Optimal. Leaf size=791 \[ -\frac {3 b \sin (c) \text {Ci}(d x)}{a^4}+\frac {3 b \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}+\frac {3 b \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^4}-\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 {d^2 \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^3}+\frac {d^2 \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 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^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 {b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\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 {Ci}(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 {b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {9 \sqrt {b} d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\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 (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\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}} \]
[Out]
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Rubi [A] time = 1.88, antiderivative size = 791, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3334
Rule 3341
Rule 3345
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*}
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Mathematica [C] time = 2.87, size = 995, normalized size = 1.26 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.86, size = 766, normalized size = 0.97 \[ \frac {{\left (16 i \, {\left (a b^{2} d^{2} + 6 \, b^{3}\right )} x^{6} + 32 i \, {\left (a^{2} b d^{2} + 6 \, a b^{2}\right )} x^{4} + 16 i \, {\left (a^{3} d^{2} + 6 \, a^{2} b\right )} x^{2}\right )} {\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} + {\left (-16 i \, {\left (a b^{2} d^{2} + 6 \, b^{3}\right )} x^{6} - 32 i \, {\left (a^{2} b d^{2} + 6 \, a b^{2}\right )} x^{4} - 16 i \, {\left (a^{3} d^{2} + 6 \, a^{2} b\right )} x^{2}\right )} {\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} + {\left (-2 i \, {\left (a b^{2} d^{2} + 24 \, b^{3}\right )} x^{6} - 4 i \, {\left (a^{2} b d^{2} + 24 \, a b^{2}\right )} x^{4} - 2 i \, {\left (a^{3} d^{2} + 24 \, a^{2} b\right )} x^{2} + 2 \, {\left (9 i \, b^{3} x^{6} + 18 i \, a b^{2} x^{4} + 9 i \, a^{2} b x^{2}\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 (-2 i \, {\left (a b^{2} d^{2} + 24 \, b^{3}\right )} x^{6} - 4 i \, {\left (a^{2} b d^{2} + 24 \, a b^{2}\right )} x^{4} - 2 i \, {\left (a^{3} d^{2} + 24 \, a^{2} b\right )} x^{2} + 2 \, {\left (-9 i \, b^{3} x^{6} - 18 i \, a b^{2} x^{4} - 9 i \, a^{2} b x^{2}\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 (2 i \, {\left (a b^{2} d^{2} + 24 \, b^{3}\right )} x^{6} + 4 i \, {\left (a^{2} b d^{2} + 24 \, a b^{2}\right )} x^{4} + 2 i \, {\left (a^{3} d^{2} + 24 \, a^{2} b\right )} x^{2} + 2 \, {\left (-9 i \, b^{3} x^{6} - 18 i \, a b^{2} x^{4} - 9 i \, a^{2} b x^{2}\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 (2 i \, {\left (a b^{2} d^{2} + 24 \, b^{3}\right )} x^{6} + 4 i \, {\left (a^{2} b d^{2} + 24 \, a b^{2}\right )} x^{4} + 2 i \, {\left (a^{3} d^{2} + 24 \, a^{2} b\right )} x^{2} + 2 \, {\left (9 i \, b^{3} x^{6} + 18 i \, a b^{2} x^{4} + 9 i \, a^{2} b x^{2}\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 (3 \, a b^{2} d x^{5} + 7 \, a^{2} b d x^{3} + 4 \, a^{3} d x\right )} \cos \left (d x + c\right ) - 16 \, {\left (6 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + 2 \, a^{3}\right )} \sin \left (d x + c\right )}{64 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 701, normalized size = 0.89 \[ d^{2} \left (-\frac {\sin \left (d x +c \right ) \left (6 b^{2} \left (d x +c \right )^{4}-24 c \left (d x +c \right )^{3} b^{2}+9 \left (d x +c \right )^{2} a b \,d^{2}+36 b^{2} c^{2} \left (d x +c \right )^{2}-18 \left (d x +c \right ) a b c \,d^{2}-24 \left (d x +c \right ) b^{2} c^{3}+2 a^{2} d^{4}+9 a b \,c^{2} d^{2}+6 b^{2} c^{4}\right )}{4 a^{3} x^{2} d^{2} \left (\left (d x +c \right )^{2} b -2 \left (d x +c \right ) b c +a \,d^{2}+b \,c^{2}\right )^{2}}-\frac {\cos \left (d x +c \right ) \left (3 \left (d x +c \right )^{2} b -6 \left (d x +c \right ) b c +4 a \,d^{2}+3 b \,c^{2}\right )}{8 a^{3} x d \left (\left (d x +c \right )^{2} b -2 \left (d x +c \right ) b c +a \,d^{2}+b \,c^{2}\right )}+\frac {\left (a \,d^{2}+24 b \right ) \left (\Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{16 d^{2} a^{4}}+\frac {\left (a \,d^{2}+24 b \right ) \left (\Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{16 d^{2} a^{4}}-\frac {\left (a \,d^{2}+6 b \right ) \left (\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )\right )}{2 a^{4} d^{2}}+\frac {-\frac {9 \Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{16}+\frac {9 \Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{16}}{a^{3} \left (\frac {d \sqrt {-a b}+c b}{b}-c \right )}+\frac {\frac {9 \Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{16}+\frac {9 \Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{16}}{a^{3} \left (-\frac {d \sqrt {-a b}-c b}{b}-c \right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sin \left (c+d\,x\right )}{x^3\,{\left (b\,x^2+a\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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