Optimal. Leaf size=791 \[ \text{result too large to display} \]
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Rubi [A] time = 1.87678, antiderivative size = 791, normalized size of antiderivative = 1., number of steps used = 46, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {3345, 3297, 3303, 3299, 3302, 3341, 3334} \[ \frac{d^2 \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a^3}+\frac{d^2 \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^3}-\frac{3 b \sin (c) \text{CosIntegral}(d x)}{a^4}+\frac{3 b \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}+\frac{3 b \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}-\frac{d^2 \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^3}+\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a^3}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{3 b \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}+\frac{3 b \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{d^2 \sin (c) \text{CosIntegral}(d x)}{2 a^3}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^3}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{9 \sqrt{b} d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{7/2}}+\frac{9 \sqrt{b} d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{7/2}}-\frac{9 \sqrt{b} d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{7/2}}-\frac{9 \sqrt{b} d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 3345
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rule 3341
Rule 3334
Rubi steps
\begin{align*} \int \frac{\sin (c+d x)}{x^3 \left (a+b x^2\right )^3} \, dx &=\int \left (\frac{\sin (c+d x)}{a^3 x^3}-\frac{3 b \sin (c+d x)}{a^4 x}+\frac{b^2 x \sin (c+d x)}{a^2 \left (a+b x^2\right )^3}+\frac{2 b^2 x \sin (c+d x)}{a^3 \left (a+b x^2\right )^2}+\frac{3 b^2 x \sin (c+d x)}{a^4 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\sin (c+d x)}{x^3} \, dx}{a^3}-\frac{(3 b) \int \frac{\sin (c+d x)}{x} \, dx}{a^4}+\frac{\left (3 b^2\right ) \int \frac{x \sin (c+d x)}{a+b x^2} \, dx}{a^4}+\frac{\left (2 b^2\right ) \int \frac{x \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a^3}+\frac{b^2 \int \frac{x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx}{a^2}\\ &=-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}+\frac{\left (3 b^2\right ) \int \left (-\frac{\sin (c+d x)}{2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{a^4}+\frac{d \int \frac{\cos (c+d x)}{x^2} \, dx}{2 a^3}+\frac{(b d) \int \frac{\cos (c+d x)}{a+b x^2} \, dx}{a^3}+\frac{(b d) \int \frac{\cos (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 a^2}-\frac{(3 b \cos (c)) \int \frac{\sin (d x)}{x} \, dx}{a^4}-\frac{(3 b \sin (c)) \int \frac{\cos (d x)}{x} \, dx}{a^4}\\ &=-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{\left (3 b^{3/2}\right ) \int \frac{\sin (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 a^4}+\frac{\left (3 b^{3/2}\right ) \int \frac{\sin (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 a^4}+\frac{(b d) \int \left (\frac{\sqrt{-a} \cos (c+d x)}{2 a \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{-a} \cos (c+d x)}{2 a \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{a^3}+\frac{(b d) \int \left (-\frac{b \cos (c+d x)}{4 a \left (\sqrt{-a} \sqrt{b}-b x\right )^2}-\frac{b \cos (c+d x)}{4 a \left (\sqrt{-a} \sqrt{b}+b x\right )^2}-\frac{b \cos (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{4 a^2}-\frac{d^2 \int \frac{\sin (c+d x)}{x} \, dx}{2 a^3}\\ &=-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}+\frac{(b d) \int \frac{\cos (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 (-a)^{7/2}}+\frac{(b d) \int \frac{\cos (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 (-a)^{7/2}}-\frac{\left (b^2 d\right ) \int \frac{\cos (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^2} \, dx}{16 a^3}-\frac{\left (b^2 d\right ) \int \frac{\cos (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^2} \, dx}{16 a^3}-\frac{\left (b^2 d\right ) \int \frac{\cos (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^3}-\frac{\left (d^2 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{2 a^3}+\frac{\left (3 b^{3/2} \cos \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} x} \, dx}{2 a^4}+\frac{\left (3 b^{3/2} \cos \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} x} \, dx}{2 a^4}-\frac{\left (d^2 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{2 a^3}+\frac{\left (3 b^{3/2} \sin \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} x} \, dx}{2 a^4}-\frac{\left (3 b^{3/2} \sin \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} x} \, dx}{2 a^4}\\ &=-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}-\frac{d^2 \text{Ci}(d x) \sin (c)}{2 a^3}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^3}-\frac{3 b \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}+\frac{3 b \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}-\frac{\left (b^2 d\right ) \int \left (-\frac{\sqrt{-a} \cos (c+d x)}{2 a b \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{\sqrt{-a} \cos (c+d x)}{2 a b \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{8 a^3}-\frac{\left (b d^2\right ) \int \frac{\sin (c+d x)}{\sqrt{-a} \sqrt{b}-b x} \, dx}{16 a^3}+\frac{\left (b d^2\right ) \int \frac{\sin (c+d x)}{\sqrt{-a} \sqrt{b}+b x} \, dx}{16 a^3}+\frac{\left (b 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} x} \, dx}{2 (-a)^{7/2}}+\frac{\left (b 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} x} \, dx}{2 (-a)^{7/2}}-\frac{\left (b 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} x} \, dx}{2 (-a)^{7/2}}+\frac{\left (b 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} x} \, dx}{2 (-a)^{7/2}}\\ &=-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{\sqrt{b} d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{7/2}}+\frac{\sqrt{b} d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{7/2}}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}-\frac{d^2 \text{Ci}(d x) \sin (c)}{2 a^3}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^3}-\frac{3 b \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}-\frac{\sqrt{b} d \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{7/2}}+\frac{3 b \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}-\frac{\sqrt{b} d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{7/2}}+\frac{(b d) \int \frac{\cos (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 (-a)^{7/2}}+\frac{(b d) \int \frac{\cos (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 (-a)^{7/2}}+\frac{\left (b d^2 \cos \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}{16 a^3}+\frac{\left (b d^2 \cos \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}{16 a^3}+\frac{\left (b d^2 \sin \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}{16 a^3}-\frac{\left (b d^2 \sin \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}{16 a^3}\\ &=-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{\sqrt{b} d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{7/2}}+\frac{\sqrt{b} d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{7/2}}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}-\frac{d^2 \text{Ci}(d x) \sin (c)}{2 a^3}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}+\frac{d^2 \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a^3}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}+\frac{d^2 \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a^3}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^3}-\frac{3 b \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}-\frac{d^2 \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^3}-\frac{\sqrt{b} d \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{7/2}}+\frac{3 b \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}+\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a^3}-\frac{\sqrt{b} d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{7/2}}+\frac{\left (b 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} x} \, dx}{16 (-a)^{7/2}}+\frac{\left (b 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} x} \, dx}{16 (-a)^{7/2}}-\frac{\left (b 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} x} \, dx}{16 (-a)^{7/2}}+\frac{\left (b 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} x} \, dx}{16 (-a)^{7/2}}\\ &=-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{9 \sqrt{b} d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{7/2}}+\frac{9 \sqrt{b} d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{7/2}}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}-\frac{d^2 \text{Ci}(d x) \sin (c)}{2 a^3}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}+\frac{d^2 \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a^3}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}+\frac{d^2 \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a^3}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^3}-\frac{3 b \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}-\frac{d^2 \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^3}-\frac{9 \sqrt{b} d \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{7/2}}+\frac{3 b \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}+\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a^3}-\frac{9 \sqrt{b} d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{7/2}}\\ \end{align*}
Mathematica [C] time = 2.72224, size = 995, normalized size = 1.26 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.055, size = 701, normalized size = 0.9 \begin{align*} \text{result too large to display} \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 )}^{3} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.40066, size = 1685, normalized size = 2.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{3} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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