Optimal. Leaf size=271 \[ \frac {\sin \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Ci}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\sin \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \text {Ci}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}+\frac {\cos \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Si}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\cos \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \text {Si}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}} \]
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Rubi [A] time = 0.80, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, 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 3299
Rule 3302
Rule 3303
Rule 6728
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*}
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Mathematica [A] time = 0.56, size = 238, normalized size = 0.88 \[ \frac {\sin \left (a+\frac {b \left (\sqrt {d^2-4 c e}-d\right )}{2 e}\right ) \text {Ci}\left (\frac {b \left (d+2 e x-\sqrt {d^2-4 c e}\right )}{2 e}\right )-\sin \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \text {Ci}\left (\frac {b \left (d+2 e x+\sqrt {d^2-4 c e}\right )}{2 e}\right )-\cos \left (a+\frac {b \left (\sqrt {d^2-4 c e}-d\right )}{2 e}\right ) \text {Si}\left (\frac {b \left (\sqrt {d^2-4 c e}-d\right )}{2 e}-b x\right )-\cos \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \text {Si}\left (\frac {b \left (d+2 e x+\sqrt {d^2-4 c e}\right )}{2 e}\right )}{\sqrt {d^2-4 c e}} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 2.02, size = 434, normalized size = 1.60 \[ -\frac {e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {-2 i \, b e x - i \, b d - e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) e^{\left (\frac {i \, b d - 2 i \, a e + e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )} - e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {-2 i \, b e x - i \, b d + e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) e^{\left (\frac {i \, b d - 2 i \, a e - e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )} + e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 i \, b e x + i \, b d - e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) e^{\left (\frac {-i \, b d + 2 i \, a e + e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )} - e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 i \, b e x + i \, b d + e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) e^{\left (\frac {-i \, b d + 2 i \, a e - e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )}}{2 \, {\left (b d^{2} - 4 \, b c e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b x + a\right )}{e x^{2} + d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 320, normalized size = 1.18 \[ b \left (\frac {\Si \left (b x +a -\frac {2 a e -d b +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right ) \cos \left (\frac {2 a e -d b +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )+\Ci \left (b x +a -\frac {2 a e -d b +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right ) \sin \left (\frac {2 a e -d b +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{\sqrt {-4 b^{2} c e +b^{2} d^{2}}}-\frac {\Si \left (b x +a +\frac {-2 a e +d b +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right ) \cos \left (\frac {-2 a e +d b +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )-\Ci \left (b x +a +\frac {-2 a e +d b +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right ) \sin \left (\frac {-2 a e +d b +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{\sqrt {-4 b^{2} c e +b^{2} d^{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b x + a\right )}{e x^{2} + d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sin \left (a+b\,x\right )}{e\,x^2+d\,x+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (a + b x \right )}}{c + d x + e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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