Optimal. Leaf size=117 \[ \frac{\sqrt{\frac{\pi }{2}} \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt{b}}+\frac{\sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt{b}} \]
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Rubi [A] time = 0.0586227, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6720, 3353, 3352, 3351} \[ \frac{\sqrt{\frac{\pi }{2}} \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt{b}}+\frac{\sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3353
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int \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 \sin \left (a+b x^2\right ) \, dx\\ &=\left (\cos (a) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \sin \left (b x^2\right ) \, dx+\left (\csc \left (a+b x^2\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \cos \left (b x^2\right ) \, dx\\ &=\frac{\sqrt{\frac{\pi }{2}} \cos (a) \csc \left (a+b x^2\right ) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt{b}}+\frac{\sqrt{\frac{\pi }{2}} \csc \left (a+b x^2\right ) C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.12134, size = 80, normalized size = 0.68 \[ \frac{\sqrt{\frac{\pi }{2}} \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (\sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )+\cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.083, size = 157, normalized size = 1.3 \begin{align*}{\frac{\sqrt{\pi }{{\rm e}^{i \left ( b{x}^{2}+2\,a \right ) }}}{4\,{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-4}\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}}}}-{\frac{{{\rm e}^{ib{x}^{2}}}\sqrt{\pi }}{4\,{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-4}\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 = 1.71268, size = 444, normalized size = 3.79 \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 = 1.81931, size = 356, normalized size = 3.04 \begin{align*} -\frac{4^{\frac{1}{3}}{\left (4^{\frac{2}{3}} \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{S}\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{C}\left (\sqrt{2} x \sqrt{\frac{b}{\pi }}\right ) \sin \left (b x^{2} + a\right ) \sin \left (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}}}{8 \,{\left (b \cos \left (b x^{2} + a\right )^{2} - b\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 \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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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