Optimal. Leaf size=555 \[ -\frac{\sqrt [3]{-1} b d \cos \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac{b d \cos \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac{(-1)^{2/3} b d \cos \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{\sqrt [3]{-1} b d \sin \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{b d \sin \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{\sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{(-1)^{2/3} b d \sin \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{\sin \left (a+b \sqrt [3]{c+d x}\right )}{f (e+f x)} \]
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Rubi [A] time = 2.12414, antiderivative size = 555, 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} \[ -\frac{\sqrt [3]{-1} b d \cos \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac{b d \cos \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac{(-1)^{2/3} b d \cos \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{\sqrt [3]{-1} b d \sin \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{b d \sin \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{\sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{(-1)^{2/3} b d \sin \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{\sin \left (a+b \sqrt [3]{c+d x}\right )}{f (e+f x)} \]
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+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 (e-\frac{c f}{d}+\frac{f x^3}{d}\right )^2} \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=-\frac{\sin \left (a+b \sqrt [3]{c+d x}\right )}{f (e+f x)}+\frac{b \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{e-\frac{c f}{d}+\frac{f x^3}{d}} \, dx,x,\sqrt [3]{c+d x}\right )}{f}\\ &=-\frac{\sin \left (a+b \sqrt [3]{c+d x}\right )}{f (e+f x)}+\frac{b \operatorname{Subst}\left (\int \left (-\frac{\sqrt [3]{d e-c f} \cos (a+b x)}{3 \left (e-\frac{c f}{d}\right ) \left (-\sqrt [3]{d e-c f}-\sqrt [3]{f} x\right )}-\frac{\sqrt [3]{d e-c f} \cos (a+b x)}{3 \left (e-\frac{c f}{d}\right ) \left (-\sqrt [3]{d e-c f}+\sqrt [3]{-1} \sqrt [3]{f} x\right )}-\frac{\sqrt [3]{d e-c f} \cos (a+b x)}{3 \left (e-\frac{c f}{d}\right ) \left (-\sqrt [3]{d e-c f}-(-1)^{2/3} \sqrt [3]{f} x\right )}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{f}\\ &=-\frac{\sin \left (a+b \sqrt [3]{c+d x}\right )}{f (e+f x)}-\frac{(b d) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{-\sqrt [3]{d e-c f}-\sqrt [3]{f} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 f (d e-c f)^{2/3}}-\frac{(b d) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{-\sqrt [3]{d e-c f}+\sqrt [3]{-1} \sqrt [3]{f} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 f (d e-c f)^{2/3}}-\frac{(b d) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{-\sqrt [3]{d e-c f}-(-1)^{2/3} \sqrt [3]{f} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 f (d e-c f)^{2/3}}\\ &=-\frac{\sin \left (a+b \sqrt [3]{c+d x}\right )}{f (e+f x)}-\frac{\left (b d \cos \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b x\right )}{-\sqrt [3]{d e-c f}-\sqrt [3]{f} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 f (d e-c f)^{2/3}}-\frac{\left (b d \cos \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b x\right )}{-\sqrt [3]{d e-c f}-(-1)^{2/3} \sqrt [3]{f} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 f (d e-c f)^{2/3}}-\frac{\left (b d \cos \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b x\right )}{-\sqrt [3]{d e-c f}+\sqrt [3]{-1} \sqrt [3]{f} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 f (d e-c f)^{2/3}}+\frac{\left (b d \sin \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b x\right )}{-\sqrt [3]{d e-c f}-\sqrt [3]{f} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 f (d e-c f)^{2/3}}-\frac{\left (b d \sin \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b x\right )}{-\sqrt [3]{d e-c f}-(-1)^{2/3} \sqrt [3]{f} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 f (d e-c f)^{2/3}}+\frac{\left (b d \sin \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b x\right )}{-\sqrt [3]{d e-c f}+\sqrt [3]{-1} \sqrt [3]{f} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 f (d e-c f)^{2/3}}\\ &=-\frac{\sqrt [3]{-1} b d \cos \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac{b d \cos \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Ci}\left (\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac{(-1)^{2/3} b d \cos \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{\sin \left (a+b \sqrt [3]{c+d x}\right )}{f (e+f x)}-\frac{\sqrt [3]{-1} b d \sin \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{b d \sin \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{(-1)^{2/3} b d \sin \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}\\ \end{align*}
Mathematica [C] time = 1.11035, size = 180, normalized size = 0.32 \[ \frac{b d \text{RootSum}\left [\text{$\#$1}^3 f-c f+d e\& ,\frac{e^{-i \text{$\#$1} b-i a} \text{Ei}\left (-i b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )}{\text{$\#$1}^2}\& \right ]+b d \text{RootSum}\left [\text{$\#$1}^3 f-c f+d e\& ,\frac{e^{i \text{$\#$1} b+i a} \text{Ei}\left (i b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )}{\text{$\#$1}^2}\& \right ]+\frac{3 i f e^{-i \left (a+b \sqrt [3]{c+d x}\right )} \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right )}{e+f x}}{6 f^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.067, size = 1175, normalized size = 2.1 \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(-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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Fricas [C] time = 2.12379, size = 1789, normalized size = 3.22 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (a + b \sqrt [3]{c + d x} \right )}}{\left (e + f x\right )^{2}}\, 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 ({\left (d x + c\right )}^{\frac{1}{3}} b + a\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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