Optimal. Leaf size=270 \[ -\frac {b \sin (c) \text {Ci}(d x)}{a^2}+\frac {b \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {b \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {b \cos (c) \text {Si}(d x)}{a^2}-\frac {b \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}-\frac {d^2 \sin (c) \text {Ci}(d x)}{2 a}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}-\frac {\sin (c+d x)}{2 a x^2}-\frac {d \cos (c+d x)}{2 a x} \]
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Rubi [A] time = 0.51, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3345, 3297, 3303, 3299, 3302} \[ -\frac {b \sin (c) \text {CosIntegral}(d x)}{a^2}+\frac {b \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2}+\frac {b \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {b \cos (c) \text {Si}(d x)}{a^2}-\frac {b \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}-\frac {d^2 \sin (c) \text {CosIntegral}(d x)}{2 a}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}-\frac {\sin (c+d x)}{2 a x^2}-\frac {d \cos (c+d x)}{2 a x} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3345
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{x^3 \left (a+b x^2\right )} \, dx &=\int \left (\frac {\sin (c+d x)}{a x^3}-\frac {b \sin (c+d x)}{a^2 x}+\frac {b^2 x \sin (c+d x)}{a^2 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\sin (c+d x)}{x^3} \, dx}{a}-\frac {b \int \frac {\sin (c+d x)}{x} \, dx}{a^2}+\frac {b^2 \int \frac {x \sin (c+d x)}{a+b x^2} \, dx}{a^2}\\ &=-\frac {\sin (c+d x)}{2 a x^2}+\frac {b^2 \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^2}+\frac {d \int \frac {\cos (c+d x)}{x^2} \, dx}{2 a}-\frac {(b \cos (c)) \int \frac {\sin (d x)}{x} \, dx}{a^2}-\frac {(b \sin (c)) \int \frac {\cos (d x)}{x} \, dx}{a^2}\\ &=-\frac {d \cos (c+d x)}{2 a x}-\frac {b \text {Ci}(d x) \sin (c)}{a^2}-\frac {\sin (c+d x)}{2 a x^2}-\frac {b \cos (c) \text {Si}(d x)}{a^2}-\frac {b^{3/2} \int \frac {\sin (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 a^2}+\frac {b^{3/2} \int \frac {\sin (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 a^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x} \, dx}{2 a}\\ &=-\frac {d \cos (c+d x)}{2 a x}-\frac {b \text {Ci}(d x) \sin (c)}{a^2}-\frac {\sin (c+d x)}{2 a x^2}-\frac {b \cos (c) \text {Si}(d x)}{a^2}-\frac {\left (d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{2 a}+\frac {\left (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^2}+\frac {\left (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^2}-\frac {\left (d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{2 a}+\frac {\left (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^2}-\frac {\left (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^2}\\ &=-\frac {d \cos (c+d x)}{2 a x}-\frac {b \text {Ci}(d x) \sin (c)}{a^2}-\frac {d^2 \text {Ci}(d x) \sin (c)}{2 a}+\frac {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^2}+\frac {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^2}-\frac {\sin (c+d x)}{2 a x^2}-\frac {b \cos (c) \text {Si}(d x)}{a^2}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}-\frac {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^2}+\frac {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^2}\\ \end {align*}
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Mathematica [C] time = 0.69, size = 247, normalized size = 0.91 \[ -\frac {x^2 \sin (c) \left (a d^2+2 b\right ) \text {Ci}(d x)-b x^2 \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )-b x^2 \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x-\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )-b x^2 \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+b x^2 \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+a d^2 x^2 \cos (c) \text {Si}(d x)+a \sin (c+d x)+a d x \cos (c+d x)+2 b x^2 \cos (c) \text {Si}(d x)}{2 a^2 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.91, size = 231, normalized size = 0.86 \[ \frac {i \, {\left (a d^{2} + 2 \, b\right )} x^{2} {\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} - i \, {\left (a d^{2} + 2 \, b\right )} x^{2} {\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} - i \, b x^{2} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - i \, b x^{2} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + i \, b x^{2} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + i \, b x^{2} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 2 \, a d x \cos \left (d x + c\right ) - 2 \, a \sin \left (d x + c\right )}{4 \, a^{2} x^{2}} \]
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 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 259, normalized size = 0.96 \[ d^{2} \left (-\frac {\sin \left (d x +c \right )}{2 a \,x^{2} d^{2}}-\frac {\cos \left (d x +c \right )}{2 a x d}+\frac {b \left (\Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{2 d^{2} a^{2}}+\frac {b \left (\Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{2 d^{2} a^{2}}-\frac {\left (a \,d^{2}+2 b \right ) \left (\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )\right )}{2 a^{2} d^{2}}\right ) \]
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 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{3}}\,{d x} \]
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 \[ \int \frac {\sin \left (c+d\,x\right )}{x^3\,\left (b\,x^2+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (c + d x \right )}}{x^{3} \left (a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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