Optimal. Leaf size=337 \[ \frac{2 \sqrt{2 \pi } b^{3/2} f^2 (d e-c f) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{c+d x}\right )}{d^4}+\frac{b^2 f^3 \text{Si}\left (\frac{b}{(c+d x)^2}\right )}{4 d^4}-\frac{3 b f (d e-c f)^2 \text{CosIntegral}\left (\frac{b}{(c+d x)^2}\right )}{2 d^4}+\frac{f^2 (c+d x)^3 (d e-c f) \sin \left (\frac{b}{(c+d x)^2}\right )}{d^4}+\frac{2 b f^2 (c+d x) (d e-c f) \cos \left (\frac{b}{(c+d x)^2}\right )}{d^4}-\frac{\sqrt{2 \pi } \sqrt{b} (d e-c f)^3 \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{c+d x}\right )}{d^4}+\frac{3 f (c+d x)^2 (d e-c f)^2 \sin \left (\frac{b}{(c+d x)^2}\right )}{2 d^4}+\frac{(c+d x) (d e-c f)^3 \sin \left (\frac{b}{(c+d x)^2}\right )}{d^4}+\frac{f^3 (c+d x)^4 \sin \left (\frac{b}{(c+d x)^2}\right )}{4 d^4}+\frac{b f^3 (c+d x)^2 \cos \left (\frac{b}{(c+d x)^2}\right )}{4 d^4} \]
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Rubi [A] time = 0.420275, antiderivative size = 337, normalized size of antiderivative = 1., 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 = {3433, 3359, 3387, 3352, 3379, 3297, 3302, 3409, 3388, 3351, 3299} \[ \frac{2 \sqrt{2 \pi } b^{3/2} f^2 (d e-c f) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{c+d x}\right )}{d^4}+\frac{b^2 f^3 \text{Si}\left (\frac{b}{(c+d x)^2}\right )}{4 d^4}-\frac{3 b f (d e-c f)^2 \text{CosIntegral}\left (\frac{b}{(c+d x)^2}\right )}{2 d^4}+\frac{f^2 (c+d x)^3 (d e-c f) \sin \left (\frac{b}{(c+d x)^2}\right )}{d^4}+\frac{2 b f^2 (c+d x) (d e-c f) \cos \left (\frac{b}{(c+d x)^2}\right )}{d^4}-\frac{\sqrt{2 \pi } \sqrt{b} (d e-c f)^3 \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{c+d x}\right )}{d^4}+\frac{3 f (c+d x)^2 (d e-c f)^2 \sin \left (\frac{b}{(c+d x)^2}\right )}{2 d^4}+\frac{(c+d x) (d e-c f)^3 \sin \left (\frac{b}{(c+d x)^2}\right )}{d^4}+\frac{f^3 (c+d x)^4 \sin \left (\frac{b}{(c+d x)^2}\right )}{4 d^4}+\frac{b f^3 (c+d x)^2 \cos \left (\frac{b}{(c+d x)^2}\right )}{4 d^4} \]
Antiderivative was successfully verified.
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Rule 3433
Rule 3359
Rule 3387
Rule 3352
Rule 3379
Rule 3297
Rule 3302
Rule 3409
Rule 3388
Rule 3351
Rule 3299
Rubi steps
\begin{align*} \int (e+f x)^3 \sin \left (\frac{b}{(c+d x)^2}\right ) \, dx &=\frac{\operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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 \operatorname{Subst}\left (\int \sin \left (\frac{b}{x^2}\right ) \, dx,x,c+d x\right )}{d^4}\\ &=-\frac{f^3 \operatorname{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 ) \operatorname{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 ) \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.869559, size = 440, normalized size = 1.31 \[ \frac{8 \sqrt{2 \pi } b^{3/2} d e f^2 S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{c+d x}\right )-8 \sqrt{2 \pi } b^{3/2} c f^3 S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{c+d x}\right )+b^2 f^3 \text{Si}\left (\frac{b}{(c+d x)^2}\right )-6 c^2 d^2 e^2 f \sin \left (\frac{b}{(c+d x)^2}\right )+4 c^3 d e f^2 \sin \left (\frac{b}{(c+d x)^2}\right )+c^4 \left (-f^3\right ) \sin \left (\frac{b}{(c+d x)^2}\right )-7 b c^2 f^3 \cos \left (\frac{b}{(c+d x)^2}\right )-6 b f (d e-c f)^2 \text{CosIntegral}\left (\frac{b}{(c+d x)^2}\right )+6 d^4 e^2 f x^2 \sin \left (\frac{b}{(c+d x)^2}\right )+4 c d^3 e^3 \sin \left (\frac{b}{(c+d x)^2}\right )+4 d^4 e^3 x \sin \left (\frac{b}{(c+d x)^2}\right )+4 d^4 e f^2 x^3 \sin \left (\frac{b}{(c+d x)^2}\right )+8 b d^2 e f^2 x \cos \left (\frac{b}{(c+d x)^2}\right )+d^4 f^3 x^4 \sin \left (\frac{b}{(c+d x)^2}\right )+b d^2 f^3 x^2 \cos \left (\frac{b}{(c+d x)^2}\right )+8 b c d e f^2 \cos \left (\frac{b}{(c+d x)^2}\right )-4 \sqrt{2 \pi } \sqrt{b} (d e-c f)^3 \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{c+d x}\right )-6 b c d f^3 x \cos \left (\frac{b}{(c+d x)^2}\right )}{4 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 365, normalized size = 1.1 \begin{align*}{\frac{1}{{d}^{4}} \left ( - \left ({c}^{3}{f}^{3}-3\,{c}^{2}de{f}^{2}+3\,c{d}^{2}{e}^{2}f-{d}^{3}{e}^{3} \right ) \left ( dx+c \right ) \sin \left ({\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) + \left ({c}^{3}{f}^{3}-3\,{c}^{2}de{f}^{2}+3\,c{d}^{2}{e}^{2}f-{d}^{3}{e}^{3} \right ) \sqrt{b}\sqrt{2}\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi } \left ( dx+c \right ) }\sqrt{b}} \right ) -{\frac{ \left ( -3\,{c}^{2}{f}^{3}+6\,cde{f}^{2}-3\,{d}^{2}{e}^{2}f \right ) \left ( dx+c \right ) ^{2}}{2}\sin \left ({\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) }+{\frac{ \left ( -3\,{c}^{2}{f}^{3}+6\,cde{f}^{2}-3\,{d}^{2}{e}^{2}f \right ) b}{2}{\it Ci} \left ({\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) }-{\frac{ \left ( 3\,c{f}^{3}-3\,{f}^{2}ed \right ) \left ( dx+c \right ) ^{3}}{3}\sin \left ({\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) }+{\frac{ \left ( 6\,c{f}^{3}-6\,{f}^{2}ed \right ) b}{3} \left ( - \left ( dx+c \right ) \cos \left ({\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) -\sqrt{b}\sqrt{2}\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi } \left ( dx+c \right ) }\sqrt{b}} \right ) \right ) }+{\frac{{f}^{3} \left ( dx+c \right ) ^{4}}{4}\sin \left ({\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) }-{\frac{{f}^{3}b}{2} \left ( -{\frac{ \left ( dx+c \right ) ^{2}}{2}\cos \left ({\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) }-{\frac{b}{2}{\it Si} \left ({\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{-{\left (\frac{4 \, c^{3} e f^{2} \sin \left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{d^{3}} - \frac{3 \, c^{4} f^{3} \sin \left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{d^{4}} - 2 \, \int -\frac{2 \,{\left (3 \,{\left (b d^{3} e^{2} f - 2 \, b c d^{2} e f^{2} + b c^{2} d f^{3}\right )} x^{2} + 2 \,{\left (b d^{3} e^{3} - 3 \, b c^{2} d e f^{2} + 2 \, b c^{3} f^{3}\right )} x\right )} \cos \left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) -{\left (b^{2} d f^{3} x^{2} + 2 \,{\left (4 \, b^{2} d e f^{2} - 3 \, b^{2} c f^{3}\right )} x\right )} \sin \left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \,{\left (d^{5} x^{3} + 3 \, c d^{4} x^{2} + 3 \, c^{2} d^{3} x + c^{3} d^{2}\right )}}\,{d x} - 2 \, \int -\frac{2 \,{\left (3 \,{\left (b d^{3} e^{2} f - 2 \, b c d^{2} e f^{2} + b c^{2} d f^{3}\right )} x^{2} + 2 \,{\left (b d^{3} e^{3} - 3 \, b c^{2} d e f^{2} + 2 \, b c^{3} f^{3}\right )} x\right )} \cos \left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) -{\left (b^{2} d f^{3} x^{2} + 2 \,{\left (4 \, b^{2} d e f^{2} - 3 \, b^{2} c f^{3}\right )} x\right )} \sin \left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \,{\left ({\left (d^{5} x^{3} + 3 \, c d^{4} x^{2} + 3 \, c^{2} d^{3} x + c^{3} d^{2}\right )} \cos \left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} +{\left (d^{5} x^{3} + 3 \, c d^{4} x^{2} + 3 \, c^{2} d^{3} x + c^{3} d^{2}\right )} \sin \left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2}\right )}}\,{d x}\right )} d^{3} -{\left (b d f^{3} x^{2} + 2 \,{\left (4 \, b d e f^{2} - 3 \, b c f^{3}\right )} x\right )} \cos \left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) -{\left (d^{3} f^{3} x^{4} + 4 \, d^{3} e f^{2} x^{3} + 6 \, d^{3} e^{2} f x^{2} + 4 \, d^{3} e^{3} x\right )} \sin \left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{4 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06195, size = 1023, normalized size = 3.04 \begin{align*} \frac{b^{2} f^{3} \operatorname{Si}\left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 4 \, \sqrt{2} \pi{\left (d^{4} e^{3} - 3 \, c d^{3} e^{2} f + 3 \, c^{2} d^{2} e f^{2} - c^{3} d f^{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} \pi{\left (b d^{2} e f^{2} - b c d f^{3}\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} + 8 \, b c d e f^{2} - 7 \, b c^{2} f^{3} + 2 \,{\left (4 \, b d^{2} e f^{2} - 3 \, b c d f^{3}\right )} x\right )} \cos \left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 3 \,{\left (b d^{2} e^{2} f - 2 \, b c d e f^{2} + b c^{2} f^{3}\right )} \operatorname{Ci}\left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 3 \,{\left (b d^{2} e^{2} f - 2 \, b c d e f^{2} + b c^{2} f^{3}\right )} \operatorname{Ci}\left (-\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) +{\left (d^{4} f^{3} x^{4} + 4 \, d^{4} e f^{2} x^{3} + 6 \, d^{4} e^{2} f x^{2} + 4 \, d^{4} e^{3} x + 4 \, c d^{3} e^{3} - 6 \, c^{2} d^{2} e^{2} f + 4 \, c^{3} d e f^{2} - c^{4} f^{3}\right )} \sin \left (\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{4 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{3} \sin \left (\frac{b}{{\left (d x + c\right )}^{2}}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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