Optimal. Leaf size=371 \[ \frac {2 \sqrt {2 \pi } b^{3/2} f^2 \sin (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{3 d^3}+\frac {2 \sqrt {2 \pi } b^{3/2} f^2 \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{3 d^3}-\frac {b f \cos (a) (d e-c f) \text {Ci}\left (\frac {b}{(c+d x)^2}\right )}{d^3}-\frac {\sqrt {2 \pi } \sqrt {b} \cos (a) (d e-c f)^2 C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^3}+\frac {\sqrt {2 \pi } \sqrt {b} \sin (a) (d e-c f)^2 S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^3}+\frac {b f \sin (a) (d e-c f) \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f (c+d x)^2 (d e-c f) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {(c+d x) (d e-c f)^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{3 d^3}+\frac {2 b f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^2}\right )}{3 d^3} \]
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Rubi [A] time = 0.48, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3433, 3359, 3387, 3354, 3352, 3351, 3379, 3297, 3303, 3299, 3302, 3409, 3388, 3353} \[ \frac {2 \sqrt {2 \pi } b^{3/2} f^2 \sin (a) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{c+d x}\right )}{3 d^3}+\frac {2 \sqrt {2 \pi } b^{3/2} f^2 \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{3 d^3}-\frac {b f \cos (a) (d e-c f) \text {CosIntegral}\left (\frac {b}{(c+d x)^2}\right )}{d^3}-\frac {\sqrt {2 \pi } \sqrt {b} \cos (a) (d e-c f)^2 \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{c+d x}\right )}{d^3}+\frac {\sqrt {2 \pi } \sqrt {b} \sin (a) (d e-c f)^2 S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^3}+\frac {b f \sin (a) (d e-c f) \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f (c+d x)^2 (d e-c f) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {(c+d x) (d e-c f)^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{3 d^3}+\frac {2 b f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^2}\right )}{3 d^3} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3351
Rule 3352
Rule 3353
Rule 3354
Rule 3359
Rule 3379
Rule 3387
Rule 3388
Rule 3409
Rule 3433
Rubi steps
\begin {align*} \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^2}\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^2}\right )+2 d e f \left (1-\frac {c f}{d e}\right ) x \sin \left (a+\frac {b}{x^2}\right )+f^2 x^2 \sin \left (a+\frac {b}{x^2}\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^2}\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^2}\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^2}\right ) \, dx,x,c+d x\right )}{d^3}\\ &=-\frac {f^2 \operatorname {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^4} \, dx,x,\frac {1}{c+d x}\right )}{d^3}-\frac {(f (d e-c f)) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,\frac {1}{(c+d x)^2}\right )}{d^3}-\frac {(d e-c f)^2 \operatorname {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d^3}\\ &=\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{3 d^3}-\frac {\left (2 b f^2\right ) \operatorname {Subst}\left (\int \frac {\cos \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{3 d^3}-\frac {(b f (d e-c f)) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,\frac {1}{(c+d x)^2}\right )}{d^3}-\frac {\left (2 b (d e-c f)^2\right ) \operatorname {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{d^3}\\ &=\frac {2 b f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^2}\right )}{3 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{3 d^3}+\frac {\left (4 b^2 f^2\right ) \operatorname {Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{3 d^3}-\frac {(b f (d e-c f) \cos (a)) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{(c+d x)^2}\right )}{d^3}-\frac {\left (2 b (d e-c f)^2 \cos (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{d^3}+\frac {(b f (d e-c f) \sin (a)) \operatorname {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{(c+d x)^2}\right )}{d^3}+\frac {\left (2 b (d e-c f)^2 \sin (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{d^3}\\ &=\frac {2 b f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^2}\right )}{3 d^3}-\frac {b f (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^2}\right )}{d^3}-\frac {\sqrt {b} (d e-c f)^2 \sqrt {2 \pi } \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^3}+\frac {\sqrt {b} (d e-c f)^2 \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right ) \sin (a)}{d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{3 d^3}+\frac {b f (d e-c f) \sin (a) \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {\left (4 b^2 f^2 \cos (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{3 d^3}+\frac {\left (4 b^2 f^2 \sin (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{3 d^3}\\ &=\frac {2 b f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^2}\right )}{3 d^3}-\frac {b f (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^2}\right )}{d^3}-\frac {\sqrt {b} (d e-c f)^2 \sqrt {2 \pi } \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^3}+\frac {2 b^{3/2} f^2 \sqrt {2 \pi } \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{3 d^3}+\frac {2 b^{3/2} f^2 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right ) \sin (a)}{3 d^3}+\frac {\sqrt {b} (d e-c f)^2 \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right ) \sin (a)}{d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{3 d^3}+\frac {b f (d e-c f) \sin (a) \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 1.54, size = 467, normalized size = 1.26 \[ \frac {2 \sqrt {2 \pi } b^{3/2} f^2 \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+c^3 f^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )-3 c^2 d e f \sin \left (a+\frac {b}{(c+d x)^2}\right )+3 \sqrt {2 \pi } \sqrt {b} c^2 f^2 \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+3 b f \cos (a) (c f-d e) \text {Ci}\left (\frac {b}{(c+d x)^2}\right )+3 d^3 e^2 x \sin \left (a+\frac {b}{(c+d x)^2}\right )+3 d^3 e f x^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )+d^3 f^2 x^3 \sin \left (a+\frac {b}{(c+d x)^2}\right )+3 \sqrt {2 \pi } \sqrt {b} d^2 e^2 \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+3 c d^2 e^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )+\sqrt {2 \pi } \sqrt {b} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right ) \left (2 b f^2 \sin (a)-3 \cos (a) (d e-c f)^2\right )-6 \sqrt {2 \pi } \sqrt {b} c d e f \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+3 b d e f \sin (a) \text {Si}\left (\frac {b}{(c+d x)^2}\right )-3 b c f^2 \sin (a) \text {Si}\left (\frac {b}{(c+d x)^2}\right )+2 b c f^2 \cos \left (a+\frac {b}{(c+d x)^2}\right )+2 b d f^2 x \cos \left (a+\frac {b}{(c+d x)^2}\right )}{3 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 430, normalized size = 1.16 \[ \frac {2 \, \sqrt {2} {\left (2 \, \pi b d f^{2} \sin \relax (a) - 3 \, \pi {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \cos \relax (a)\right )} \sqrt {\frac {b}{\pi d^{2}}} \operatorname {C}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) + 2 \, \sqrt {2} {\left (2 \, \pi b d f^{2} \cos \relax (a) + 3 \, \pi {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \sin \relax (a)\right )} \sqrt {\frac {b}{\pi d^{2}}} \operatorname {S}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) + 6 \, {\left (b d e f - b c f^{2}\right )} \sin \relax (a) \operatorname {Si}\left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 3 \, {\left ({\left (b d e f - b c f^{2}\right )} \operatorname {Ci}\left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + {\left (b d e f - b c f^{2}\right )} \operatorname {Ci}\left (-\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )\right )} \cos \relax (a) + 4 \, {\left (b d f^{2} x + b c f^{2}\right )} \cos \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 2 \, {\left (d^{3} f^{2} x^{3} + 3 \, d^{3} e f x^{2} + 3 \, d^{3} e^{2} x + 3 \, c d^{2} e^{2} - 3 \, c^{2} d e f + c^{3} f^{2}\right )} \sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{6 \, d^{3}} \]
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 \[ \int {\left (f x + e\right )}^{2} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{2}}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 302, normalized size = 0.81 \[ -\frac {-\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (d x +c \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )+\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )-\sin \relax (a ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )\right )-\frac {\left (-2 c \,f^{2}+2 d e f \right ) \left (d x +c \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )}{2}+\left (-2 c \,f^{2}+2 d e f \right ) b \left (\frac {\cos \relax (a ) \Ci \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-\frac {\sin \relax (a ) \Si \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}\right )-\frac {f^{2} \left (d x +c \right )^{3} \sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )}{3}+\frac {2 f^{2} b \left (-\left (d x +c \right ) \cos \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )-\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )+\sin \relax (a ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )\right )\right )}{3}}{d^{3}} \]
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 \[ \frac {2 \, b f^{2} x \cos \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + {\left (\frac {c^{3} f^{2} \sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{d^{3}} - 2 \, \int \frac {2 \, b^{2} f^{2} x \sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 3 \, {\left ({\left (b d^{2} e f - b c d f^{2}\right )} x^{2} + {\left (b d^{2} e^{2} - b c^{2} f^{2}\right )} x\right )} \cos \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, {\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d\right )}}\,{d x} - 2 \, \int \frac {2 \, b^{2} f^{2} x \sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 3 \, {\left ({\left (b d^{2} e f - b c d f^{2}\right )} x^{2} + {\left (b d^{2} e^{2} - b c^{2} f^{2}\right )} x\right )} \cos \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, {\left ({\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d\right )} \cos \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + {\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d\right )} \sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2}\right )}}\,{d x}\right )} d^{2} + {\left (d^{2} f^{2} x^{3} + 3 \, d^{2} e f x^{2} + 3 \, d^{2} e^{2} x\right )} \sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{3 \, d^{2}} \]
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 \[ \int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^2}\right )\,{\left (e+f\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e + f x\right )^{2} \sin {\left (a + \frac {b}{c^{2} + 2 c d x + d^{2} x^{2}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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