Integrand size = 17, antiderivative size = 70 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=-\frac {b \cos (c+d x)}{d}-\frac {a d \cos (c+d x)}{2 x}-\frac {1}{2} a d^2 \operatorname {CosIntegral}(d x) \sin (c)-\frac {a \sin (c+d x)}{2 x^2}-\frac {1}{2} a d^2 \cos (c) \text {Si}(d x) \]
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Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3420, 2718, 3378, 3384, 3380, 3383} \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=-\frac {1}{2} a d^2 \sin (c) \operatorname {CosIntegral}(d x)-\frac {1}{2} a d^2 \cos (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{2 x^2}-\frac {a d \cos (c+d x)}{2 x}-\frac {b \cos (c+d x)}{d} \]
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Rule 2718
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3420
Rubi steps \begin{align*} \text {integral}& = \int \left (b \sin (c+d x)+\frac {a \sin (c+d x)}{x^3}\right ) \, dx \\ & = a \int \frac {\sin (c+d x)}{x^3} \, dx+b \int \sin (c+d x) \, dx \\ & = -\frac {b \cos (c+d x)}{d}-\frac {a \sin (c+d x)}{2 x^2}+\frac {1}{2} (a d) \int \frac {\cos (c+d x)}{x^2} \, dx \\ & = -\frac {b \cos (c+d x)}{d}-\frac {a d \cos (c+d x)}{2 x}-\frac {a \sin (c+d x)}{2 x^2}-\frac {1}{2} \left (a d^2\right ) \int \frac {\sin (c+d x)}{x} \, dx \\ & = -\frac {b \cos (c+d x)}{d}-\frac {a d \cos (c+d x)}{2 x}-\frac {a \sin (c+d x)}{2 x^2}-\frac {1}{2} \left (a d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx-\frac {1}{2} \left (a d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx \\ & = -\frac {b \cos (c+d x)}{d}-\frac {a d \cos (c+d x)}{2 x}-\frac {1}{2} a d^2 \operatorname {CosIntegral}(d x) \sin (c)-\frac {a \sin (c+d x)}{2 x^2}-\frac {1}{2} a d^2 \cos (c) \text {Si}(d x) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=\frac {1}{2} \left (-\frac {2 b \cos (c+d x)}{d}-\frac {a d \cos (c+d x)}{x}-a d^2 \operatorname {CosIntegral}(d x) \sin (c)-\frac {a \sin (c+d x)}{x^2}-a d^2 \cos (c) \text {Si}(d x)\right ) \]
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Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(d^{2} \left (a \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}\right )-\frac {b \cos \left (d x +c \right )}{d^{3}}\right )\) | \(65\) |
default | \(d^{2} \left (a \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}\right )-\frac {b \cos \left (d x +c \right )}{d^{3}}\right )\) | \(65\) |
risch | \(-\frac {i d^{2} \cos \left (c \right ) a \,\operatorname {Ei}_{1}\left (-i d x \right )}{4}+\frac {i d^{2} \cos \left (c \right ) a \,\operatorname {Ei}_{1}\left (i d x \right )}{4}+\frac {d^{2} \sin \left (c \right ) a \,\operatorname {Ei}_{1}\left (-i d x \right )}{4}+\frac {d^{2} \sin \left (c \right ) a \,\operatorname {Ei}_{1}\left (i d x \right )}{4}-\frac {i \left (-2 i a \,d^{6} x^{3}-4 i b \,d^{4} x^{4}\right ) \cos \left (d x +c \right )}{4 d^{5} x^{4}}-\frac {a \sin \left (d x +c \right )}{2 x^{2}}\) | \(112\) |
meijerg | \(\frac {b \sin \left (c \right ) \sin \left (d x \right )}{d}+\frac {b \sqrt {\pi }\, \cos \left (c \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (d x \right )}{\sqrt {\pi }}\right )}{d}+\frac {a \sqrt {\pi }\, \sin \left (c \right ) d^{2} \left (-\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+\ln \left (d^{2}\right )\right )}{\sqrt {\pi }}+\frac {-6 d^{2} x^{2}+4}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \gamma }{\sqrt {\pi }}+\frac {4 \ln \left (2\right )}{\sqrt {\pi }}+\frac {4 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}-\frac {4 \cos \left (d x \right )}{\sqrt {\pi }\, d^{2} x^{2}}+\frac {4 \sin \left (d x \right )}{\sqrt {\pi }\, d x}-\frac {4 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{8}+\frac {a \sqrt {\pi }\, \cos \left (c \right ) d^{2} \left (-\frac {4 \cos \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 \sin \left (d x \right )}{d^{2} x^{2} \sqrt {\pi }}-\frac {4 \,\operatorname {Si}\left (d x \right )}{\sqrt {\pi }}\right )}{8}\) | \(211\) |
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Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=-\frac {a d^{3} x^{2} \operatorname {Ci}\left (d x\right ) \sin \left (c\right ) + a d^{3} x^{2} \cos \left (c\right ) \operatorname {Si}\left (d x\right ) + a d \sin \left (d x + c\right ) + {\left (a d^{2} x + 2 \, b x^{2}\right )} \cos \left (d x + c\right )}{2 \, d x^{2}} \]
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\[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=\int \frac {\left (a + b x^{3}\right ) \sin {\left (c + d x \right )}}{x^{3}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 1146, normalized size of antiderivative = 16.37 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=\text {Too large to display} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.32 (sec) , antiderivative size = 564, normalized size of antiderivative = 8.06 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=\frac {a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{3} x^{2} \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, a d^{3} x^{2} \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{3} x^{2} \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} + a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} - 2 \, a d^{3} x^{2} \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x\right )^{2} + a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} - a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{3} x^{2} \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, a d^{3} x^{2} \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{3} x^{2} \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{2} x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) + a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) - 2 \, a d^{3} x^{2} \operatorname {Si}\left (d x\right ) - 4 \, b x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{2} x \tan \left (\frac {1}{2} \, d x\right )^{2} + 8 \, a d^{2} x \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) + 2 \, a d^{2} x \tan \left (\frac {1}{2} \, c\right )^{2} + 4 \, b x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} + 16 \, b x^{2} \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) + 4 \, a d \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 4 \, b x^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 4 \, a d \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, a d^{2} x - 4 \, b x^{2} - 4 \, a d \tan \left (\frac {1}{2} \, d x\right ) - 4 \, a d \tan \left (\frac {1}{2} \, c\right )}{4 \, {\left (d x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} + d x^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d x^{2}\right )}} \]
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Timed out. \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=\int \frac {\sin \left (c+d\,x\right )\,\left (b\,x^3+a\right )}{x^3} \,d x \]
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