Optimal. Leaf size=856 \[ \frac {\text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac {\cos (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cos (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {3 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^2 b}-\frac {3 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}-\frac {3 \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^2 b}+\frac {3 \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}-\frac {3 \text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {3 \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )^2}-\frac {3 \cos \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 {3 \cos \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}} \]
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Rubi [A] time = 1.18, antiderivative size = 856, normalized size of antiderivative = 1.00, number of steps used = 28, 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 {\text {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac {\text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac {\cos (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cos (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {3 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^2 b}-\frac {3 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}-\frac {3 \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^2 b}+\frac {3 \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}-\frac {3 \text {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {3 \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )^2}-\frac {3 \cos \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 {3 \cos \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}} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3333
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx &=\int \left (-\frac {b^{3/2} \sin (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}-b x\right )^3}-\frac {3 b \sin (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b^{3/2} \sin (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}+b x\right )^3}-\frac {3 b \sin (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {3 b \sin (c+d x)}{8 a^2 \left (-a b-b^2 x^2\right )}\right ) \, dx\\ &=-\frac {(3 b) \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 a^2}-\frac {(3 b) \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 a^2}-\frac {(3 b) \int \frac {\sin (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^2}-\frac {b^{3/2} \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^3} \, dx}{8 (-a)^{3/2}}-\frac {b^{3/2} \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^3} \, dx}{8 (-a)^{3/2}}\\ &=-\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )^2}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {(3 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}{8 a^2}+\frac {(3 d) \int \frac {\cos (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a^2}-\frac {(3 d) \int \frac {\cos (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a^2}+\frac {\left (\sqrt {b} d\right ) \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 (-a)^{3/2}}-\frac {\left (\sqrt {b} d\right ) \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 (-a)^{3/2}}\\ &=\frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )^2}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 \int \frac {\sin (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{5/2}}-\frac {3 \int \frac {\sin (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{5/2}}+\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 (-a)^{3/2} \sqrt {b}}+\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 (-a)^{3/2} \sqrt {b}}-\frac {\left (3 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}{16 a^2}+\frac {\left (3 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}{16 a^2}+\frac {\left (3 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}{16 a^2}+\frac {\left (3 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}{16 a^2}\\ &=\frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 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^2 b}-\frac {3 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^2 b}-\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )^2}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 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^2 b}+\frac {3 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^2 b}-\frac {\left (3 \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}{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)^{3/2} \sqrt {b}}+\frac {\left (3 \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}{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)^{3/2} \sqrt {b}}-\frac {\left (3 \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}{16 (-a)^{5/2}}+\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)^{3/2} \sqrt {b}}-\frac {\left (3 \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}{16 (-a)^{5/2}}+\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)^{3/2} \sqrt {b}}\\ &=\frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 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^2 b}-\frac {3 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^2 b}-\frac {3 \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\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)^{3/2} b^{3/2}}+\frac {3 \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\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)^{3/2} b^{3/2}}-\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )^2}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 \cos \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 {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)^{3/2} b^{3/2}}-\frac {3 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^2 b}-\frac {3 \cos \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 {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)^{3/2} b^{3/2}}+\frac {3 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^2 b}\\ \end {align*}
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Mathematica [C] time = 2.52, size = 932, normalized size = 1.09 \[ \frac {\frac {6 b^{5/2} \cos (d x) \sin (c) x^3}{\left (b x^2+a\right )^2}+\frac {6 b^{5/2} \cos (c) \sin (d x) x^3}{\left (b x^2+a\right )^2}+\frac {2 a b^{3/2} d \cos (c) \cos (d x) x^2}{\left (b x^2+a\right )^2}-\frac {2 a b^{3/2} d \sin (c) \sin (d x) x^2}{\left (b x^2+a\right )^2}+\frac {10 a b^{3/2} \cos (d x) \sin (c) x}{\left (b x^2+a\right )^2}+\frac {10 a b^{3/2} \cos (c) \sin (d x) x}{\left (b x^2+a\right )^2}+\frac {2 a^2 \sqrt {b} d \cos (c) \cos (d x)}{\left (b x^2+a\right )^2}+\frac {i \text {Ci}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right ) \left (3 i \sqrt {a} \sqrt {b} d \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )+\left (a d^2+3 b\right ) \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )}{\sqrt {a}}-\frac {i \text {Ci}\left (d \left (x-\frac {i \sqrt {a}}{\sqrt {b}}\right )\right ) \left (\left (a d^2+3 b\right ) \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )-3 i \sqrt {a} \sqrt {b} d \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )}{\sqrt {a}}-\frac {2 a^2 \sqrt {b} d \sin (c) \sin (d x)}{\left (b x^2+a\right )^2}+i \sqrt {a} d^2 \cos (c) \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+\frac {3 i b \cos (c) \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )}{\sqrt {a}}+3 \sqrt {b} d \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sin (c) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )-3 i \sqrt {b} d \cos (c) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )-\sqrt {a} d^2 \sin (c) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )-\frac {3 b \sin (c) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )}{\sqrt {a}}+i \sqrt {a} d^2 \cos (c) \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\frac {3 i b \cos (c) \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )}{\sqrt {a}}-3 \sqrt {b} d \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sin (c) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-3 i \sqrt {b} d \cos (c) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\sqrt {a} d^2 \sin (c) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\frac {3 b \sin (c) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )}{\sqrt {a}}}{16 a^2 b^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.79, size = 611, normalized size = 0.71 \[ -\frac {{\left (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} - {\left (a^{3} d^{2} + {\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \, {\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} + {\left (a^{3} d^{2} + {\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \, {\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} - {\left (a^{3} d^{2} + {\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \, {\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} + {\left (a^{3} d^{2} + {\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \, {\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 4 \, {\left (a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cos \left (d x + c\right ) - 4 \, {\left (3 \, a b^{2} d x^{3} + 5 \, a^{2} b d x\right )} \sin \left (d x + c\right )}{32 \, {\left (a^{3} b^{3} d x^{4} + 2 \, a^{4} b^{2} d x^{2} + a^{5} b d\right )}} \]
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 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 602, normalized size = 0.70 \[ d^{5} \left (\frac {\sin \left (d x +c \right ) \left (3 \left (d x +c \right )^{3} b -9 c \left (d x +c \right )^{2} b +5 \left (d x +c \right ) a \,d^{2}+9 \left (d x +c \right ) b \,c^{2}-5 a c \,d^{2}-3 b \,c^{3}\right )}{8 a^{2} d^{4} \left (\left (d x +c \right )^{2} b -2 \left (d x +c \right ) b c +a \,d^{2}+b \,c^{2}\right )^{2}}+\frac {\cos \left (d x +c \right )}{8 a b \,d^{2} \left (\left (d x +c \right )^{2} b -2 \left (d x +c \right ) b c +a \,d^{2}+b \,c^{2}\right )}+\frac {\left (a \,d^{2}+3 b \right ) \left (\Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{16 a^{2} b^{2} d^{4} \left (\frac {d \sqrt {-a b}+c b}{b}-c \right )}+\frac {\left (a \,d^{2}+3 b \right ) \left (\Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{16 a^{2} b^{2} d^{4} \left (-\frac {d \sqrt {-a b}-c b}{b}-c \right )}-\frac {3 \left (-\Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{16 a^{2} b \,d^{4}}-\frac {3 \left (\Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )+\Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{16 a^{2} b \,d^{4}}\right ) \]
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 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \]
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 \[ \int \frac {\sin \left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^3} \,d x \]
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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