Optimal. Leaf size=73 \[ \frac{\sin (a) \text{CosIntegral}\left (b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{n}+\frac{\cos (a) \text{Si}\left (b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{n} \]
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Rubi [A] time = 0.150866, 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} \[ \frac{\sin (a) \text{CosIntegral}\left (b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{n}+\frac{\cos (a) \text{Si}\left (b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{n} \]
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^n\right )}}{x} \, dx &=\left (\csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}\right ) \int \frac{\sin \left (a+b x^n\right )}{x} \, dx\\ &=\left (\cos (a) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}\right ) \int \frac{\sin \left (b x^n\right )}{x} \, dx+\left (\csc \left (a+b x^n\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}\right ) \int \frac{\cos \left (b x^n\right )}{x} \, dx\\ &=\frac{\text{Ci}\left (b x^n\right ) \csc \left (a+b x^n\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{n}+\frac{\cos (a) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \text{Si}\left (b x^n\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0744186, size = 47, normalized size = 0.64 \[ \frac{\csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left (\sin (a) \text{CosIntegral}\left (b x^n\right )+\cos (a) \text{Si}\left (b x^n\right )\right )}{n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.144, size = 280, normalized size = 3.8 \begin{align*} -{\frac{{\it Ei} \left ( 1,-ib{x}^{n} \right ){{\rm e}^{i \left ( b{x}^{n}+2\,a \right ) }}}{ \left ( 2\,{{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-2 \right ) n}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( a+b{x}^{n} \right ) }}}}-{\frac{{\frac{i}{2}}{{\rm e}^{ib{x}^{n}}}\pi \,{\it csgn} \left ( b{x}^{n} \right ) }{ \left ({{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-1 \right ) n}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( a+b{x}^{n} \right ) }}}}+{\frac{i{{\rm e}^{ib{x}^{n}}}{\it Si} \left ( b{x}^{n} \right ) }{ \left ({{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-1 \right ) n}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( a+b{x}^{n} \right ) }}}}+{\frac{{{\rm e}^{ib{x}^{n}}}{\it Ei} \left ( 1,-ib{x}^{n} \right ) }{ \left ( 2\,{{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-2 \right ) n}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( a+b{x}^{n} \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77753, size = 293, normalized size = 4.01 \begin{align*} -\frac{4^{\frac{1}{3}}{\left (4^{\frac{2}{3}} \operatorname{Ci}\left (b x^{n}\right ) \sin \left (a\right ) + 4^{\frac{2}{3}} \operatorname{Ci}\left (-b x^{n}\right ) \sin \left (a\right ) + 2 \cdot 4^{\frac{2}{3}} \cos \left (a\right ) \operatorname{Si}\left (b x^{n}\right )\right )} \left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac{1}{3}} \sin \left (b x^{n} + a\right )}{8 \,{\left (n \cos \left (b x^{n} + a\right )^{2} - n\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^{n} \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^{n} + 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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