Optimal. Leaf size=107 \[ \frac{a^2 x^6}{6}+\frac{2 a b \sin \left (c+d x^3\right )}{3 d^2}-\frac{2 a b x^3 \cos \left (c+d x^3\right )}{3 d}+\frac{b^2 \sin ^2\left (c+d x^3\right )}{12 d^2}-\frac{b^2 x^3 \sin \left (c+d x^3\right ) \cos \left (c+d x^3\right )}{6 d}+\frac{b^2 x^6}{12} \]
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Rubi [A] time = 0.132765, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3379, 3317, 3296, 2637, 3310, 30} \[ \frac{a^2 x^6}{6}+\frac{2 a b \sin \left (c+d x^3\right )}{3 d^2}-\frac{2 a b x^3 \cos \left (c+d x^3\right )}{3 d}+\frac{b^2 \sin ^2\left (c+d x^3\right )}{12 d^2}-\frac{b^2 x^3 \sin \left (c+d x^3\right ) \cos \left (c+d x^3\right )}{6 d}+\frac{b^2 x^6}{12} \]
Antiderivative was successfully verified.
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Rule 3379
Rule 3317
Rule 3296
Rule 2637
Rule 3310
Rule 30
Rubi steps
\begin{align*} \int x^5 \left (a+b \sin \left (c+d x^3\right )\right )^2 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x (a+b \sin (c+d x))^2 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (a^2 x+2 a b x \sin (c+d x)+b^2 x \sin ^2(c+d x)\right ) \, dx,x,x^3\right )\\ &=\frac{a^2 x^6}{6}+\frac{1}{3} (2 a b) \operatorname{Subst}\left (\int x \sin (c+d x) \, dx,x,x^3\right )+\frac{1}{3} b^2 \operatorname{Subst}\left (\int x \sin ^2(c+d x) \, dx,x,x^3\right )\\ &=\frac{a^2 x^6}{6}-\frac{2 a b x^3 \cos \left (c+d x^3\right )}{3 d}-\frac{b^2 x^3 \cos \left (c+d x^3\right ) \sin \left (c+d x^3\right )}{6 d}+\frac{b^2 \sin ^2\left (c+d x^3\right )}{12 d^2}+\frac{1}{6} b^2 \operatorname{Subst}\left (\int x \, dx,x,x^3\right )+\frac{(2 a b) \operatorname{Subst}\left (\int \cos (c+d x) \, dx,x,x^3\right )}{3 d}\\ &=\frac{a^2 x^6}{6}+\frac{b^2 x^6}{12}-\frac{2 a b x^3 \cos \left (c+d x^3\right )}{3 d}+\frac{2 a b \sin \left (c+d x^3\right )}{3 d^2}-\frac{b^2 x^3 \cos \left (c+d x^3\right ) \sin \left (c+d x^3\right )}{6 d}+\frac{b^2 \sin ^2\left (c+d x^3\right )}{12 d^2}\\ \end{align*}
Mathematica [A] time = 0.281155, size = 92, normalized size = 0.86 \[ \frac{4 a^2 d^2 x^6+16 a b \sin \left (c+d x^3\right )-16 a b d x^3 \cos \left (c+d x^3\right )-2 b^2 d x^3 \sin \left (2 \left (c+d x^3\right )\right )-b^2 \cos \left (2 \left (c+d x^3\right )\right )+2 b^2 d^2 x^6}{24 d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.095, size = 137, normalized size = 1.3 \begin{align*}{\frac{{a}^{2}{x}^{6}}{6}}+{\frac{{b}^{2}{x}^{6}}{6}}-{\frac{{b}^{2}}{2} \left ({\frac{{x}^{6}}{6}}+{\frac{1}{1+ \left ( \tan \left ( d{x}^{3}+c \right ) \right ) ^{2}} \left ({\frac{1}{6\,{d}^{2}}}+{\frac{{x}^{3}\tan \left ( d{x}^{3}+c \right ) }{3\,d}} \right ) } \right ) }+{\frac{1}{2} \left ({\frac{8\,ab}{3\,{d}^{2}}\tan \left ({\frac{d{x}^{3}}{2}}+{\frac{c}{2}} \right ) }-{\frac{4\,ab{x}^{3}}{3\,d}}+{\frac{4\,ab{x}^{3}}{3\,d} \left ( \tan \left ({\frac{d{x}^{3}}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{d{x}^{3}}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01646, size = 117, normalized size = 1.09 \begin{align*} \frac{1}{6} \, a^{2} x^{6} - \frac{2 \,{\left (d x^{3} \cos \left (d x^{3} + c\right ) - \sin \left (d x^{3} + c\right )\right )} a b}{3 \, d^{2}} + \frac{{\left (2 \, d^{2} x^{6} - 2 \, d x^{3} \sin \left (2 \, d x^{3} + 2 \, c\right ) - \cos \left (2 \, d x^{3} + 2 \, c\right )\right )} b^{2}}{24 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76182, size = 189, normalized size = 1.77 \begin{align*} \frac{{\left (2 \, a^{2} + b^{2}\right )} d^{2} x^{6} - 8 \, a b d x^{3} \cos \left (d x^{3} + c\right ) - b^{2} \cos \left (d x^{3} + c\right )^{2} - 2 \,{\left (b^{2} d x^{3} \cos \left (d x^{3} + c\right ) - 4 \, a b\right )} \sin \left (d x^{3} + c\right )}{12 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.30871, size = 143, normalized size = 1.34 \begin{align*} \begin{cases} \frac{a^{2} x^{6}}{6} - \frac{2 a b x^{3} \cos{\left (c + d x^{3} \right )}}{3 d} + \frac{2 a b \sin{\left (c + d x^{3} \right )}}{3 d^{2}} + \frac{b^{2} x^{6} \sin ^{2}{\left (c + d x^{3} \right )}}{12} + \frac{b^{2} x^{6} \cos ^{2}{\left (c + d x^{3} \right )}}{12} - \frac{b^{2} x^{3} \sin{\left (c + d x^{3} \right )} \cos{\left (c + d x^{3} \right )}}{6 d} - \frac{b^{2} \cos ^{2}{\left (c + d x^{3} \right )}}{12 d^{2}} & \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 = 1.10524, size = 166, normalized size = 1.55 \begin{align*} \frac{\frac{4 \,{\left ({\left (d x^{3} + c\right )}^{2} - 2 \,{\left (d x^{3} + c\right )} c\right )} a^{2}}{d} - \frac{16 \,{\left (d x^{3} \cos \left (d x^{3} + c\right ) - \sin \left (d x^{3} + c\right )\right )} a b}{d} - \frac{{\left (2 \, d x^{3} \sin \left (2 \, d x^{3} + 2 \, c\right ) - 2 \,{\left (d x^{3} + c\right )}^{2} + 4 \,{\left (d x^{3} + c\right )} c + \cos \left (2 \, d x^{3} + 2 \, c\right )\right )} b^{2}}{d}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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