Optimal. Leaf size=73 \[ \frac{1}{2} \sin (a) \text{CosIntegral}\left (b x^2\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}+\frac{1}{2} \cos (a) \text{Si}\left (b x^2\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \]
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Rubi [A] time = 0.120985, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6720, 3377, 3376, 3375} \[ \frac{1}{2} \sin (a) \text{CosIntegral}\left (b x^2\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}+\frac{1}{2} \cos (a) \text{Si}\left (b x^2\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3377
Rule 3376
Rule 3375
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x} \, dx &=\left (\csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \frac{\sin \left (a+b x^2\right )}{x} \, dx\\ &=\left (\cos (a) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \frac{\sin \left (b x^2\right )}{x} \, dx+\left (\csc \left (a+b x^2\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \frac{\cos \left (b x^2\right )}{x} \, dx\\ &=\frac{1}{2} \text{Ci}\left (b x^2\right ) \csc \left (a+b x^2\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}+\frac{1}{2} \cos (a) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \text{Si}\left (b x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0584489, size = 47, normalized size = 0.64 \[ \frac{1}{2} \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (\sin (a) \text{CosIntegral}\left (b x^2\right )+\cos (a) \text{Si}\left (b x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.083, size = 268, normalized size = 3.7 \begin{align*} -{\frac{{\it Ei} \left ( 1,-ib{x}^{2} \right ){{\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 ) }}}}-{\frac{{\frac{i}{4}}{{\rm e}^{ib{x}^{2}}}\pi \,{\it csgn} \left ( b{x}^{2} \right ) }{{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1}\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}{2}}{{\rm e}^{ib{x}^{2}}}{\it Si} \left ( b{x}^{2} \right ) }{{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1}\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{{{\rm e}^{ib{x}^{2}}}{\it Ei} \left ( 1,-ib{x}^{2} \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 ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.69687, size = 63, normalized size = 0.86 \begin{align*} \frac{1}{8} \,{\left ({\left (i \,{\rm Ei}\left (i \, b x^{2}\right ) - i \,{\rm Ei}\left (-i \, b x^{2}\right )\right )} \cos \left (a\right ) -{\left ({\rm Ei}\left (i \, b x^{2}\right ) +{\rm Ei}\left (-i \, b x^{2}\right )\right )} \sin \left (a\right )\right )} c^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74076, size = 285, normalized size = 3.9 \begin{align*} -\frac{4^{\frac{1}{3}}{\left (2 \cdot 4^{\frac{2}{3}} \cos \left (a\right ) \operatorname{Si}\left (b x^{2}\right ) +{\left (4^{\frac{2}{3}} \operatorname{Ci}\left (b x^{2}\right ) + 4^{\frac{2}{3}} \operatorname{Ci}\left (-b x^{2}\right )\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}} \sin \left (b x^{2} + a\right )}{16 \,{\left (\cos \left (b x^{2} + a\right )^{2} - 1\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 \frac{\sqrt [3]{c \sin ^{3}{\left (a + b x^{2} \right )}}}{x}\, 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 \frac{\left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac{1}{3}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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