Optimal. Leaf size=223 \[ \frac {3 \sqrt {\frac {\pi }{2}} f^2 (d e-c f) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b^{3/2} d^4}+\frac {f^3 \sin \left (b (c+d x)^2\right )}{2 b^2 d^4}-\frac {3 f^2 (c+d x) (d e-c f) \cos \left (b (c+d x)^2\right )}{2 b d^4}+\frac {\sqrt {\frac {\pi }{2}} (d e-c f)^3 S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^4}-\frac {3 f (d e-c f)^2 \cos \left (b (c+d x)^2\right )}{2 b d^4}-\frac {f^3 (c+d x)^2 \cos \left (b (c+d x)^2\right )}{2 b d^4} \]
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Rubi [A] time = 0.31, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3433, 3351, 3379, 2638, 3385, 3352, 3296, 2637} \[ \frac {3 \sqrt {\frac {\pi }{2}} f^2 (d e-c f) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} (c+d x)\right )}{2 b^{3/2} d^4}+\frac {f^3 \sin \left (b (c+d x)^2\right )}{2 b^2 d^4}-\frac {3 f^2 (c+d x) (d e-c f) \cos \left (b (c+d x)^2\right )}{2 b d^4}+\frac {\sqrt {\frac {\pi }{2}} (d e-c f)^3 S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^4}-\frac {3 f (d e-c f)^2 \cos \left (b (c+d x)^2\right )}{2 b d^4}-\frac {f^3 (c+d x)^2 \cos \left (b (c+d x)^2\right )}{2 b d^4} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 3351
Rule 3352
Rule 3379
Rule 3385
Rule 3433
Rubi steps
\begin {align*} \int (e+f x)^3 \sin \left (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 (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 (b x^2\right )+3 d e f^2 \left (1-\frac {c f}{d e}\right ) x^2 \sin \left (b x^2\right )+f^3 x^3 \sin \left (b x^2\right )\right ) \, dx,x,c+d x\right )}{d^4}\\ &=\frac {f^3 \operatorname {Subst}\left (\int x^3 \sin \left (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 (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 (b x^2\right ) \, dx,x,c+d x\right )}{d^4}+\frac {(d e-c f)^3 \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,c+d x\right )}{d^4}\\ &=-\frac {3 f^2 (d e-c f) (c+d x) \cos \left (b (c+d x)^2\right )}{2 b d^4}+\frac {(d e-c f)^3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^4}+\frac {f^3 \operatorname {Subst}\left (\int x \sin (b x) \, dx,x,(c+d x)^2\right )}{2 d^4}+\frac {\left (3 f^2 (d e-c f)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,c+d x\right )}{2 b d^4}+\frac {\left (3 f (d e-c f)^2\right ) \operatorname {Subst}\left (\int \sin (b x) \, dx,x,(c+d x)^2\right )}{2 d^4}\\ &=-\frac {3 f (d e-c f)^2 \cos \left (b (c+d x)^2\right )}{2 b d^4}-\frac {3 f^2 (d e-c f) (c+d x) \cos \left (b (c+d x)^2\right )}{2 b d^4}-\frac {f^3 (c+d x)^2 \cos \left (b (c+d x)^2\right )}{2 b d^4}+\frac {3 f^2 (d e-c f) \sqrt {\frac {\pi }{2}} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b^{3/2} d^4}+\frac {(d e-c f)^3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^4}+\frac {f^3 \operatorname {Subst}\left (\int \cos (b x) \, dx,x,(c+d x)^2\right )}{2 b d^4}\\ &=-\frac {3 f (d e-c f)^2 \cos \left (b (c+d x)^2\right )}{2 b d^4}-\frac {3 f^2 (d e-c f) (c+d x) \cos \left (b (c+d x)^2\right )}{2 b d^4}-\frac {f^3 (c+d x)^2 \cos \left (b (c+d x)^2\right )}{2 b d^4}+\frac {3 f^2 (d e-c f) \sqrt {\frac {\pi }{2}} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b^{3/2} d^4}+\frac {(d e-c f)^3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^4}+\frac {f^3 \sin \left (b (c+d x)^2\right )}{2 b^2 d^4}\\ \end {align*}
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Mathematica [A] time = 1.08, size = 173, normalized size = 0.78 \[ \frac {4 \sqrt {2 \pi } b^{3/2} (d e-c f)^3 S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )-4 b f \cos \left (b (c+d x)^2\right ) \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )-6 \sqrt {2 \pi } \sqrt {b} f^2 (c f-d e) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )+4 f^3 \sin \left (b (c+d x)^2\right )}{8 b^2 d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 255, normalized size = 1.14 \[ \frac {2 \, d f^{3} \sin \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right ) + 3 \, \sqrt {2} \pi {\left (d e f^{2} - c f^{3}\right )} \sqrt {\frac {b d^{2}}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) + 2 \, \sqrt {2} \pi {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \sqrt {\frac {b d^{2}}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) - 2 \, {\left (b d^{3} f^{3} x^{2} + 3 \, b d^{3} e^{2} f - 3 \, b c d^{2} e f^{2} + b c^{2} d f^{3} + {\left (3 \, b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x\right )} \cos \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )}{4 \, b^{2} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.73, size = 1023, normalized size = 4.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 586, normalized size = 2.63 \[ -\frac {f^{3} x^{2} \cos \left (d^{2} x^{2} b +2 c d x b +b \,c^{2}\right )}{2 d^{2} b}-\frac {f^{3} c \left (-\frac {x \cos \left (d^{2} x^{2} b +2 c d x b +b \,c^{2}\right )}{2 d^{2} b}-\frac {c \left (-\frac {\cos \left (d^{2} x^{2} b +2 c d x b +b \,c^{2}\right )}{2 d^{2} b}-\frac {c \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )}{2 d \sqrt {d^{2} b}}\right )}{d}+\frac {\sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )}{4 d^{2} b \sqrt {d^{2} b}}\right )}{d}+\frac {f^{3} \left (\frac {\sin \left (d^{2} x^{2} b +2 c d x b +b \,c^{2}\right )}{2 d^{2} b}-\frac {c \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )}{2 d \sqrt {d^{2} b}}\right )}{d^{2} b}-\frac {3 e \,f^{2} x \cos \left (d^{2} x^{2} b +2 c d x b +b \,c^{2}\right )}{2 d^{2} b}-\frac {3 e \,f^{2} c \left (-\frac {\cos \left (d^{2} x^{2} b +2 c d x b +b \,c^{2}\right )}{2 d^{2} b}-\frac {c \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )}{2 d \sqrt {d^{2} b}}\right )}{d}+\frac {3 e \,f^{2} \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )}{4 d^{2} b \sqrt {d^{2} b}}-\frac {3 e^{2} f \cos \left (d^{2} x^{2} b +2 c d x b +b \,c^{2}\right )}{2 d^{2} b}-\frac {3 e^{2} f c \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )}{2 d \sqrt {d^{2} b}}+\frac {\sqrt {2}\, \sqrt {\pi }\, e^{3} \mathrm {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )}{2 \sqrt {d^{2} b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.28, size = 972, normalized size = 4.36 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.71, size = 231, normalized size = 1.04 \[ \frac {f^3\,\sin \left (b\,{\left (c+d\,x\right )}^2\right )}{2\,b^2\,d^4}-\frac {\cos \left (b\,{\left (c+d\,x\right )}^2\right )\,\left (c^2\,f^3-3\,c\,d\,e\,f^2+3\,d^2\,e^2\,f\right )}{2\,b\,d^4}-\frac {f^3\,x^2\,\cos \left (b\,{\left (c+d\,x\right )}^2\right )}{2\,b\,d^2}+\frac {x\,\cos \left (b\,{\left (c+d\,x\right )}^2\right )\,\left (c\,f^3-3\,d\,e\,f^2\right )}{2\,b\,d^3}-\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {S}\left (\frac {\sqrt {2}\,\sqrt {b}\,\left (c+d\,x\right )}{\sqrt {\pi }}\right )\,\left (c^3\,f^3-3\,c^2\,d\,e\,f^2+3\,c\,d^2\,e^2\,f-d^3\,e^3\right )}{2\,\sqrt {b}\,d^4}-\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\sqrt {b}\,\left (c+d\,x\right )}{\sqrt {\pi }}\right )\,\left (3\,c\,f^3-3\,d\,e\,f^2\right )}{4\,b^{3/2}\,d^4} \]
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 \[ \int \left (e + f x\right )^{3} \sin {\left (b c^{2} + 2 b c d x + b d^{2} x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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