Optimal. Leaf size=155 \[ \frac{\sqrt{\frac{\pi }{2}} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^{3/2}}-\frac{x \cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b} \]
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Rubi [A] time = 0.2145, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6720, 3385, 3354, 3352, 3351} \[ \frac{\sqrt{\frac{\pi }{2}} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^{3/2}}-\frac{x \cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3385
Rule 3354
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int x^2 \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \, dx &=\left (\csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int x^2 \sin \left (a+b x^2\right ) \, dx\\ &=-\frac{x \cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b}+\frac{\left (\csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \cos \left (a+b x^2\right ) \, dx}{2 b}\\ &=-\frac{x \cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b}+\frac{\left (\cos (a) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \cos \left (b x^2\right ) \, dx}{2 b}-\frac{\left (\csc \left (a+b x^2\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \sin \left (b x^2\right ) \, dx}{2 b}\\ &=-\frac{x \cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b}+\frac{\sqrt{\frac{\pi }{2}} \cos (a) \csc \left (a+b x^2\right ) C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \csc \left (a+b x^2\right ) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.271106, size = 105, normalized size = 0.68 \[ -\frac{\csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (-\sqrt{2 \pi } \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )+\sqrt{2 \pi } \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )+2 \sqrt{b} x \cos \left (a+b x^2\right )\right )}{4 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.123, size = 240, normalized size = 1.6 \begin{align*}{\frac{1}{2\,{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-2}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }}} \left ({\frac{-{\frac{i}{2}}x{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}}{b}}+{\frac{{\frac{i}{4}}\sqrt{\pi }{{\rm e}^{i \left ( b{x}^{2}+2\,a \right ) }}}{b}{\it Erf} \left ( \sqrt{-ib}x \right ){\frac{1}{\sqrt{-ib}}}} \right ) }-{\frac{{\frac{i}{4}}x}{ \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) b}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }}}}+{\frac{{\frac{i}{8}}{{\rm e}^{ib{x}^{2}}}\sqrt{\pi }}{ \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) b}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }}}{\it Erf} \left ( \sqrt{ib}x \right ){\frac{1}{\sqrt{ib}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.71994, size = 481, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7084, size = 425, normalized size = 2.74 \begin{align*} -\frac{4^{\frac{1}{3}}{\left (4^{\frac{2}{3}} \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{C}\left (\sqrt{2} x \sqrt{\frac{b}{\pi }}\right ) \sin \left (b x^{2} + a\right ) - 4^{\frac{2}{3}} \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \operatorname{S}\left (\sqrt{2} x \sqrt{\frac{b}{\pi }}\right ) \sin \left (b x^{2} + a\right ) \sin \left (a\right ) - 2 \cdot 4^{\frac{2}{3}} b x \cos \left (b x^{2} + a\right ) \sin \left (b x^{2} + a\right )\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac{1}{3}}}{16 \,{\left (b^{2} \cos \left (b x^{2} + a\right )^{2} - b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt [3]{c \sin ^{3}{\left (a + b x^{2} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac{1}{3}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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