Optimal. Leaf size=476 \[ -\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 (-a)^{3/2} \sqrt{b}}+\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 (-a)^{3/2} \sqrt{b}}-\frac{d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 a b}-\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 a b}-\frac{d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 a b}+\frac{d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 a b}+\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 (-a)^{3/2} \sqrt{b}}+\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 (-a)^{3/2} \sqrt{b}}-\frac{\sin (c+d x)}{4 a \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{4 a \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )} \]
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Rubi [A] time = 0.806036, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {3333, 3297, 3303, 3299, 3302} \[ -\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 (-a)^{3/2} \sqrt{b}}+\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 (-a)^{3/2} \sqrt{b}}-\frac{d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 a b}-\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 a b}-\frac{d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 a b}+\frac{d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 a b}+\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 (-a)^{3/2} \sqrt{b}}+\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 (-a)^{3/2} \sqrt{b}}-\frac{\sin (c+d x)}{4 a \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{4 a \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )} \]
Antiderivative was successfully verified.
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Rule 3333
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\sin (c+d x)}{\left (a+b x^2\right )^2} \, dx &=\int \left (-\frac{b \sin (c+d x)}{4 a \left (\sqrt{-a} \sqrt{b}-b x\right )^2}-\frac{b \sin (c+d x)}{4 a \left (\sqrt{-a} \sqrt{b}+b x\right )^2}-\frac{b \sin (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx\\ &=-\frac{b \int \frac{\sin (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^2} \, dx}{4 a}-\frac{b \int \frac{\sin (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^2} \, dx}{4 a}-\frac{b \int \frac{\sin (c+d x)}{-a b-b^2 x^2} \, dx}{2 a}\\ &=-\frac{\sin (c+d x)}{4 a \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{4 a \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{b \int \left (-\frac{\sqrt{-a} \sin (c+d x)}{2 a b \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{\sqrt{-a} \sin (c+d x)}{2 a b \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{2 a}+\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a} \sqrt{b}-b x} \, dx}{4 a}-\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a} \sqrt{b}+b x} \, dx}{4 a}\\ &=-\frac{\sin (c+d x)}{4 a \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{4 a \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{\int \frac{\sin (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{4 (-a)^{3/2}}+\frac{\int \frac{\sin (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{4 (-a)^{3/2}}-\frac{\left (d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a} \sqrt{b}+b x} \, dx}{4 a}+\frac{\left (d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a} \sqrt{b}-b x} \, dx}{4 a}+\frac{\left (d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a} \sqrt{b}+b x} \, dx}{4 a}+\frac{\left (d \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a} \sqrt{b}-b x} \, dx}{4 a}\\ &=-\frac{d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 a b}-\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 a b}-\frac{\sin (c+d x)}{4 a \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{4 a \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{d \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 a b}+\frac{d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 a b}+\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}{4 (-a)^{3/2}}-\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}{4 (-a)^{3/2}}+\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}{4 (-a)^{3/2}}+\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}{4 (-a)^{3/2}}\\ &=-\frac{d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 a b}-\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 a b}+\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 (-a)^{3/2} \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 )}{4 (-a)^{3/2} \sqrt{b}}-\frac{\sin (c+d x)}{4 a \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{4 a \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{\cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 (-a)^{3/2} \sqrt{b}}-\frac{d \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 a b}+\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 (-a)^{3/2} \sqrt{b}}+\frac{d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 a b}\\ \end{align*}
Mathematica [C] time = 0.628191, size = 585, normalized size = 1.23 \[ \frac{a^{3/2} d \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-a^{3/2} d \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+i b^{3/2} x^2 \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+i b^{3/2} x^2 \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )-\left (a+b x^2\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right ) \left (\sqrt{a} d \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )-i \sqrt{b} \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )-\left (a+b x^2\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right ) \left (i \sqrt{b} \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )+\sqrt{a} d \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )+\sqrt{a} b d x^2 \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-\sqrt{a} b d x^2 \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+i a \sqrt{b} \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+i a \sqrt{b} \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+2 \sqrt{a} b x \sin (c+d x)}{4 a^{3/2} b \left (a+b x^2\right )} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.024, size = 495, normalized size = 1. \begin{align*}{d}^{3} \left ({\frac{\sin \left ( dx+c \right ) }{ \left ( dx+c \right ) ^{2}b-2\, \left ( dx+c \right ) bc+a{d}^{2}+{c}^{2}b} \left ({\frac{dx+c}{2\,a{d}^{2}}}-{\frac{c}{2\,a{d}^{2}}} \right ) }+{\frac{1}{4\,ab{d}^{2}} \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}{4\,ab{d}^{2}} \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}{4\,ab{d}^{2}} \left ( -{\it Si} \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 ) +{\it Ci} \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 ) \right ) }-{\frac{1}{4\,ab{d}^{2}} \left ({\it Si} \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 ) +{\it Ci} \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 ) \right ) } \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 )}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.83844, size = 664, normalized size = 1.39 \begin{align*} \frac{4 \, a b d x \sin \left (d x + c\right ) -{\left (a b d^{2} x^{2} + a^{2} d^{2} -{\left (b^{2} x^{2} + a b\right )} \sqrt{\frac{a d^{2}}{b}}\right )}{\rm Ei}\left (i \, d x - \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt{\frac{a d^{2}}{b}}\right )} -{\left (a b d^{2} x^{2} + a^{2} d^{2} +{\left (b^{2} x^{2} + a b\right )} \sqrt{\frac{a d^{2}}{b}}\right )}{\rm Ei}\left (i \, d x + \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt{\frac{a d^{2}}{b}}\right )} -{\left (a b d^{2} x^{2} + a^{2} d^{2} -{\left (b^{2} x^{2} + a b\right )} \sqrt{\frac{a d^{2}}{b}}\right )}{\rm Ei}\left (-i \, d x - \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt{\frac{a d^{2}}{b}}\right )} -{\left (a b d^{2} x^{2} + a^{2} d^{2} +{\left (b^{2} x^{2} + a b\right )} \sqrt{\frac{a d^{2}}{b}}\right )}{\rm Ei}\left (-i \, d x + \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt{\frac{a d^{2}}{b}}\right )}}{8 \,{\left (a^{2} b^{2} d x^{2} + a^{3} b d\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 (c + d x \right )}}{\left (a + b x^{2}\right )^{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 )}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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