Optimal. Leaf size=122 \[ -\frac{3 b \cos (a) \sqrt [3]{c+d x} \text{CosIntegral}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac{3 b \sin (a) \sqrt [3]{c+d x} \text{Si}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac{3 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}} \]
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Rubi [A] time = 0.145402, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {3435, 3381, 3379, 3297, 3303, 3299, 3302} \[ -\frac{3 b \cos (a) \sqrt [3]{c+d x} \text{CosIntegral}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac{3 b \sin (a) \sqrt [3]{c+d x} \text{Si}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac{3 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3435
Rule 3381
Rule 3379
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{\sqrt [3]{c e+d e x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin \left (a+\frac{b}{x^{2/3}}\right )}{\sqrt [3]{e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac{\sqrt [3]{c+d x} \operatorname{Subst}\left (\int \frac{\sin \left (a+\frac{b}{x^{2/3}}\right )}{\sqrt [3]{x}} \, dx,x,c+d x\right )}{d \sqrt [3]{e (c+d x)}}\\ &=-\frac{\left (3 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^2} \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ &=\frac{3 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}-\frac{\left (3 b \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x} \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ &=\frac{3 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}-\frac{\left (3 b \sqrt [3]{c+d x} \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac{\left (3 b \sqrt [3]{c+d x} \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ &=-\frac{3 b \sqrt [3]{c+d x} \cos (a) \text{Ci}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac{3 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac{3 b \sqrt [3]{c+d x} \sin (a) \text{Si}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.212919, size = 90, normalized size = 0.74 \[ \frac{3 \left (-b \cos (a) \sqrt [3]{c+d x} \text{CosIntegral}\left (\frac{b}{(c+d x)^{2/3}}\right )+b \sin (a) \sqrt [3]{c+d x} \text{Si}\left (\frac{b}{(c+d x)^{2/3}}\right )+(c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{e (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\sin \left ( a+{b \left ( dx+c \right ) ^{-{\frac{2}{3}}}} \right ){\frac{1}{\sqrt [3]{dex+ce}}}}\, dx \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sin \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{1}{3}} b}{d x + c}\right )}{{\left (d e x + c e\right )}^{\frac{1}{3}}}, x\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{\sin{\left (a + \frac{b}{\left (c + d x\right )^{\frac{2}{3}}} \right )}}{\sqrt [3]{e \left (c + d x\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 \frac{\sin \left (a + \frac{b}{{\left (d x + c\right )}^{\frac{2}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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