Optimal. Leaf size=213 \[ -\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 \sqrt{-a} \sqrt{b}}+\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 \sqrt{-a} \sqrt{b}}-\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 \sqrt{-a} \sqrt{b}}-\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 \sqrt{-a} \sqrt{b}} \]
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Rubi [A] time = 0.239183, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3333, 3303, 3299, 3302} \[ -\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 \sqrt{-a} \sqrt{b}}+\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 \sqrt{-a} \sqrt{b}}-\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 \sqrt{-a} \sqrt{b}}-\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 \sqrt{-a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 3333
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\sin (c+d x)}{a+b x^2} \, dx &=\int \left (\frac{\sqrt{-a} \sin (c+d x)}{2 a \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{-a} \sin (c+d x)}{2 a \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{\sin (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 \sqrt{-a}}-\frac{\int \frac{\sin (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 \sqrt{-a}}\\ &=-\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \int \frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 \sqrt{-a}}+\frac{\cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \int \frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 \sqrt{-a}}-\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \int \frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 \sqrt{-a}}-\frac{\sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \int \frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 \sqrt{-a}}\\ &=-\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 \sqrt{-a} \sqrt{b}}+\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 \sqrt{-a} \sqrt{b}}-\frac{\cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 \sqrt{-a} \sqrt{b}}-\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 \sqrt{-a} \sqrt{b}}\\ \end{align*}
Mathematica [C] time = 0.223419, size = 172, normalized size = 0.81 \[ \frac{i \left (\sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-\sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )\right )}{2 \sqrt{a} \sqrt{b}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.013, size = 229, normalized size = 1.1 \begin{align*} d \left ({\frac{1}{2\,b} \left ({\it Si} \left ( dx+c-{\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) } \right ) \cos \left ({\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) } \right ) +{\it Ci} \left ( dx+c-{\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) } \right ) \sin \left ({\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) } \right ) \right ) \left ({\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) }-c \right ) ^{-1}}+{\frac{1}{2\,b} \left ({\it Si} \left ( dx+c+{\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) } \right ) \cos \left ({\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) } \right ) -{\it Ci} \left ( dx+c+{\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) } \right ) \sin \left ({\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) } \right ) \right ) \left ( -{\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) }-c \right ) ^{-1}} \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 )}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.95672, size = 377, normalized size = 1.77 \begin{align*} \frac{\sqrt{\frac{a d^{2}}{b}}{\rm Ei}\left (i \, d x - \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt{\frac{a d^{2}}{b}}\right )} - \sqrt{\frac{a d^{2}}{b}}{\rm Ei}\left (i \, d x + \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt{\frac{a d^{2}}{b}}\right )} + \sqrt{\frac{a d^{2}}{b}}{\rm Ei}\left (-i \, d x - \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt{\frac{a d^{2}}{b}}\right )} - \sqrt{\frac{a d^{2}}{b}}{\rm Ei}\left (-i \, d x + \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt{\frac{a d^{2}}{b}}\right )}}{4 \, a d} \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 (c + d x \right )}}{a + b 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 (d x + c\right )}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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