Optimal. Leaf size=153 \[ \frac {3 \sqrt {\frac {\pi }{2}} \sin (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} \sin (3 a) C\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}}+\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) S\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}} \]
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Rubi [A] time = 0.08, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3357, 3353, 3352, 3351} \[ \frac {3 \sqrt {\frac {\pi }{2}} \sin (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} x\right )}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} \sin (3 a) \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {b} x\right )}{4 \sqrt {b}}+\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) S\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3353
Rule 3357
Rubi steps
\begin {align*} \int \sin ^3\left (a+b x^2\right ) \, dx &=\int \left (\frac {3}{4} \sin \left (a+b x^2\right )-\frac {1}{4} \sin \left (3 a+3 b x^2\right )\right ) \, dx\\ &=-\left (\frac {1}{4} \int \sin \left (3 a+3 b x^2\right ) \, dx\right )+\frac {3}{4} \int \sin \left (a+b x^2\right ) \, dx\\ &=\frac {1}{4} (3 \cos (a)) \int \sin \left (b x^2\right ) \, dx-\frac {1}{4} \cos (3 a) \int \sin \left (3 b x^2\right ) \, dx+\frac {1}{4} (3 \sin (a)) \int \cos \left (b x^2\right ) \, dx-\frac {1}{4} \sin (3 a) \int \cos \left (3 b x^2\right ) \, dx\\ &=\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) S\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}}+\frac {3 \sqrt {\frac {\pi }{2}} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} C\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)}{4 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 117, normalized size = 0.76 \[ \frac {\sqrt {\frac {\pi }{6}} \left (3 \sqrt {3} \sin (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )-\sin (3 a) C\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )+3 \sqrt {3} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )-\cos (3 a) S\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )\right )}{4 \sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 120, normalized size = 0.78 \[ -\frac {\sqrt {6} \pi \sqrt {\frac {b}{\pi }} \cos \left (3 \, a\right ) \operatorname {S}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) - 9 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) + \sqrt {6} \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (3 \, a\right ) - 9 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \relax (a)}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.93, size = 185, normalized size = 1.21 \[ -\frac {i \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {6} \sqrt {b} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (3 i \, a\right )}}{48 \, \sqrt {b} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} + \frac {3 i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{16 \, {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {3 i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{16 \, {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {i \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {6} \sqrt {b} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-3 i \, a\right )}}{48 \, \sqrt {b} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 99, normalized size = 0.65 \[ \frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \mathrm {S}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \relax (a ) \FresnelC \left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{8 \sqrt {b}}-\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (3 a \right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )+\sin \left (3 a \right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )\right )}{24 \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.19, size = 112, normalized size = 0.73 \[ \frac {3 \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i + 1\right ) \, \cos \left (3 \, a\right ) + \left (i - 1\right ) \, \sin \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3 i \, b} x\right ) + {\left (\left (i - 1\right ) \, \cos \left (3 \, a\right ) - \left (i + 1\right ) \, \sin \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-3 i \, b} x\right )\right )} b^{\frac {3}{2}} + \sqrt {2} \sqrt {\pi } {\left ({\left (\left (27 i + 27\right ) \, \cos \relax (a) - \left (27 i - 27\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {i \, b} x\right ) + {\left (-\left (27 i - 27\right ) \, \cos \relax (a) + \left (27 i + 27\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {-i \, b} x\right )\right )} b^{\frac {3}{2}}}{288 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (b\,x^2+a\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.36, size = 129, normalized size = 0.84 \[ \frac {3 \sqrt {2} \sqrt {\pi } \left (\sin {\relax (a )} C\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right ) + \cos {\relax (a )} S\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right )\right ) \sqrt {\frac {1}{b}}}{8} - \frac {\sqrt {6} \sqrt {\pi } \left (\sin {\left (3 a \right )} C\left (\frac {\sqrt {6} \sqrt {b} x}{\sqrt {\pi }}\right ) + \cos {\left (3 a \right )} S\left (\frac {\sqrt {6} \sqrt {b} x}{\sqrt {\pi }}\right )\right ) \sqrt {\frac {1}{b}}}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
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