Optimal. Leaf size=150 \[ -\frac {f (d e-c f) \cos \left (b (c+d x)^2\right )}{b d^3}-\frac {f^2 (c+d x) \cos \left (b (c+d x)^2\right )}{2 b d^3}+\frac {f^2 \sqrt {\frac {\pi }{2}} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b^{3/2} d^3}+\frac {(d e-c f)^2 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^3} \]
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Rubi [A]
time = 0.11, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3514, 3432,
3460, 2718, 3466, 3433} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} f^2 \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} (c+d x)\right )}{2 b^{3/2} d^3}+\frac {\sqrt {\frac {\pi }{2}} (d e-c f)^2 S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^3}-\frac {f (d e-c f) \cos \left (b (c+d x)^2\right )}{b d^3}-\frac {f^2 (c+d x) \cos \left (b (c+d x)^2\right )}{2 b d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3432
Rule 3433
Rule 3460
Rule 3466
Rule 3514
Rubi steps
\begin {align*} \int (e+f x)^2 \sin \left (b (c+d x)^2\right ) \, dx &=\frac {\text {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 (b x^2\right )+2 d e f \left (1-\frac {c f}{d e}\right ) x \sin \left (b x^2\right )+f^2 x^2 \sin \left (b x^2\right )\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac {f^2 \text {Subst}\left (\int x^2 \sin \left (b x^2\right ) \, dx,x,c+d x\right )}{d^3}+\frac {(2 f (d e-c f)) \text {Subst}\left (\int x \sin \left (b x^2\right ) \, dx,x,c+d x\right )}{d^3}+\frac {(d e-c f)^2 \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,c+d x\right )}{d^3}\\ &=-\frac {f^2 (c+d x) \cos \left (b (c+d x)^2\right )}{2 b d^3}+\frac {(d e-c f)^2 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^3}+\frac {f^2 \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,c+d x\right )}{2 b d^3}+\frac {(f (d e-c f)) \text {Subst}\left (\int \sin (b x) \, dx,x,(c+d x)^2\right )}{d^3}\\ &=-\frac {f (d e-c f) \cos \left (b (c+d x)^2\right )}{b d^3}-\frac {f^2 (c+d x) \cos \left (b (c+d x)^2\right )}{2 b d^3}+\frac {f^2 \sqrt {\frac {\pi }{2}} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b^{3/2} d^3}+\frac {(d e-c f)^2 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^3}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 117, normalized size = 0.78 \begin {gather*} \frac {-2 \sqrt {b} f (2 d e-c f+d f x) \cos \left (b (c+d x)^2\right )+f^2 \sqrt {2 \pi } C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )+2 b (d e-c f)^2 \sqrt {2 \pi } S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{4 b^{3/2} d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(290\) vs.
\(2(127)=254\).
time = 0.08, size = 291, normalized size = 1.94
method | result | size |
default | \(-\frac {f^{2} x \cos \left (d^{2} x^{2} b +2 c d x b +b \,c^{2}\right )}{2 b \,d^{2}}-\frac {f^{2} c \left (-\frac {\cos \left (d^{2} x^{2} b +2 c d x b +b \,c^{2}\right )}{2 b \,d^{2}}-\frac {c \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )}{2 d \sqrt {b \,d^{2}}}\right )}{d}+\frac {f^{2} \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )}{4 b \,d^{2} \sqrt {b \,d^{2}}}-\frac {e f \cos \left (d^{2} x^{2} b +2 c d x b +b \,c^{2}\right )}{b \,d^{2}}-\frac {e f c \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )}{d \sqrt {b \,d^{2}}}+\frac {\sqrt {2}\, \sqrt {\pi }\, e^{2} \mathrm {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )}{2 \sqrt {b \,d^{2}}}\) | \(291\) |
risch | \(\frac {i e^{2} \sqrt {\pi }\, \erf \left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d \sqrt {-i b}}+\frac {i f^{2} c^{2} \sqrt {\pi }\, \erf \left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d^{3} \sqrt {-i b}}-\frac {f^{2} \sqrt {\pi }\, \erf \left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{8 b \,d^{3} \sqrt {-i b}}-\frac {i e f c \sqrt {\pi }\, \erf \left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{2 d^{2} \sqrt {-i b}}+\frac {i e^{2} \sqrt {\pi }\, \erf \left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d \sqrt {i b}}+\frac {i f^{2} c^{2} \sqrt {\pi }\, \erf \left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d^{3} \sqrt {i b}}+\frac {f^{2} \sqrt {\pi }\, \erf \left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{8 b \,d^{3} \sqrt {i b}}-\frac {i e f c \sqrt {\pi }\, \erf \left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{2 d^{2} \sqrt {i b}}+2 \left (\frac {i f^{2} \left (\frac {i x}{2 d^{2} b}-\frac {i c}{2 d^{3} b}\right )}{2}-\frac {e f}{2 b \,d^{2}}\right ) \cos \left (b \left (d x +c \right )^{2}\right )\) | \(386\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 1.08, size = 564, normalized size = 3.76 \begin {gather*} \frac {{\left (4 \, b c d x {\left (e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}\right )} + e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}\right )}\right )} + 4 \, b c^{2} {\left (e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}\right )} + e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}\right )}\right )} - \sqrt {b d^{2} x^{2} + 2 \, b c d x + b c^{2}} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}}\right ) - 1\right )} + \left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}}\right ) - 1\right )}\right )} b c^{2} - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}\right ) + \left (i + 1\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, -i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}\right )\right )}\right )} f^{2}}{8 \, {\left (b^{2} d^{4} x + b^{2} c d^{3}\right )}} - \frac {{\left (2 \, d x {\left (e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}\right )} + e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}\right )}\right )} - \sqrt {b d^{2} x^{2} + 2 \, b c d x + b c^{2}} {\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}}\right ) - 1\right )} + \left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}}\right ) - 1\right )}\right )} c + 2 \, c {\left (e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}\right )} + e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}\right )}\right )}\right )} f e}{4 \, {\left (b d^{3} x + b c d^{2}\right )}} + \frac {\sqrt {2} \sqrt {\pi } {\left (\left (i + 1\right ) \, \operatorname {erf}\left (\frac {i \, b d x + i \, b c}{\sqrt {i \, b}}\right ) + \left (i - 1\right ) \, \operatorname {erf}\left (\frac {i \, b d x + i \, b c}{\sqrt {-i \, b}}\right )\right )} e^{2}}{8 \, \sqrt {b} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 165, normalized size = 1.10 \begin {gather*} \frac {\sqrt {2} \pi \sqrt {\frac {b d^{2}}{\pi }} f^{2} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) + 2 \, \sqrt {2} {\left (\pi b c^{2} f^{2} - 2 \, \pi b c d f e + \pi b d^{2} e^{2}\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^{2} f^{2} x - b c d f^{2} + 2 \, b d^{2} f e\right )} \cos \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )}{4 \, b^{2} 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 )^{2} \sin {\left (b c^{2} + 2 b c d x + b d^{2} x^{2} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 4.59, size = 669, normalized size = 4.46 \begin {gather*} -\frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e^{2}}{4 \, \sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )}} + \frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e^{2}}{4 \, \sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )}} - \frac {-\frac {i \, \sqrt {2} \sqrt {\pi } c f \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e}{\sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )}} + \frac {f e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2} + 1\right )}}{b d}}{2 \, d} - \frac {\frac {i \, \sqrt {2} \sqrt {\pi } c f \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e}{\sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )}} + \frac {f e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2} + 1\right )}}{b d}}{2 \, d} - \frac {\frac {i \, \sqrt {2} \sqrt {\pi } {\left (2 \, b c^{2} f^{2} - i \, f^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right )}{\sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} b} + \frac {2 i \, {\left (d f^{2} {\left (-i \, x - \frac {i \, c}{d}\right )} + 2 i \, c f^{2}\right )} e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}\right )}}{b d}}{8 \, d^{2}} - \frac {-\frac {i \, \sqrt {2} \sqrt {\pi } {\left (2 \, b c^{2} f^{2} + i \, f^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right )}{\sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} b} + \frac {2 i \, {\left (d f^{2} {\left (-i \, x - \frac {i \, c}{d}\right )} + 2 i \, c f^{2}\right )} e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}\right )}}{b d}}{8 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 136, normalized size = 0.91 \begin {gather*} \frac {\cos \left (b\,{\left (c+d\,x\right )}^2\right )\,\left (c\,f^2-2\,d\,e\,f\right )}{2\,b\,d^3}-\frac {f^2\,x\,\cos \left (b\,{\left (c+d\,x\right )}^2\right )}{2\,b\,d^2}+\frac {\sqrt {2}\,f^2\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\sqrt {b}\,\left (c+d\,x\right )}{\sqrt {\pi }}\right )}{4\,b^{3/2}\,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^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2\right )}{2\,\sqrt {b}\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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