Optimal. Leaf size=730 \[ -\frac{d^2 \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a^2 b}-\frac{d^2 \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^2 b}-\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^3}-\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^3}+\frac{d^2 \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^2 b}-\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a^2 b}+\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^3}-\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^3}+\frac{\sin (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{\sin (c) \text{CosIntegral}(d x)}{a^3}+\frac{\cos (c) \text{Si}(d x)}{a^3}-\frac{5 d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{5/2} \sqrt{b}}+\frac{5 d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{5/2} \sqrt{b}}-\frac{5 d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{5/2} \sqrt{b}}-\frac{5 d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{5/2} \sqrt{b}}+\frac{\sin (c+d x)}{4 a \left (a+b x^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.83024, antiderivative size = 730, normalized size of antiderivative = 1., number of steps used = 41, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {3345, 3303, 3299, 3302, 3341, 3334, 3297} \[ -\frac{d^2 \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a^2 b}-\frac{d^2 \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^2 b}-\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^3}-\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^3}+\frac{d^2 \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^2 b}-\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a^2 b}+\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^3}-\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^3}+\frac{\sin (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{\sin (c) \text{CosIntegral}(d x)}{a^3}+\frac{\cos (c) \text{Si}(d x)}{a^3}-\frac{5 d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{5/2} \sqrt{b}}+\frac{5 d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{5/2} \sqrt{b}}-\frac{5 d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{5/2} \sqrt{b}}-\frac{5 d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{5/2} \sqrt{b}}+\frac{\sin (c+d x)}{4 a \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3345
Rule 3303
Rule 3299
Rule 3302
Rule 3341
Rule 3334
Rule 3297
Rubi steps
\begin{align*} \int \frac{\sin (c+d x)}{x \left (a+b x^2\right )^3} \, dx &=\int \left (\frac{\sin (c+d x)}{a^3 x}-\frac{b x \sin (c+d x)}{a \left (a+b x^2\right )^3}-\frac{b x \sin (c+d x)}{a^2 \left (a+b x^2\right )^2}-\frac{b x \sin (c+d x)}{a^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\sin (c+d x)}{x} \, dx}{a^3}-\frac{b \int \frac{x \sin (c+d x)}{a+b x^2} \, dx}{a^3}-\frac{b \int \frac{x \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a^2}-\frac{b \int \frac{x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx}{a}\\ &=\frac{\sin (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac{\sin (c+d x)}{2 a^2 \left (a+b x^2\right )}-\frac{b \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^3}-\frac{d \int \frac{\cos (c+d x)}{a+b x^2} \, dx}{2 a^2}-\frac{d \int \frac{\cos (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 a}+\frac{\cos (c) \int \frac{\sin (d x)}{x} \, dx}{a^3}+\frac{\sin (c) \int \frac{\cos (d x)}{x} \, dx}{a^3}\\ &=\frac{\text{Ci}(d x) \sin (c)}{a^3}+\frac{\sin (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac{\sin (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac{\cos (c) \text{Si}(d x)}{a^3}+\frac{\sqrt{b} \int \frac{\sin (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 a^3}-\frac{\sqrt{b} \int \frac{\sin (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 a^3}-\frac{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}{2 a^2}-\frac{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}\\ &=\frac{\text{Ci}(d x) \sin (c)}{a^3}+\frac{\sin (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac{\sin (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac{\cos (c) \text{Si}(d x)}{a^3}+\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{4 (-a)^{5/2}}+\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{4 (-a)^{5/2}}+\frac{(b d) \int \frac{\cos (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^2} \, dx}{16 a^2}+\frac{(b d) \int \frac{\cos (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^2} \, dx}{16 a^2}+\frac{(b d) \int \frac{\cos (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^2}-\frac{\left (\sqrt{b} \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^3}-\frac{\left (\sqrt{b} \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^3}-\frac{\left (\sqrt{b} \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^3}+\frac{\left (\sqrt{b} \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^3}\\ &=\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^3}-\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^3}+\frac{\sin (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac{\sin (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac{\cos (c) \text{Si}(d x)}{a^3}+\frac{\cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^3}-\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^3}+\frac{(b d) \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^2}+\frac{d^2 \int \frac{\sin (c+d x)}{\sqrt{-a} \sqrt{b}-b x} \, dx}{16 a^2}-\frac{d^2 \int \frac{\sin (c+d x)}{\sqrt{-a} \sqrt{b}+b x} \, dx}{16 a^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} x} \, dx}{4 (-a)^{5/2}}+\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} x} \, dx}{4 (-a)^{5/2}}-\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} x} \, dx}{4 (-a)^{5/2}}+\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} x} \, dx}{4 (-a)^{5/2}}\\ &=\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}-\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)^{5/2} \sqrt{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)^{5/2} \sqrt{b}}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^3}-\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^3}+\frac{\sin (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac{\sin (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac{\cos (c) \text{Si}(d x)}{a^3}+\frac{\cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^3}-\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)^{5/2} \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 a^3}-\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)^{5/2} \sqrt{b}}+\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 (-a)^{5/2}}+\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 (-a)^{5/2}}-\frac{\left (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^2}-\frac{\left (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^2}-\frac{\left (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^2}+\frac{\left (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^2}\\ &=\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}-\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)^{5/2} \sqrt{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)^{5/2} \sqrt{b}}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^3}-\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^2 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 a^3}-\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^2 b}+\frac{\sin (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac{\sin (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac{\cos (c) \text{Si}(d x)}{a^3}+\frac{\cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^3}+\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^2 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)^{5/2} \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 a^3}-\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^2 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)^{5/2} \sqrt{b}}+\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} x} \, dx}{16 (-a)^{5/2}}+\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} x} \, dx}{16 (-a)^{5/2}}-\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} x} \, dx}{16 (-a)^{5/2}}+\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} x} \, dx}{16 (-a)^{5/2}}\\ &=\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{5 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)^{5/2} \sqrt{b}}+\frac{5 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)^{5/2} \sqrt{b}}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^3}-\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^2 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 a^3}-\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^2 b}+\frac{\sin (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac{\sin (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac{\cos (c) \text{Si}(d x)}{a^3}+\frac{\cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^3}+\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^2 b}-\frac{5 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)^{5/2} \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 a^3}-\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^2 b}-\frac{5 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)^{5/2} \sqrt{b}}\\ \end{align*}
Mathematica [C] time = 7.93089, size = 1384, normalized size = 1.9 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.042, size = 584, normalized size = 0.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 2.12876, size = 1455, normalized size = 1.99 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]