Optimal. Leaf size=163 \[ \frac{a^2 x^6}{6}+\frac{2 a b x^2 \sin \left (c+d x^2\right )}{d^2}+\frac{2 a b \cos \left (c+d x^2\right )}{d^3}-\frac{a b x^4 \cos \left (c+d x^2\right )}{d}+\frac{b^2 x^2 \sin ^2\left (c+d x^2\right )}{4 d^2}+\frac{b^2 \sin \left (c+d x^2\right ) \cos \left (c+d x^2\right )}{8 d^3}-\frac{b^2 x^4 \sin \left (c+d x^2\right ) \cos \left (c+d x^2\right )}{4 d}-\frac{b^2 x^2}{8 d^2}+\frac{b^2 x^6}{12} \]
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Rubi [A] time = 0.246997, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {3379, 3317, 3296, 2638, 3311, 30, 2635, 8} \[ \frac{a^2 x^6}{6}+\frac{2 a b x^2 \sin \left (c+d x^2\right )}{d^2}+\frac{2 a b \cos \left (c+d x^2\right )}{d^3}-\frac{a b x^4 \cos \left (c+d x^2\right )}{d}+\frac{b^2 x^2 \sin ^2\left (c+d x^2\right )}{4 d^2}+\frac{b^2 \sin \left (c+d x^2\right ) \cos \left (c+d x^2\right )}{8 d^3}-\frac{b^2 x^4 \sin \left (c+d x^2\right ) \cos \left (c+d x^2\right )}{4 d}-\frac{b^2 x^2}{8 d^2}+\frac{b^2 x^6}{12} \]
Antiderivative was successfully verified.
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Rule 3379
Rule 3317
Rule 3296
Rule 2638
Rule 3311
Rule 30
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int x^5 \left (a+b \sin \left (c+d x^2\right )\right )^2 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b \sin (c+d x))^2 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^2 x^2+2 a b x^2 \sin (c+d x)+b^2 x^2 \sin ^2(c+d x)\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 x^6}{6}+(a b) \operatorname{Subst}\left (\int x^2 \sin (c+d x) \, dx,x,x^2\right )+\frac{1}{2} b^2 \operatorname{Subst}\left (\int x^2 \sin ^2(c+d x) \, dx,x,x^2\right )\\ &=\frac{a^2 x^6}{6}-\frac{a b x^4 \cos \left (c+d x^2\right )}{d}-\frac{b^2 x^4 \cos \left (c+d x^2\right ) \sin \left (c+d x^2\right )}{4 d}+\frac{b^2 x^2 \sin ^2\left (c+d x^2\right )}{4 d^2}+\frac{1}{4} b^2 \operatorname{Subst}\left (\int x^2 \, dx,x,x^2\right )-\frac{b^2 \operatorname{Subst}\left (\int \sin ^2(c+d x) \, dx,x,x^2\right )}{4 d^2}+\frac{(2 a b) \operatorname{Subst}\left (\int x \cos (c+d x) \, dx,x,x^2\right )}{d}\\ &=\frac{a^2 x^6}{6}+\frac{b^2 x^6}{12}-\frac{a b x^4 \cos \left (c+d x^2\right )}{d}+\frac{2 a b x^2 \sin \left (c+d x^2\right )}{d^2}+\frac{b^2 \cos \left (c+d x^2\right ) \sin \left (c+d x^2\right )}{8 d^3}-\frac{b^2 x^4 \cos \left (c+d x^2\right ) \sin \left (c+d x^2\right )}{4 d}+\frac{b^2 x^2 \sin ^2\left (c+d x^2\right )}{4 d^2}-\frac{(2 a b) \operatorname{Subst}\left (\int \sin (c+d x) \, dx,x,x^2\right )}{d^2}-\frac{b^2 \operatorname{Subst}\left (\int 1 \, dx,x,x^2\right )}{8 d^2}\\ &=-\frac{b^2 x^2}{8 d^2}+\frac{a^2 x^6}{6}+\frac{b^2 x^6}{12}+\frac{2 a b \cos \left (c+d x^2\right )}{d^3}-\frac{a b x^4 \cos \left (c+d x^2\right )}{d}+\frac{2 a b x^2 \sin \left (c+d x^2\right )}{d^2}+\frac{b^2 \cos \left (c+d x^2\right ) \sin \left (c+d x^2\right )}{8 d^3}-\frac{b^2 x^4 \cos \left (c+d x^2\right ) \sin \left (c+d x^2\right )}{4 d}+\frac{b^2 x^2 \sin ^2\left (c+d x^2\right )}{4 d^2}\\ \end{align*}
Mathematica [A] time = 0.389886, size = 122, normalized size = 0.75 \[ \frac{8 a^2 d^3 x^6-48 a b \left (d^2 x^4-2\right ) \cos \left (c+d x^2\right )+96 a b d x^2 \sin \left (c+d x^2\right )-6 b^2 d^2 x^4 \sin \left (2 \left (c+d x^2\right )\right )+3 b^2 \sin \left (2 \left (c+d x^2\right )\right )-6 b^2 d x^2 \cos \left (2 \left (c+d x^2\right )\right )+4 b^2 d^3 x^6}{48 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 140, normalized size = 0.9 \begin{align*}{\frac{{a}^{2}{x}^{6}}{6}}+{\frac{{b}^{2}{x}^{6}}{12}}-{\frac{{b}^{2}}{2} \left ({\frac{{x}^{4}\sin \left ( 2\,d{x}^{2}+2\,c \right ) }{4\,d}}-{\frac{1}{d} \left ( -{\frac{{x}^{2}\cos \left ( 2\,d{x}^{2}+2\,c \right ) }{4\,d}}+{\frac{\sin \left ( 2\,d{x}^{2}+2\,c \right ) }{8\,{d}^{2}}} \right ) } \right ) }+2\,ab \left ( -1/2\,{\frac{{x}^{4}\cos \left ( d{x}^{2}+c \right ) }{d}}+2\,{\frac{1}{d} \left ( 1/2\,{\frac{{x}^{2}\sin \left ( d{x}^{2}+c \right ) }{d}}+1/2\,{\frac{\cos \left ( d{x}^{2}+c \right ) }{{d}^{2}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03411, size = 143, normalized size = 0.88 \begin{align*} \frac{1}{6} \, a^{2} x^{6} + \frac{{\left (2 \, d x^{2} \sin \left (d x^{2} + c\right ) -{\left (d^{2} x^{4} - 2\right )} \cos \left (d x^{2} + c\right )\right )} a b}{d^{3}} + \frac{{\left (4 \, d^{3} x^{6} - 6 \, d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) - 3 \,{\left (2 \, d^{2} x^{4} - 1\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} b^{2}}{48 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05386, size = 265, normalized size = 1.63 \begin{align*} \frac{2 \,{\left (2 \, a^{2} + b^{2}\right )} d^{3} x^{6} - 6 \, b^{2} d x^{2} \cos \left (d x^{2} + c\right )^{2} + 3 \, b^{2} d x^{2} - 24 \,{\left (a b d^{2} x^{4} - 2 \, a b\right )} \cos \left (d x^{2} + c\right ) + 3 \,{\left (16 \, a b d x^{2} -{\left (2 \, b^{2} d^{2} x^{4} - b^{2}\right )} \cos \left (d x^{2} + c\right )\right )} \sin \left (d x^{2} + c\right )}{24 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.29998, size = 209, normalized size = 1.28 \begin{align*} \begin{cases} \frac{a^{2} x^{6}}{6} - \frac{a b x^{4} \cos{\left (c + d x^{2} \right )}}{d} + \frac{2 a b x^{2} \sin{\left (c + d x^{2} \right )}}{d^{2}} + \frac{2 a b \cos{\left (c + d x^{2} \right )}}{d^{3}} + \frac{b^{2} x^{6} \sin ^{2}{\left (c + d x^{2} \right )}}{12} + \frac{b^{2} x^{6} \cos ^{2}{\left (c + d x^{2} \right )}}{12} - \frac{b^{2} x^{4} \sin{\left (c + d x^{2} \right )} \cos{\left (c + d x^{2} \right )}}{4 d} + \frac{b^{2} x^{2} \sin ^{2}{\left (c + d x^{2} \right )}}{8 d^{2}} - \frac{b^{2} x^{2} \cos ^{2}{\left (c + d x^{2} \right )}}{8 d^{2}} + \frac{b^{2} \sin{\left (c + d x^{2} \right )} \cos{\left (c + d x^{2} \right )}}{8 d^{3}} & \text{for}\: d \neq 0 \\\frac{x^{6} \left (a + b \sin{\left (c \right )}\right )^{2}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11926, size = 244, normalized size = 1.5 \begin{align*} \frac{8 \, a^{2} d x^{6} + 48 \,{\left (\frac{2 \, x^{2} \sin \left (d x^{2} + c\right )}{d} - \frac{{\left ({\left (d x^{2} + c\right )}^{2} - 2 \,{\left (d x^{2} + c\right )} c + c^{2} - 2\right )} \cos \left (d x^{2} + c\right )}{d^{2}}\right )} a b -{\left (\frac{6 \, x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right )}{d} + \frac{3 \,{\left (2 \,{\left (d x^{2} + c\right )}^{2} - 4 \,{\left (d x^{2} + c\right )} c + 2 \, c^{2} - 1\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )}{d^{2}} - \frac{4 \,{\left ({\left (d x^{2} + c\right )}^{3} - 3 \,{\left (d x^{2} + c\right )}^{2} c + 3 \,{\left (d x^{2} + c\right )} c^{2}\right )}}{d^{2}}\right )} b^{2}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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