Optimal. Leaf size=271 \[ \frac{\sin \left (a-\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}\right ) \text{CosIntegral}\left (\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}-\frac{\sin \left (a-\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}\right ) \text{CosIntegral}\left (\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}+\frac{\cos \left (a-\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}\right ) \text{Si}\left (\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}-\frac{\cos \left (a-\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}\right ) \text{Si}\left (\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}} \]
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Rubi [A] time = 0.801635, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {6728, 3303, 3299, 3302} \[ \frac{\sin \left (a-\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}\right ) \text{CosIntegral}\left (\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}-\frac{\sin \left (a-\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}\right ) \text{CosIntegral}\left (\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}+\frac{\cos \left (a-\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}\right ) \text{Si}\left (\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}-\frac{\cos \left (a-\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}\right ) \text{Si}\left (\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}} \]
Antiderivative was successfully verified.
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Rule 6728
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\sin (a+b x)}{c+d x+e x^2} \, dx &=\int \left (\frac{2 e \sin (a+b x)}{\sqrt{d^2-4 c e} \left (d-\sqrt{d^2-4 c e}+2 e x\right )}-\frac{2 e \sin (a+b x)}{\sqrt{d^2-4 c e} \left (d+\sqrt{d^2-4 c e}+2 e x\right )}\right ) \, dx\\ &=\frac{(2 e) \int \frac{\sin (a+b x)}{d-\sqrt{d^2-4 c e}+2 e x} \, dx}{\sqrt{d^2-4 c e}}-\frac{(2 e) \int \frac{\sin (a+b x)}{d+\sqrt{d^2-4 c e}+2 e x} \, dx}{\sqrt{d^2-4 c e}}\\ &=\frac{\left (2 e \cos \left (a-\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}\right )\right ) \int \frac{\sin \left (\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{d-\sqrt{d^2-4 c e}+2 e x} \, dx}{\sqrt{d^2-4 c e}}-\frac{\left (2 e \cos \left (a-\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}\right )\right ) \int \frac{\sin \left (\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{d+\sqrt{d^2-4 c e}+2 e x} \, dx}{\sqrt{d^2-4 c e}}+\frac{\left (2 e \sin \left (a-\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}\right )\right ) \int \frac{\cos \left (\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{d-\sqrt{d^2-4 c e}+2 e x} \, dx}{\sqrt{d^2-4 c e}}-\frac{\left (2 e \sin \left (a-\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}\right )\right ) \int \frac{\cos \left (\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{d+\sqrt{d^2-4 c e}+2 e x} \, dx}{\sqrt{d^2-4 c e}}\\ &=\frac{\text{Ci}\left (\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}+b x\right ) \sin \left (a-\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}\right )}{\sqrt{d^2-4 c e}}-\frac{\text{Ci}\left (\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}+b x\right ) \sin \left (a-\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}\right )}{\sqrt{d^2-4 c e}}+\frac{\cos \left (a-\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}\right ) \text{Si}\left (\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}-\frac{\cos \left (a-\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}\right ) \text{Si}\left (\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}\\ \end{align*}
Mathematica [A] time = 0.578755, size = 238, normalized size = 0.88 \[ \frac{\sin \left (a+\frac{b \left (\sqrt{d^2-4 c e}-d\right )}{2 e}\right ) \text{CosIntegral}\left (\frac{b \left (-\sqrt{d^2-4 c e}+d+2 e x\right )}{2 e}\right )-\sin \left (a-\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}\right ) \text{CosIntegral}\left (\frac{b \left (\sqrt{d^2-4 c e}+d+2 e x\right )}{2 e}\right )-\cos \left (a+\frac{b \left (\sqrt{d^2-4 c e}-d\right )}{2 e}\right ) \text{Si}\left (\frac{b \left (\sqrt{d^2-4 c e}-d\right )}{2 e}-b x\right )-\cos \left (a-\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}\right ) \text{Si}\left (\frac{b \left (d+2 e x+\sqrt{d^2-4 c e}\right )}{2 e}\right )}{\sqrt{d^2-4 c e}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.015, size = 320, normalized size = 1.2 \begin{align*} b \left ({ \left ({\it Si} \left ( bx+a-{\frac{1}{2\,e} \left ( 2\,ae-db+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) } \right ) \cos \left ({\frac{1}{2\,e} \left ( 2\,ae-db+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) } \right ) +{\it Ci} \left ( bx+a-{\frac{1}{2\,e} \left ( 2\,ae-db+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) } \right ) \sin \left ({\frac{1}{2\,e} \left ( 2\,ae-db+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) } \right ) \right ){\frac{1}{\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}}}}}-{ \left ({\it Si} \left ( bx+a+{\frac{1}{2\,e} \left ( -2\,ae+db+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) } \right ) \cos \left ({\frac{1}{2\,e} \left ( -2\,ae+db+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) } \right ) -{\it Ci} \left ( bx+a+{\frac{1}{2\,e} \left ( -2\,ae+db+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) } \right ) \sin \left ({\frac{1}{2\,e} \left ( -2\,ae+db+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) } \right ) \right ){\frac{1}{\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}}}}} \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 (b x + a\right )}{e x^{2} + d x + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.42761, size = 950, normalized size = 3.51 \begin{align*} -\frac{e \sqrt{-\frac{b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}{\rm Ei}\left (\frac{-2 i \, b e x - i \, b d - e \sqrt{-\frac{b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) e^{\left (\frac{i \, b d - 2 i \, a e + e \sqrt{-\frac{b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )} - e \sqrt{-\frac{b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}{\rm Ei}\left (\frac{-2 i \, b e x - i \, b d + e \sqrt{-\frac{b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) e^{\left (\frac{i \, b d - 2 i \, a e - e \sqrt{-\frac{b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )} + e \sqrt{-\frac{b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}{\rm Ei}\left (\frac{2 i \, b e x + i \, b d - e \sqrt{-\frac{b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) e^{\left (\frac{-i \, b d + 2 i \, a e + e \sqrt{-\frac{b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )} - e \sqrt{-\frac{b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}{\rm Ei}\left (\frac{2 i \, b e x + i \, b d + e \sqrt{-\frac{b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) e^{\left (\frac{-i \, b d + 2 i \, a e - e \sqrt{-\frac{b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )}}{2 \,{\left (b d^{2} - 4 \, b c e\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 + b x \right )}}{c + d x + e x^{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 (b x + a\right )}{e x^{2} + d x + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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