {Subst}\left (\int \sin \left (\frac {b}{x^2}\right ) \, dx,x,c+d x\right )}{d^2}\\ &=-\frac {f \operatorname {Subst}\left (\int \frac {\sin (b x)}{x^2} \, dx,x,\frac {1}{(c+d x)^2}\right )}{2 d^2}-\frac {(d e-c f) \operatorname {Subst}\left (\int \frac {\sin \left (b x^2\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(d e-c f) (c+d x) \sin \left (\frac {b}{(c+d x)^2}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (\frac {b}{(c+d x)^2}\right )}{2 d^2}-\frac {(b f) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{(c+d x)^2}\right )}{2 d^2}-\frac {(2 b (d e-c f)) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=-\frac {b f \text {Ci}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2}-\frac {\sqrt {b} (d e-c f) \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^2}+\frac {(d e-c f) (c+d x) \sin \left (\frac {b}{(c+d x)^2}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (\frac {b}{(c+d x)^2}\right )}{2 d^2}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 95, normalized size = 0.79 \[ -\frac {b f \text {Ci}\left (\frac {b}{(c+d x)^2}\right )+2 \sqrt {2 \pi } \sqrt {b} (d e-c f) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+(c+d x) \sin \left (\frac {b}{(c+d x)^2}\right ) (c f-2 d e-d f x)}{2 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 155, normalized size = 1.29 \[ -\frac {4 \, \sqrt {2} \pi {\left (d^{2} e - c d f\right )} \sqrt {\frac {b}{\pi d^{2}}} \operatorname {C}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) + b f \operatorname {Ci}\left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + b f \operatorname {Ci}\left (-\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 2 \, {\left (d^{2} f x^{2} + 2 \, d^{2} e x + 2 \, c d e - c^{2} f\right )} \sin \left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{4 \, d^{2}} \]
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 \[ \int {\left (f x + e\right )} \sin \left (\frac {b}{{\left (d x + c\right )}^{2}}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 101, normalized size = 0.84 \[ \frac {-\left (c f -d e \right ) \left (d x +c \right ) \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )+\left (c f -d e \right ) \sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )+\frac {f \left (d x +c \right )^{2} \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-\frac {f b \Ci \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}}{d^{2}} \]
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 \[ \frac {1}{2} \, {\left (f x^{2} + 2 \, e x\right )} \sin \left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \int \frac {{\left (b d f x^{2} + 2 \, b d e x\right )} \cos \left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}}\,{d x} + \int \frac {{\left (b d f x^{2} + 2 \, b d e x\right )} \cos \left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, {\left ({\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \cos \left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \sin \left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sin \left (\frac {b}{{\left (c+d\,x\right )}^2}\right )\,\left (e+f\,x\right ) \,d x \]
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 \[ \int \left (e + f x\right ) \sin {\left (\frac {b}{c^{2} + 2 c d x + d^{2} x^{2}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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