Optimal. Leaf size=83 \[ \frac {\sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d}+\frac {\sqrt {\frac {\pi }{2}} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \sin (a)}{\sqrt {b} d} \]
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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3434, 3433,
3432} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} \sin (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} (c+d x)\right )}{\sqrt {b} d}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3433
Rule 3434
Rubi steps
\begin {align*} \int \sin \left (a+b (c+d x)^2\right ) \, dx &=\cos (a) \int \sin \left (b (c+d x)^2\right ) \, dx+\sin (a) \int \cos \left (b (c+d x)^2\right ) \, dx\\ &=\frac {\sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d}+\frac {\sqrt {\frac {\pi }{2}} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \sin (a)}{\sqrt {b} d}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 67, normalized size = 0.81 \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} \left (\cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )+C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \sin (a)\right )}{\sqrt {b} d} \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. \(135\) vs.
\(2(65)=130\).
time = 0.04, size = 136, normalized size = 1.64
method | result | size |
risch | \(\frac {i \sqrt {\pi }\, {\mathrm e}^{-i a} \erf \left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d \sqrt {i b}}+\frac {i \sqrt {\pi }\, {\mathrm e}^{i a} \erf \left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d \sqrt {-i b}}\) | \(87\) |
default | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {b^{2} c^{2} d^{2}-b \,d^{2} \left (b \,c^{2}+a \right )}{b \,d^{2}}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )-\sin \left (\frac {b^{2} c^{2} d^{2}-b \,d^{2} \left (b \,c^{2}+a \right )}{b \,d^{2}}\right ) \FresnelC \left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )\right )}{2 \sqrt {b \,d^{2}}}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.32, size = 69, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i + 1\right ) \, \cos \left (a\right ) + \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\frac {i \, b d x + i \, b c}{\sqrt {i \, b}}\right ) + {\left (-\left (i - 1\right ) \, \cos \left (a\right ) + \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\frac {i \, b d x + i \, b c}{\sqrt {-i \, b}}\right )\right )}}{8 \, \sqrt {b} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 89, normalized size = 1.07 \begin {gather*} \frac {\sqrt {2} \pi \sqrt {\frac {b d^{2}}{\pi }} \cos \left (a\right ) \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) + \sqrt {2} \pi \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 ) \sin \left (a\right )}{2 \, b d^{2}} \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 \sin {\left (a + b \left (c + d x\right )^{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 = 4.38, size = 151, normalized size = 1.82 \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^{\left (i \, a\right )}}{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^{\left (-i \, a\right )}}{4 \, \sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 95, normalized size = 1.14 \begin {gather*} \frac {\sqrt {2}\,\sqrt {\pi }\,\cos \left (a\right )\,\mathrm {S}\left (\frac {\sqrt {2}\,\sqrt {\frac {1}{b\,d^2}}\,\left (b\,x\,d^2+b\,c\,d\right )}{\sqrt {\pi }}\right )\,\sqrt {\frac {1}{b\,d^2}}}{2}+\frac {\sqrt {2}\,\sqrt {\pi }\,\sin \left (a\right )\,\mathrm {C}\left (\frac {\sqrt {2}\,\sqrt {\frac {1}{b\,d^2}}\,\left (b\,x\,d^2+b\,c\,d\right )}{\sqrt {\pi }}\right )\,\sqrt {\frac {1}{b\,d^2}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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