Optimal. Leaf size=117 \[ -\frac {\cos ^3\left (a+b x^2\right )}{27 b^3}+\frac {7 \cos \left (a+b x^2\right )}{9 b^3}+\frac {x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}+\frac {2 x^2 \sin \left (a+b x^2\right )}{3 b^2}-\frac {x^4 \cos \left (a+b x^2\right )}{3 b}-\frac {x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{6 b} \]
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Rubi [A] time = 0.13, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3379, 3311, 3296, 2638, 2633} \[ \frac {x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}+\frac {2 x^2 \sin \left (a+b x^2\right )}{3 b^2}-\frac {\cos ^3\left (a+b x^2\right )}{27 b^3}+\frac {7 \cos \left (a+b x^2\right )}{9 b^3}-\frac {x^4 \cos \left (a+b x^2\right )}{3 b}-\frac {x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{6 b} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 2638
Rule 3296
Rule 3311
Rule 3379
Rubi steps
\begin {align*} \int x^5 \sin ^3\left (a+b x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 \sin ^3(a+b x) \, dx,x,x^2\right )\\ &=-\frac {x^4 \cos \left (a+b x^2\right ) \sin ^2\left (a+b x^2\right )}{6 b}+\frac {x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}+\frac {1}{3} \operatorname {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,x^2\right )-\frac {\operatorname {Subst}\left (\int \sin ^3(a+b x) \, dx,x,x^2\right )}{9 b^2}\\ &=-\frac {x^4 \cos \left (a+b x^2\right )}{3 b}-\frac {x^4 \cos \left (a+b x^2\right ) \sin ^2\left (a+b x^2\right )}{6 b}+\frac {x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}+\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (a+b x^2\right )\right )}{9 b^3}+\frac {2 \operatorname {Subst}\left (\int x \cos (a+b x) \, dx,x,x^2\right )}{3 b}\\ &=\frac {\cos \left (a+b x^2\right )}{9 b^3}-\frac {x^4 \cos \left (a+b x^2\right )}{3 b}-\frac {\cos ^3\left (a+b x^2\right )}{27 b^3}+\frac {2 x^2 \sin \left (a+b x^2\right )}{3 b^2}-\frac {x^4 \cos \left (a+b x^2\right ) \sin ^2\left (a+b x^2\right )}{6 b}+\frac {x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}-\frac {2 \operatorname {Subst}\left (\int \sin (a+b x) \, dx,x,x^2\right )}{3 b^2}\\ &=\frac {7 \cos \left (a+b x^2\right )}{9 b^3}-\frac {x^4 \cos \left (a+b x^2\right )}{3 b}-\frac {\cos ^3\left (a+b x^2\right )}{27 b^3}+\frac {2 x^2 \sin \left (a+b x^2\right )}{3 b^2}-\frac {x^4 \cos \left (a+b x^2\right ) \sin ^2\left (a+b x^2\right )}{6 b}+\frac {x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 75, normalized size = 0.64 \[ \frac {-81 \left (b^2 x^4-2\right ) \cos \left (a+b x^2\right )+\left (9 b^2 x^4-2\right ) \cos \left (3 \left (a+b x^2\right )\right )-6 b x^2 \left (\sin \left (3 \left (a+b x^2\right )\right )-27 \sin \left (a+b x^2\right )\right )}{216 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 79, normalized size = 0.68 \[ \frac {{\left (9 \, b^{2} x^{4} - 2\right )} \cos \left (b x^{2} + a\right )^{3} - 3 \, {\left (9 \, b^{2} x^{4} - 14\right )} \cos \left (b x^{2} + a\right ) - 6 \, {\left (b x^{2} \cos \left (b x^{2} + a\right )^{2} - 7 \, b x^{2}\right )} \sin \left (b x^{2} + a\right )}{54 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 122, normalized size = 1.04 \[ -\frac {\frac {6 \, x^{2} \sin \left (3 \, b x^{2} + 3 \, a\right )}{b} - \frac {162 \, x^{2} \sin \left (b x^{2} + a\right )}{b} - \frac {{\left (9 \, {\left (b x^{2} + a\right )}^{2} - 18 \, {\left (b x^{2} + a\right )} a + 9 \, a^{2} - 2\right )} \cos \left (3 \, b x^{2} + 3 \, a\right )}{b^{2}} + \frac {81 \, {\left ({\left (b x^{2} + a\right )}^{2} - 2 \, {\left (b x^{2} + a\right )} a + a^{2} - 2\right )} \cos \left (b x^{2} + a\right )}{b^{2}}}{216 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 113, normalized size = 0.97 \[ -\frac {3 x^{4} \cos \left (b \,x^{2}+a \right )}{8 b}+\frac {\frac {3 x^{2} \sin \left (b \,x^{2}+a \right )}{4 b}+\frac {3 \cos \left (b \,x^{2}+a \right )}{4 b^{2}}}{b}+\frac {x^{4} \cos \left (3 b \,x^{2}+3 a \right )}{24 b}-\frac {\frac {x^{2} \sin \left (3 b \,x^{2}+3 a \right )}{6 b}+\frac {\cos \left (3 b \,x^{2}+3 a \right )}{18 b^{2}}}{6 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 79, normalized size = 0.68 \[ -\frac {6 \, b x^{2} \sin \left (3 \, b x^{2} + 3 \, a\right ) - 162 \, b x^{2} \sin \left (b x^{2} + a\right ) - {\left (9 \, b^{2} x^{4} - 2\right )} \cos \left (3 \, b x^{2} + 3 \, a\right ) + 81 \, {\left (b^{2} x^{4} - 2\right )} \cos \left (b x^{2} + a\right )}{216 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.97, size = 94, normalized size = 0.80 \[ \frac {\frac {3\,\cos \left (b\,x^2+a\right )}{4}-\frac {\cos \left (3\,b\,x^2+3\,a\right )}{108}+b\,\left (\frac {3\,x^2\,\sin \left (b\,x^2+a\right )}{4}-\frac {x^2\,\sin \left (3\,b\,x^2+3\,a\right )}{36}\right )+b^2\,\left (\frac {x^4\,\cos \left (3\,b\,x^2+3\,a\right )}{24}-\frac {3\,x^4\,\cos \left (b\,x^2+a\right )}{8}\right )}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.33, size = 143, normalized size = 1.22 \[ \begin {cases} - \frac {x^{4} \sin ^{2}{\left (a + b x^{2} \right )} \cos {\left (a + b x^{2} \right )}}{2 b} - \frac {x^{4} \cos ^{3}{\left (a + b x^{2} \right )}}{3 b} + \frac {7 x^{2} \sin ^{3}{\left (a + b x^{2} \right )}}{9 b^{2}} + \frac {2 x^{2} \sin {\left (a + b x^{2} \right )} \cos ^{2}{\left (a + b x^{2} \right )}}{3 b^{2}} + \frac {7 \sin ^{2}{\left (a + b x^{2} \right )} \cos {\left (a + b x^{2} \right )}}{9 b^{3}} + \frac {20 \cos ^{3}{\left (a + b x^{2} \right )}}{27 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{6} \sin ^{3}{\relax (a )}}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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