Optimal. Leaf size=69 \[ -\frac {f \cos \left (b (c+d x)^2\right )}{2 b d^2}+\frac {(d e-c f) \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3514, 3432,
3460, 2718} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} (d e-c f) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^2}-\frac {f \cos \left (b (c+d x)^2\right )}{2 b d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3432
Rule 3460
Rule 3514
Rubi steps
\begin {align*} \int (e+f x) \sin \left (b (c+d x)^2\right ) \, dx &=\frac {\text {Subst}\left (\int \left (d e \left (1-\frac {c f}{d e}\right ) \sin \left (b x^2\right )+f x \sin \left (b x^2\right )\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac {f \text {Subst}\left (\int x \sin \left (b x^2\right ) \, dx,x,c+d x\right )}{d^2}+\frac {(d e-c f) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac {(d e-c f) \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^2}+\frac {f \text {Subst}\left (\int \sin (b x) \, dx,x,(c+d x)^2\right )}{2 d^2}\\ &=-\frac {f \cos \left (b (c+d x)^2\right )}{2 b d^2}+\frac {(d e-c f) \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^2}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 66, normalized size = 0.96 \begin {gather*} \frac {-f \cos \left (b (c+d x)^2\right )+\sqrt {b} (d e-c f) \sqrt {2 \pi } S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b d^2} \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. \(119\) vs.
\(2(58)=116\).
time = 0.04, size = 120, normalized size = 1.74
method | result | size |
default | \(-\frac {f \cos \left (d^{2} x^{2} b +2 c d x b +b \,c^{2}\right )}{2 b \,d^{2}}-\frac {f c \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )}{2 d \sqrt {b \,d^{2}}}+\frac {\sqrt {2}\, \sqrt {\pi }\, e \,\mathrm {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )}{2 \sqrt {b \,d^{2}}}\) | \(120\) |
risch | \(\frac {i e \sqrt {\pi }\, \erf \left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d \sqrt {-i b}}-\frac {i f c \sqrt {\pi }\, \erf \left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d^{2} \sqrt {-i b}}+\frac {i e \sqrt {\pi }\, \erf \left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d \sqrt {i b}}-\frac {i f c \sqrt {\pi }\, \erf \left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d^{2} \sqrt {i b}}-\frac {f \cos \left (b \left (d x +c \right )^{2}\right )}{2 b \,d^{2}}\) | \(177\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.76, size = 272, normalized size = 3.94 \begin {gather*} \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}{8 \, \sqrt {b} d} - \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}{8 \, {\left (b d^{3} x + b c d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 81, normalized size = 1.17 \begin {gather*} -\frac {d f \cos \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right ) + \sqrt {2} {\left (\pi c f - \pi d e\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 \, b d^{3}} \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 ) \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 = 2.72, size = 367, normalized size = 5.32 \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}{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}{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 )}{\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}\right )}}{b d}}{4 \, 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 )}{\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}\right )}}{b d}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 58, normalized size = 0.84 \begin {gather*} -\frac {f\,\cos \left (b\,{\left (c+d\,x\right )}^2\right )}{2\,b\,d^2}-\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {S}\left (\frac {\sqrt {2}\,\sqrt {b}\,\left (c+d\,x\right )}{\sqrt {\pi }}\right )\,\left (c\,f-d\,e\right )}{2\,\sqrt {b}\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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