Optimal. Leaf size=99 \[ \frac{a^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{a^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^3}+\frac{a \cos (c+d x)}{b^2 d}+\frac{\sin (c+d x)}{b d^2}-\frac{x \cos (c+d x)}{b d} \]
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Rubi [A] time = 0.262161, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {6742, 2638, 3296, 2637, 3303, 3299, 3302} \[ \frac{a^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{a^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^3}+\frac{a \cos (c+d x)}{b^2 d}+\frac{\sin (c+d x)}{b d^2}-\frac{x \cos (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2638
Rule 3296
Rule 2637
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{x^2 \sin (c+d x)}{a+b x} \, dx &=\int \left (-\frac{a \sin (c+d x)}{b^2}+\frac{x \sin (c+d x)}{b}+\frac{a^2 \sin (c+d x)}{b^2 (a+b x)}\right ) \, dx\\ &=-\frac{a \int \sin (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{\sin (c+d x)}{a+b x} \, dx}{b^2}+\frac{\int x \sin (c+d x) \, dx}{b}\\ &=\frac{a \cos (c+d x)}{b^2 d}-\frac{x \cos (c+d x)}{b d}+\frac{\int \cos (c+d x) \, dx}{b d}+\frac{\left (a^2 \cos \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sin \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b^2}+\frac{\left (a^2 \sin \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cos \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b^2}\\ &=\frac{a \cos (c+d x)}{b^2 d}-\frac{x \cos (c+d x)}{b d}+\frac{a^2 \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{b^3}+\frac{\sin (c+d x)}{b d^2}+\frac{a^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.321318, size = 87, normalized size = 0.88 \[ \frac{a^2 d^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (d \left (\frac{a}{b}+x\right )\right )+a^2 d^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (d \left (\frac{a}{b}+x\right )\right )+b (d (a-b x) \cos (c+d x)+b \sin (c+d x))}{b^3 d^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 315, normalized size = 3.2 \begin{align*}{\frac{1}{{d}^{3}} \left ({\frac{ \left ( -da+cb+b \right ) d \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{b}^{2}}}+{\frac{ \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) d}{{b}^{2}} \left ({\frac{1}{b}{\it Si} \left ( dx+c+{\frac{da-cb}{b}} \right ) \cos \left ({\frac{da-cb}{b}} \right ) }-{\frac{1}{b}{\it Ci} \left ( dx+c+{\frac{da-cb}{b}} \right ) \sin \left ({\frac{da-cb}{b}} \right ) } \right ) }+2\,{\frac{dc\cos \left ( dx+c \right ) }{b}}+2\,{\frac{ \left ( da-cb \right ) dc}{b} \left ({\frac{1}{b}{\it Si} \left ( dx+c+{\frac{da-cb}{b}} \right ) \cos \left ({\frac{da-cb}{b}} \right ) }-{\frac{1}{b}{\it Ci} \left ( dx+c+{\frac{da-cb}{b}} \right ) \sin \left ({\frac{da-cb}{b}} \right ) } \right ) }+d{c}^{2} \left ({\frac{1}{b}{\it Si} \left ( dx+c+{\frac{da-cb}{b}} \right ) \cos \left ({\frac{da-cb}{b}} \right ) }-{\frac{1}{b}{\it Ci} \left ( dx+c+{\frac{da-cb}{b}} \right ) \sin \left ({\frac{da-cb}{b}} \right ) } \right ) \right ) } \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 [A] time = 1.70749, size = 319, normalized size = 3.22 \begin{align*} \frac{2 \, a^{2} d^{2} \cos \left (-\frac{b c - a d}{b}\right ) \operatorname{Si}\left (\frac{b d x + a d}{b}\right ) + 2 \, b^{2} \sin \left (d x + c\right ) - 2 \,{\left (b^{2} d x - a b d\right )} \cos \left (d x + c\right ) -{\left (a^{2} d^{2} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) + a^{2} d^{2} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right )\right )} \sin \left (-\frac{b c - a d}{b}\right )}{2 \, b^{3} d^{2}} \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{x^{2} \sin{\left (c + d x \right )}}{a + b x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.16545, size = 911, normalized size = 9.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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