Integrand size = 14, antiderivative size = 51 \[ \int x^3 \cos ^2\left (a+b x^2\right ) \, dx=\frac {x^4}{8}+\frac {\cos ^2\left (a+b x^2\right )}{8 b^2}+\frac {x^2 \cos \left (a+b x^2\right ) \sin \left (a+b x^2\right )}{4 b} \]
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Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3461, 3391, 30} \[ \int x^3 \cos ^2\left (a+b x^2\right ) \, dx=\frac {\cos ^2\left (a+b x^2\right )}{8 b^2}+\frac {x^2 \sin \left (a+b x^2\right ) \cos \left (a+b x^2\right )}{4 b}+\frac {x^4}{8} \]
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Rule 30
Rule 3391
Rule 3461
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x \cos ^2(a+b x) \, dx,x,x^2\right ) \\ & = \frac {\cos ^2\left (a+b x^2\right )}{8 b^2}+\frac {x^2 \cos \left (a+b x^2\right ) \sin \left (a+b x^2\right )}{4 b}+\frac {1}{4} \text {Subst}\left (\int x \, dx,x,x^2\right ) \\ & = \frac {x^4}{8}+\frac {\cos ^2\left (a+b x^2\right )}{8 b^2}+\frac {x^2 \cos \left (a+b x^2\right ) \sin \left (a+b x^2\right )}{4 b} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.78 \[ \int x^3 \cos ^2\left (a+b x^2\right ) \, dx=\frac {\cos \left (2 \left (a+b x^2\right )\right )+2 b x^2 \left (b x^2+\sin \left (2 \left (a+b x^2\right )\right )\right )}{16 b^2} \]
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Time = 0.52 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {x^{4}}{8}+\frac {x^{2} \sin \left (2 b \,x^{2}+2 a \right )}{8 b}+\frac {\cos \left (2 b \,x^{2}+2 a \right )}{16 b^{2}}\) | \(42\) |
risch | \(\frac {x^{4}}{8}+\frac {x^{2} \sin \left (2 b \,x^{2}+2 a \right )}{8 b}+\frac {\cos \left (2 b \,x^{2}+2 a \right )}{16 b^{2}}\) | \(42\) |
parallelrisch | \(\frac {2 x^{4} b^{2}+2 x^{2} \sin \left (2 b \,x^{2}+2 a \right ) b +\cos \left (2 b \,x^{2}+2 a \right )-1}{16 b^{2}}\) | \(44\) |
norman | \(\frac {\frac {x^{4}}{8}+\frac {x^{4} \left (\tan ^{2}\left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )\right )}{4}+\frac {x^{4} \left (\tan ^{4}\left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )\right )}{8}+\frac {x^{2} \tan \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )}{2 b}-\frac {x^{2} \left (\tan ^{3}\left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )\right )}{2 b}-\frac {\tan ^{2}\left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )}{2 b^{2}}}{\left (1+\tan ^{2}\left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )\right )^{2}}\) | \(119\) |
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Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.88 \[ \int x^3 \cos ^2\left (a+b x^2\right ) \, dx=\frac {b^{2} x^{4} + 2 \, b x^{2} \cos \left (b x^{2} + a\right ) \sin \left (b x^{2} + a\right ) + \cos \left (b x^{2} + a\right )^{2}}{8 \, b^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.53 \[ \int x^3 \cos ^2\left (a+b x^2\right ) \, dx=\begin {cases} \frac {x^{4} \sin ^{2}{\left (a + b x^{2} \right )}}{8} + \frac {x^{4} \cos ^{2}{\left (a + b x^{2} \right )}}{8} + \frac {x^{2} \sin {\left (a + b x^{2} \right )} \cos {\left (a + b x^{2} \right )}}{4 b} + \frac {\cos ^{2}{\left (a + b x^{2} \right )}}{8 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{4} \cos ^{2}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82 \[ \int x^3 \cos ^2\left (a+b x^2\right ) \, dx=\frac {2 \, b^{2} x^{4} + 2 \, b x^{2} \sin \left (2 \, b x^{2} + 2 \, a\right ) + \cos \left (2 \, b x^{2} + 2 \, a\right )}{16 \, b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.49 \[ \int x^3 \cos ^2\left (a+b x^2\right ) \, dx=-\frac {{\left (2 \, b x^{2} + 2 \, a + \sin \left (2 \, b x^{2} + 2 \, a\right )\right )} a}{8 \, b^{2}} + \frac {2 \, {\left (b x^{2} + a\right )}^{2} + 2 \, {\left (b x^{2} + a\right )} \sin \left (2 \, b x^{2} + 2 \, a\right ) + \cos \left (2 \, b x^{2} + 2 \, a\right )}{16 \, b^{2}} \]
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Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80 \[ \int x^3 \cos ^2\left (a+b x^2\right ) \, dx=\frac {\cos \left (2\,b\,x^2+2\,a\right )}{16\,b^2}+\frac {x^4}{8}+\frac {x^2\,\sin \left (2\,b\,x^2+2\,a\right )}{8\,b} \]
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