Optimal. Leaf size=213 \[ \frac{\cos \left (a+\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos \left (a-\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{-c}}{\sqrt{d}}+b x\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\sin \left (a+\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Si}\left (\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\sin \left (a-\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Si}\left (x b+\frac{\sqrt{-c} b}{\sqrt{d}}\right )}{2 \sqrt{-c} \sqrt{d}} \]
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Rubi [A] time = 0.3079, 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 = {3334, 3303, 3299, 3302} \[ \frac{\cos \left (a+\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos \left (a-\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{-c}}{\sqrt{d}}+b x\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\sin \left (a+\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Si}\left (\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\sin \left (a-\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Si}\left (x b+\frac{\sqrt{-c} b}{\sqrt{d}}\right )}{2 \sqrt{-c} \sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 3334
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\cos (a+b x)}{c+d x^2} \, dx &=\int \left (\frac{\sqrt{-c} \cos (a+b x)}{2 c \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\sqrt{-c} \cos (a+b x)}{2 c \left (\sqrt{-c}+\sqrt{d} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{\cos (a+b x)}{\sqrt{-c}-\sqrt{d} x} \, dx}{2 \sqrt{-c}}-\frac{\int \frac{\cos (a+b x)}{\sqrt{-c}+\sqrt{d} x} \, dx}{2 \sqrt{-c}}\\ &=-\frac{\cos \left (a-\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \int \frac{\cos \left (\frac{b \sqrt{-c}}{\sqrt{d}}+b x\right )}{\sqrt{-c}+\sqrt{d} x} \, dx}{2 \sqrt{-c}}-\frac{\cos \left (a+\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \int \frac{\cos \left (\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right )}{\sqrt{-c}-\sqrt{d} x} \, dx}{2 \sqrt{-c}}+\frac{\sin \left (a-\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \int \frac{\sin \left (\frac{b \sqrt{-c}}{\sqrt{d}}+b x\right )}{\sqrt{-c}+\sqrt{d} x} \, dx}{2 \sqrt{-c}}-\frac{\sin \left (a+\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \int \frac{\sin \left (\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right )}{\sqrt{-c}-\sqrt{d} x} \, dx}{2 \sqrt{-c}}\\ &=\frac{\cos \left (a+\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Ci}\left (\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos \left (a-\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Ci}\left (\frac{b \sqrt{-c}}{\sqrt{d}}+b x\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\sin \left (a+\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Si}\left (\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\sin \left (a-\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Si}\left (\frac{b \sqrt{-c}}{\sqrt{d}}+b x\right )}{2 \sqrt{-c} \sqrt{d}}\\ \end{align*}
Mathematica [C] time = 0.302956, size = 172, normalized size = 0.81 \[ -\frac{i \left (\cos \left (a+\frac{i b \sqrt{c}}{\sqrt{d}}\right ) \text{CosIntegral}\left (b \left (x-\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )-\cos \left (a-\frac{i b \sqrt{c}}{\sqrt{d}}\right ) \text{CosIntegral}\left (b \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )+\sin \left (a-\frac{i b \sqrt{c}}{\sqrt{d}}\right ) \text{Si}\left (b \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )+\sin \left (a+\frac{i b \sqrt{c}}{\sqrt{d}}\right ) \text{Si}\left (\frac{i b \sqrt{c}}{\sqrt{d}}-b x\right )\right )}{2 \sqrt{c} \sqrt{d}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.016, size = 229, normalized size = 1.1 \begin{align*} b \left ({\frac{1}{2\,d} \left ( -{\it Si} \left ( bx+a-{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) } \right ) \sin \left ({\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) } \right ) +{\it Ci} \left ( bx+a-{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) } \right ) \cos \left ({\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) } \right ) \right ) \left ({\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) }-a \right ) ^{-1}}+{\frac{1}{2\,d} \left ({\it Si} \left ( bx+a+{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) } \right ) \sin \left ({\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) } \right ) +{\it Ci} \left ( bx+a+{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) } \right ) \cos \left ({\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) } \right ) \right ) \left ( -{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) }-a \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{\cos \left (b x + a\right )}{d x^{2} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.49606, size = 398, normalized size = 1.87 \begin{align*} \frac{2 i \, \sqrt{\frac{b^{2} c}{d}}{\rm Ei}\left (i \, b x - \sqrt{\frac{b^{2} c}{d}}\right ) e^{\left (i \, a + \sqrt{\frac{b^{2} c}{d}}\right )} - 2 i \, \sqrt{\frac{b^{2} c}{d}}{\rm Ei}\left (i \, b x + \sqrt{\frac{b^{2} c}{d}}\right ) e^{\left (i \, a - \sqrt{\frac{b^{2} c}{d}}\right )} - 2 i \, \sqrt{\frac{b^{2} c}{d}}{\rm Ei}\left (-i \, b x - \sqrt{\frac{b^{2} c}{d}}\right ) e^{\left (-i \, a + \sqrt{\frac{b^{2} c}{d}}\right )} + 2 i \, \sqrt{\frac{b^{2} c}{d}}{\rm Ei}\left (-i \, b x + \sqrt{\frac{b^{2} c}{d}}\right ) e^{\left (-i \, a - \sqrt{\frac{b^{2} c}{d}}\right )}}{8 \, b c} \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{\cos{\left (a + b x \right )}}{c + d 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{\cos \left (b x + a\right )}{d x^{2} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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