Optimal. Leaf size=566 \[ -\frac{b d \cos \left (a+\frac{b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text{CosIntegral}\left (\frac{b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}-\frac{(-1)^{2/3} b d \cos \left (a+\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}+\frac{\sqrt [3]{-1} b d \cos \left (a-\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}-\frac{b d \sin \left (a+\frac{b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text{Si}\left (\frac{b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}-\frac{(-1)^{2/3} b d \sin \left (a+\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text{Si}\left (\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}-\frac{\sqrt [3]{-1} b d \sin \left (a-\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{f} b}{\sqrt [3]{c f-d e}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}+\frac{(c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{(e+f x) (d e-c f)} \]
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Rubi [A] time = 2.62981, antiderivative size = 566, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3431, 3341, 3334, 3303, 3299, 3302} \[ -\frac{b d \cos \left (a+\frac{b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text{CosIntegral}\left (\frac{b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}-\frac{(-1)^{2/3} b d \cos \left (a+\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}+\frac{\sqrt [3]{-1} b d \cos \left (a-\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}-\frac{b d \sin \left (a+\frac{b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text{Si}\left (\frac{b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}-\frac{(-1)^{2/3} b d \sin \left (a+\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text{Si}\left (\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}-\frac{\sqrt [3]{-1} b d \sin \left (a-\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{f} b}{\sqrt [3]{c f-d e}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}+\frac{(c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{(e+f x) (d e-c f)} \]
Antiderivative was successfully verified.
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Rule 3431
Rule 3341
Rule 3334
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{(e+f x)^2} \, dx &=-\frac{3 \operatorname{Subst}\left (\int \frac{x^2 \sin (a+b x)}{\left (\frac{f}{d}+\left (e-\frac{c f}{d}\right ) x^3\right )^2} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=\frac{(c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{(d e-c f) (e+f x)}-\frac{b \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{\frac{f}{d}+\left (e-\frac{c f}{d}\right ) x^3} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d e-c f}\\ &=\frac{(c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{(d e-c f) (e+f x)}-\frac{b \operatorname{Subst}\left (\int \left (\frac{d \cos (a+b x)}{3 f^{2/3} \left (\sqrt [3]{f}-\sqrt [3]{-d e+c f} x\right )}+\frac{d \cos (a+b x)}{3 f^{2/3} \left (\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{-d e+c f} x\right )}+\frac{d \cos (a+b x)}{3 f^{2/3} \left (\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{-d e+c f} x\right )}\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d e-c f}\\ &=\frac{(c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{(d e-c f) (e+f x)}-\frac{(b d) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{\sqrt [3]{f}-\sqrt [3]{-d e+c f} x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (d e-c f)}-\frac{(b d) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{-d e+c f} x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (d e-c f)}-\frac{(b d) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{-d e+c f} x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (d e-c f)}\\ &=\frac{(c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{(d e-c f) (e+f x)}-\frac{\left (b d \cos \left (a+\frac{b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-b x\right )}{\sqrt [3]{f}-\sqrt [3]{-d e+c f} x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (d e-c f)}-\frac{\left (b d \cos \left (a-\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}+b x\right )}{\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{-d e+c f} x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (d e-c f)}-\frac{\left (b d \cos \left (a+\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-b x\right )}{\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{-d e+c f} x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (d e-c f)}-\frac{\left (b d \sin \left (a+\frac{b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-b x\right )}{\sqrt [3]{f}-\sqrt [3]{-d e+c f} x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (d e-c f)}+\frac{\left (b d \sin \left (a-\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}+b x\right )}{\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{-d e+c f} x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (d e-c f)}-\frac{\left (b d \sin \left (a+\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-b x\right )}{\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{-d e+c f} x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (d e-c f)}\\ &=-\frac{b d \cos \left (a+\frac{b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text{Ci}\left (\frac{b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}-\frac{(-1)^{2/3} b d \cos \left (a+\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}+\frac{\sqrt [3]{-1} b d \cos \left (a-\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}+\frac{(c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{(d e-c f) (e+f x)}-\frac{b d \sin \left (a+\frac{b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text{Si}\left (\frac{b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}-\frac{(-1)^{2/3} b d \sin \left (a+\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text{Si}\left (\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}-\frac{\sqrt [3]{-1} b d \sin \left (a-\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}\\ \end{align*}
Mathematica [C] time = 1.2292, size = 313, normalized size = 0.55 \[ \frac{(\cos (a)+i \sin (a)) \left (b d (e+f x) \text{RootSum}\left [\text{$\#$1}^3 f-c f+d e\& ,\frac{\text{Ei}\left (\frac{i b}{\sqrt [3]{c+d x}}\right )-e^{\frac{i b}{\text{$\#$1}}} \text{Ei}\left (i b \left (\frac{1}{\sqrt [3]{c+d x}}-\frac{1}{\text{$\#$1}}\right )\right )}{\text{$\#$1}}\& \right ]+(c+d x) \left (-3 f \sin \left (\frac{b}{\sqrt [3]{c+d x}}\right )+3 i f \cos \left (\frac{b}{\sqrt [3]{c+d x}}\right )\right )\right )+i \left (\cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )-i \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )\right ) \left (b d (e+f x) \left (\sin \left (\frac{b}{\sqrt [3]{c+d x}}\right )-i \cos \left (\frac{b}{\sqrt [3]{c+d x}}\right )\right ) \text{RootSum}\left [\text{$\#$1}^3 f-c f+d e\& ,\frac{\text{Ei}\left (-\frac{i b}{\sqrt [3]{c+d x}}\right )-e^{-\frac{i b}{\text{$\#$1}}} \text{Ei}\left (-i b \left (\frac{1}{\sqrt [3]{c+d x}}-\frac{1}{\text{$\#$1}}\right )\right )}{\text{$\#$1}}\& \right ]-3 c f-3 d f x\right )}{6 f (e+f x) (c f-d e)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.083, size = 1556, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (a + \frac{b}{{\left (d x + c\right )}^{\frac{1}{3}}}\right )}{{\left (f x + e\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.7331, size = 1908, normalized size = 3.37 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{3}}}\right )}{{\left (f x + e\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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