Integrand size = 14, antiderivative size = 188 \[ \int x^2 \cos ^3\left (a+b x^2\right ) \, dx=-\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{8 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{24 b^{3/2}}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)}{8 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)}{24 b^{3/2}}+\frac {3 x \sin \left (a+b x^2\right )}{8 b}+\frac {x \sin \left (3 a+3 b x^2\right )}{24 b} \]
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Time = 0.22 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3485, 3467, 3434, 3433, 3432} \[ \int x^2 \cos ^3\left (a+b x^2\right ) \, dx=-\frac {3 \sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{8 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \sin (3 a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{24 b^{3/2}}-\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{8 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{24 b^{3/2}}+\frac {3 x \sin \left (a+b x^2\right )}{8 b}+\frac {x \sin \left (3 a+3 b x^2\right )}{24 b} \]
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Rule 3432
Rule 3433
Rule 3434
Rule 3467
Rule 3485
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{4} x^2 \cos \left (a+b x^2\right )+\frac {1}{4} x^2 \cos \left (3 a+3 b x^2\right )\right ) \, dx \\ & = \frac {1}{4} \int x^2 \cos \left (3 a+3 b x^2\right ) \, dx+\frac {3}{4} \int x^2 \cos \left (a+b x^2\right ) \, dx \\ & = \frac {3 x \sin \left (a+b x^2\right )}{8 b}+\frac {x \sin \left (3 a+3 b x^2\right )}{24 b}-\frac {\int \sin \left (3 a+3 b x^2\right ) \, dx}{24 b}-\frac {3 \int \sin \left (a+b x^2\right ) \, dx}{8 b} \\ & = \frac {3 x \sin \left (a+b x^2\right )}{8 b}+\frac {x \sin \left (3 a+3 b x^2\right )}{24 b}-\frac {(3 \cos (a)) \int \sin \left (b x^2\right ) \, dx}{8 b}-\frac {\cos (3 a) \int \sin \left (3 b x^2\right ) \, dx}{24 b}-\frac {(3 \sin (a)) \int \cos \left (b x^2\right ) \, dx}{8 b}-\frac {\sin (3 a) \int \cos \left (3 b x^2\right ) \, dx}{24 b} \\ & = -\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{8 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{24 b^{3/2}}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)}{8 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)}{24 b^{3/2}}+\frac {3 x \sin \left (a+b x^2\right )}{8 b}+\frac {x \sin \left (3 a+3 b x^2\right )}{24 b} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.85 \[ \int x^2 \cos ^3\left (a+b x^2\right ) \, dx=\frac {-27 \sqrt {2 \pi } \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )-\sqrt {6 \pi } \cos (3 a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )-27 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)-\sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)+54 \sqrt {b} x \sin \left (a+b x^2\right )+6 \sqrt {b} x \sin \left (3 \left (a+b x^2\right )\right )}{144 b^{3/2}} \]
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Time = 0.46 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.69
method | result | size |
default | \(\frac {3 x \sin \left (b \,x^{2}+a \right )}{8 b}-\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {S}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \left (a \right ) \operatorname {C}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{16 b^{\frac {3}{2}}}+\frac {x \sin \left (3 b \,x^{2}+3 a \right )}{24 b}-\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (3 a \right ) \operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )+\sin \left (3 a \right ) \operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )\right )}{144 b^{\frac {3}{2}}}\) | \(130\) |
risch | \(-\frac {i {\mathrm e}^{-3 i a} \sqrt {\pi }\, \sqrt {3}\, \operatorname {erf}\left (\sqrt {3}\, \sqrt {i b}\, x \right )}{288 b \sqrt {i b}}-\frac {3 i {\mathrm e}^{-i a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {i b}\, x \right )}{32 b \sqrt {i b}}+\frac {3 i {\mathrm e}^{i a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-i b}\, x \right )}{32 b \sqrt {-i b}}+\frac {i {\mathrm e}^{3 i a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-3 i b}\, x \right )}{96 b \sqrt {-3 i b}}+\frac {3 x \sin \left (b \,x^{2}+a \right )}{8 b}+\frac {x \sin \left (3 b \,x^{2}+3 a \right )}{24 b}\) | \(155\) |
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Time = 0.26 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.79 \[ \int x^2 \cos ^3\left (a+b x^2\right ) \, dx=-\frac {\sqrt {6} \pi \sqrt {\frac {b}{\pi }} \cos \left (3 \, a\right ) \operatorname {S}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) + 27 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) + \sqrt {6} \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (3 \, a\right ) + 27 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right ) - 24 \, {\left (b x \cos \left (b x^{2} + a\right )^{2} + 2 \, b x\right )} \sin \left (b x^{2} + a\right )}{144 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (194) = 388\).
Time = 2.18 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.34 \[ \int x^2 \cos ^3\left (a+b x^2\right ) \, dx=\frac {3 b^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}} \sin {\left (a \right )} \Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4} \\ \frac {3}{2}, \frac {7}{4}, \frac {9}{4} \end {matrix}\middle | {- \frac {b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac {7}{4}\right ) \Gamma \left (\frac {9}{4}\right )} + \frac {3 b^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}} \sin {\left (3 a \right )} \Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4} \\ \frac {3}{2}, \frac {7}{4}, \frac {9}{4} \end {matrix}\middle | {- \frac {9 b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac {7}{4}\right ) \Gamma \left (\frac {9}{4}\right )} - \frac {3 \sqrt {b} x^{3} \sqrt {\frac {1}{b}} \cos {\left (a \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} \\ \frac {1}{2}, \frac {5}{4}, \frac {7}{4} \end {matrix}\middle | {- \frac {b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac {5}{4}\right ) \Gamma \left (\frac {7}{4}\right )} - \frac {\sqrt {b} x^{3} \sqrt {\frac {1}{b}} \cos {\left (3 a \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} \\ \frac {1}{2}, \frac {5}{4}, \frac {7}{4} \end {matrix}\middle | {- \frac {9 b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac {5}{4}\right ) \Gamma \left (\frac {7}{4}\right )} - \frac {3 \sqrt {2} \sqrt {\pi } x^{2} \sqrt {\frac {1}{b}} \sin {\left (a \right )} S\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right )}{8} - \frac {\sqrt {6} \sqrt {\pi } x^{2} \sqrt {\frac {1}{b}} \sin {\left (3 a \right )} S\left (\frac {\sqrt {6} \sqrt {b} x}{\sqrt {\pi }}\right )}{24} + \frac {3 \sqrt {2} \sqrt {\pi } x^{2} \sqrt {\frac {1}{b}} \cos {\left (a \right )} C\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right )}{8} + \frac {\sqrt {6} \sqrt {\pi } x^{2} \sqrt {\frac {1}{b}} \cos {\left (3 a \right )} C\left (\frac {\sqrt {6} \sqrt {b} x}{\sqrt {\pi }}\right )}{24} \]
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Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.76 \[ \int x^2 \cos ^3\left (a+b x^2\right ) \, dx=\frac {24 \, b^{2} x \sin \left (3 \, b x^{2} + 3 \, a\right ) + 216 \, b^{2} x \sin \left (b x^{2} + a\right ) + 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i + 1\right ) \, \cos \left (3 \, a\right ) + \left (i - 1\right ) \, \sin \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3 i \, b} x\right ) + {\left (\left (i - 1\right ) \, \cos \left (3 \, a\right ) - \left (i + 1\right ) \, \sin \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-3 i \, b} x\right )\right )} b^{\frac {3}{2}} - 27 \, \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i + 1\right ) \, \cos \left (a\right ) - \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {i \, b} x\right ) + {\left (-\left (i - 1\right ) \, \cos \left (a\right ) + \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-i \, b} x\right )\right )} b^{\frac {3}{2}}}{576 \, b^{3}} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.38 \[ \int x^2 \cos ^3\left (a+b x^2\right ) \, dx=-\frac {i \, x e^{\left (3 i \, b x^{2} + 3 i \, a\right )}}{48 \, b} - \frac {3 i \, x e^{\left (i \, b x^{2} + i \, a\right )}}{16 \, b} + \frac {3 i \, x e^{\left (-i \, b x^{2} - i \, a\right )}}{16 \, b} + \frac {i \, x e^{\left (-3 i \, b x^{2} - 3 i \, a\right )}}{48 \, b} - \frac {\sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {6} \sqrt {b} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (3 i \, a\right )}}{288 \, b^{\frac {3}{2}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} - \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {2} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{32 \, b {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {2} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{32 \, b {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {\sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {6} \sqrt {b} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-3 i \, a\right )}}{288 \, b^{\frac {3}{2}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]
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Timed out. \[ \int x^2 \cos ^3\left (a+b x^2\right ) \, dx=\int x^2\,{\cos \left (b\,x^2+a\right )}^3 \,d x \]
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