Optimal. Leaf size=380 \[ \frac{\sqrt [3]{b} \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{4/3}}+\frac{(-1)^{2/3} \sqrt [3]{b} \sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{b} \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{b} \cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{b} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{4/3}}+\frac{d \cos (c) \text{CosIntegral}(d x)}{a}-\frac{d \sin (c) \text{Si}(d x)}{a}-\frac{\sin (c+d x)}{a x} \]
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Rubi [A] time = 0.608659, antiderivative size = 380, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3345, 3297, 3303, 3299, 3302} \[ \frac{\sqrt [3]{b} \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{4/3}}+\frac{(-1)^{2/3} \sqrt [3]{b} \sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{b} \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{b} \cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{b} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{4/3}}+\frac{d \cos (c) \text{CosIntegral}(d x)}{a}-\frac{d \sin (c) \text{Si}(d x)}{a}-\frac{\sin (c+d x)}{a x} \]
Antiderivative was successfully verified.
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Rule 3345
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\sin (c+d x)}{x^2 \left (a+b x^3\right )} \, dx &=\int \left (\frac{\sin (c+d x)}{a x^2}-\frac{b x \sin (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx\\ &=\frac{\int \frac{\sin (c+d x)}{x^2} \, dx}{a}-\frac{b \int \frac{x \sin (c+d x)}{a+b x^3} \, dx}{a}\\ &=-\frac{\sin (c+d x)}{a x}-\frac{b \int \left (-\frac{\sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{(-1)^{2/3} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac{\sqrt [3]{-1} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{a}+\frac{d \int \frac{\cos (c+d x)}{x} \, dx}{a}\\ &=-\frac{\sin (c+d x)}{a x}+\frac{b^{2/3} \int \frac{\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{4/3}}-\frac{\left (\sqrt [3]{-1} b^{2/3}\right ) \int \frac{\sin (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac{\left ((-1)^{2/3} b^{2/3}\right ) \int \frac{\sin (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac{(d \cos (c)) \int \frac{\cos (d x)}{x} \, dx}{a}-\frac{(d \sin (c)) \int \frac{\sin (d x)}{x} \, dx}{a}\\ &=\frac{d \cos (c) \text{Ci}(d x)}{a}-\frac{\sin (c+d x)}{a x}-\frac{d \sin (c) \text{Si}(d x)}{a}+\frac{\left (b^{2/3} \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac{\left (\sqrt [3]{-1} b^{2/3} \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac{\left ((-1)^{2/3} b^{2/3} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac{\left (b^{2/3} \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{4/3}}-\frac{\left (\sqrt [3]{-1} b^{2/3} \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac{\left ((-1)^{2/3} b^{2/3} \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}\\ &=\frac{d \cos (c) \text{Ci}(d x)}{a}+\frac{\sqrt [3]{b} \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{4/3}}+\frac{(-1)^{2/3} \sqrt [3]{b} \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{b} \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{4/3}}-\frac{\sin (c+d x)}{a x}-\frac{d \sin (c) \text{Si}(d x)}{a}-\frac{(-1)^{2/3} \sqrt [3]{b} \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{b} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.486328, size = 233, normalized size = 0.61 \[ \frac{-i x \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-i \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\& \right ]+i x \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{i \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\& \right ]+6 d x \cos (c) \text{CosIntegral}(d x)-6 d x \sin (c) \text{Si}(d x)-6 \sin (c+d x)}{6 a x} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.023, size = 116, normalized size = 0.3 \begin{align*} d \left ( -{\frac{\sin \left ( dx+c \right ) }{axd}}-{\frac{1}{3\,a}\sum _{{\it \_R1}={\it RootOf} \left ({{\it \_Z}}^{3}b-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-{c}^{3}b \right ) }{\frac{-{\it Si} \left ( -dx+{\it \_R1}-c \right ) \cos \left ({\it \_R1} \right ) +{\it Ci} \left ( dx-{\it \_R1}+c \right ) \sin \left ({\it \_R1} \right ) }{{\it \_R1}-c}}}+{\frac{-{\it Si} \left ( dx \right ) \sin \left ( c \right ) +{\it Ci} \left ( dx \right ) \cos \left ( c \right ) }{a}} \right ) \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 (d x + c\right )}{{\left (b x^{3} + a\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.42407, size = 1152, normalized size = 3.03 \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 (d x + c\right )}{{\left (b x^{3} + a\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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